"The Theory of Fractional Heat

and Porous Medium Equations"

Juan Luis Vázquez

Departamento de MatemáticasUniversidad Autónoma de Madrid

Madrid, Spain

Basque Colloquium in Mathematics and its ApplicationsBCAM, Bilbao, 24 October 2013

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 1 / 53

Outline

1 Introduction

2 Traditional porous medium

3 Nonlinear diffusion with nonlocal effectsPresentationThe modelOther models

4 TheoryMain estimatesTheory

5 Asymptotic behaviourAsymptotic behavior for the nonlocal PMERenormalized estimates

6 The second Fractional PME model

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 2 / 53

Outline

1 Introduction

2 Traditional porous medium

3 Nonlinear diffusion with nonlocal effectsPresentationThe modelOther models

4 TheoryMain estimatesTheory

5 Asymptotic behaviourAsymptotic behavior for the nonlocal PMERenormalized estimates

6 The second Fractional PME model

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 3 / 53

Recent Progress on Nonlinear Diffusion Equations

We will discuss part of our recent work on nonlinear diffusion equations.

The basic equation will be the classical heat equation. Previous work wasconcerned with equations of Porous Medium, Fast Diffusion, andp-Laplacian type.

The recent work we will discuss here concerns diffusion represented byfractional Laplacian operators. They have interesting connections withfunctional analysis

The long-time behavior offers interesting results and open questions.

The work we describe in more detail concerns a project with LuisCaffarelli, Univ. of Texas.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 4 / 53

Recent Progress on Nonlinear Diffusion Equations

We will discuss part of our recent work on nonlinear diffusion equations.

The basic equation will be the classical heat equation. Previous work wasconcerned with equations of Porous Medium, Fast Diffusion, andp-Laplacian type.

The recent work we will discuss here concerns diffusion represented byfractional Laplacian operators. They have interesting connections withfunctional analysis

The long-time behavior offers interesting results and open questions.

The work we describe in more detail concerns a project with LuisCaffarelli, Univ. of Texas.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 4 / 53

Recent Progress on Nonlinear Diffusion Equations

We will discuss part of our recent work on nonlinear diffusion equations.

The basic equation will be the classical heat equation. Previous work wasconcerned with equations of Porous Medium, Fast Diffusion, andp-Laplacian type.

The recent work we will discuss here concerns diffusion represented byfractional Laplacian operators. They have interesting connections withfunctional analysis

The long-time behavior offers interesting results and open questions.

The work we describe in more detail concerns a project with LuisCaffarelli, Univ. of Texas.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 4 / 53

Recent Progress on Nonlinear Diffusion Equations

We will discuss part of our recent work on nonlinear diffusion equations.

The basic equation will be the classical heat equation. Previous work wasconcerned with equations of Porous Medium, Fast Diffusion, andp-Laplacian type.

The recent work we will discuss here concerns diffusion represented byfractional Laplacian operators. They have interesting connections withfunctional analysis

The long-time behavior offers interesting results and open questions.

The work we describe in more detail concerns a project with LuisCaffarelli, Univ. of Texas.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 4 / 53

Outline

1 Introduction

2 Traditional porous medium

3 Nonlinear diffusion with nonlocal effectsPresentationThe modelOther models

4 TheoryMain estimatesTheory

5 Asymptotic behaviourAsymptotic behavior for the nonlocal PMERenormalized estimates

6 The second Fractional PME model

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 5 / 53

Porous Medium / Fast Diffusion Equations

The simplest model of nonlinear diffusion equation is maybe

ut = ∆um = ∇ · (c(u)∇u)

c(u) indicates density-dependent diffusivity

c(u) = mum−1[= m|u|m−1]

If m > 1 it degenerates at u = 0 , =⇒ slow diffusion

For m = 1 we get the classical Heat Equation.

On the contrary, if m < 1 it is singular at u = 0 =⇒ Fast Diffusion.

A more general model of nonlinear diffusion takes the divergence form

∂tH(u) = ∇ · ~A(x , u,Du) + B(x , t , u,Du)

with monotonicity conditions on H and ∇p ~A(x , t , u, p) and structural conditions on~A and B. This generality includes Stefan Problems, p-Laplacian flows (includingp =∞ and total variation flow p = 1) and many others.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 6 / 53

Porous Medium / Fast Diffusion Equations

The simplest model of nonlinear diffusion equation is maybe

ut = ∆um = ∇ · (c(u)∇u)

c(u) indicates density-dependent diffusivity

c(u) = mum−1[= m|u|m−1]

If m > 1 it degenerates at u = 0 , =⇒ slow diffusion

For m = 1 we get the classical Heat Equation.

On the contrary, if m < 1 it is singular at u = 0 =⇒ Fast Diffusion.

A more general model of nonlinear diffusion takes the divergence form

∂tH(u) = ∇ · ~A(x , u,Du) + B(x , t , u,Du)

with monotonicity conditions on H and ∇p ~A(x , t , u, p) and structural conditions on~A and B. This generality includes Stefan Problems, p-Laplacian flows (includingp =∞ and total variation flow p = 1) and many others.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 6 / 53

Porous Medium / Fast Diffusion Equations

The simplest model of nonlinear diffusion equation is maybe

ut = ∆um = ∇ · (c(u)∇u)

c(u) indicates density-dependent diffusivity

c(u) = mum−1[= m|u|m−1]

If m > 1 it degenerates at u = 0 , =⇒ slow diffusion

For m = 1 we get the classical Heat Equation.

On the contrary, if m < 1 it is singular at u = 0 =⇒ Fast Diffusion.

A more general model of nonlinear diffusion takes the divergence form

∂tH(u) = ∇ · ~A(x , u,Du) + B(x , t , u,Du)

with monotonicity conditions on H and ∇p ~A(x , t , u, p) and structural conditions on~A and B. This generality includes Stefan Problems, p-Laplacian flows (includingp =∞ and total variation flow p = 1) and many others.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 6 / 53

Porous Medium / Fast Diffusion Equations

The simplest model of nonlinear diffusion equation is maybe

ut = ∆um = ∇ · (c(u)∇u)

c(u) indicates density-dependent diffusivity

c(u) = mum−1[= m|u|m−1]

If m > 1 it degenerates at u = 0 , =⇒ slow diffusion

For m = 1 we get the classical Heat Equation.

On the contrary, if m < 1 it is singular at u = 0 =⇒ Fast Diffusion.

A more general model of nonlinear diffusion takes the divergence form

∂tH(u) = ∇ · ~A(x , u,Du) + B(x , t , u,Du)

with monotonicity conditions on H and ∇p ~A(x , t , u, p) and structural conditions on~A and B. This generality includes Stefan Problems, p-Laplacian flows (includingp =∞ and total variation flow p = 1) and many others.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 6 / 53

Porous Medium / Fast Diffusion Equations

The simplest model of nonlinear diffusion equation is maybe

ut = ∆um = ∇ · (c(u)∇u)

c(u) indicates density-dependent diffusivity

c(u) = mum−1[= m|u|m−1]

If m > 1 it degenerates at u = 0 , =⇒ slow diffusion

For m = 1 we get the classical Heat Equation.

On the contrary, if m < 1 it is singular at u = 0 =⇒ Fast Diffusion.

A more general model of nonlinear diffusion takes the divergence form

∂tH(u) = ∇ · ~A(x , u,Du) + B(x , t , u,Du)

with monotonicity conditions on H and ∇p ~A(x , t , u, p) and structural conditions on~A and B. This generality includes Stefan Problems, p-Laplacian flows (includingp =∞ and total variation flow p = 1) and many others.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 6 / 53

Porous Medium / Fast Diffusion Equations

The simplest model of nonlinear diffusion equation is maybe

ut = ∆um = ∇ · (c(u)∇u)

c(u) indicates density-dependent diffusivity

c(u) = mum−1[= m|u|m−1]

If m > 1 it degenerates at u = 0 , =⇒ slow diffusion

For m = 1 we get the classical Heat Equation.

On the contrary, if m < 1 it is singular at u = 0 =⇒ Fast Diffusion.

A more general model of nonlinear diffusion takes the divergence form

∂tH(u) = ∇ · ~A(x , u,Du) + B(x , t , u,Du)

with monotonicity conditions on H and ∇p ~A(x , t , u, p) and structural conditions on~A and B. This generality includes Stefan Problems, p-Laplacian flows (includingp =∞ and total variation flow p = 1) and many others.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 6 / 53

References* Well-known work starting with Barenblatt (1952), Oleinik et al (1958), and thenAronson, Bénilan, Brezis, Crandall, Caffarelli, Diaz, Di Benedetto, Friedman,Kamin, Kenig, Dalhberg, Koch, Peletier, Pierre, Wolanski, JLV.* Some recent: Carrillo-Toscani-Markowich-Otto on entropies and gradient flow;Daskalopoulos-Hamilton-Lee-Vazquez on concavity. Many works on Fastdiffusion flows (like my work with Bonforte, Dolbeault and Grillo) and logarithmicdiffusion (Daskal., del Pino, V.), on p−Laplacian flows, with recent interest on∞-Laplacian (Aronsson, Crandall, Evans,...) and 1-Laplacians (total variationflows) (Caselles et al). And more.

Book about the PME. J. L. Vázquez, "The Porous Medium Equation.Mathematical Theory", Oxford Univ. Press, 2007, xxii+624 pages.

About estimates and scaling. J. L. Vázquez, “Smoothing and Decay Estimates forNonlinear Parabolic Equations of Porous Medium Type”, Oxford Univ. Press,2006, 234 pages.About asymptotic behaviour. (Following Lyapunov and Boltzmann)J. L. Vázquez. Asymptotic behaviour for the Porous Medium Equation posed inthe whole space. Journal of Evolution Equations 3 (2003), 67–118.M. Bonforte, J. Dolbeault, G. Grillo, J. L. Vázquez, Sharp rates of decay ofsolutions to the nonlinear fast diffusion equation via functional inequalities,Proceedings Nat. Acad. Sciences, 107, no. 38 (2010) 16459–16464.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 7 / 53

References* Well-known work starting with Barenblatt (1952), Oleinik et al (1958), and thenAronson, Bénilan, Brezis, Crandall, Caffarelli, Diaz, Di Benedetto, Friedman,Kamin, Kenig, Dalhberg, Koch, Peletier, Pierre, Wolanski, JLV.* Some recent: Carrillo-Toscani-Markowich-Otto on entropies and gradient flow;Daskalopoulos-Hamilton-Lee-Vazquez on concavity. Many works on Fastdiffusion flows (like my work with Bonforte, Dolbeault and Grillo) and logarithmicdiffusion (Daskal., del Pino, V.), on p−Laplacian flows, with recent interest on∞-Laplacian (Aronsson, Crandall, Evans,...) and 1-Laplacians (total variationflows) (Caselles et al). And more.

Book about the PME. J. L. Vázquez, "The Porous Medium Equation.Mathematical Theory", Oxford Univ. Press, 2007, xxii+624 pages.

About estimates and scaling. J. L. Vázquez, “Smoothing and Decay Estimates forNonlinear Parabolic Equations of Porous Medium Type”, Oxford Univ. Press,2006, 234 pages.About asymptotic behaviour. (Following Lyapunov and Boltzmann)J. L. Vázquez. Asymptotic behaviour for the Porous Medium Equation posed inthe whole space. Journal of Evolution Equations 3 (2003), 67–118.M. Bonforte, J. Dolbeault, G. Grillo, J. L. Vázquez, Sharp rates of decay ofsolutions to the nonlinear fast diffusion equation via functional inequalities,Proceedings Nat. Acad. Sciences, 107, no. 38 (2010) 16459–16464.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 7 / 53

References* Well-known work starting with Barenblatt (1952), Oleinik et al (1958), and thenAronson, Bénilan, Brezis, Crandall, Caffarelli, Diaz, Di Benedetto, Friedman,Kamin, Kenig, Dalhberg, Koch, Peletier, Pierre, Wolanski, JLV.* Some recent: Carrillo-Toscani-Markowich-Otto on entropies and gradient flow;Daskalopoulos-Hamilton-Lee-Vazquez on concavity. Many works on Fastdiffusion flows (like my work with Bonforte, Dolbeault and Grillo) and logarithmicdiffusion (Daskal., del Pino, V.), on p−Laplacian flows, with recent interest on∞-Laplacian (Aronsson, Crandall, Evans,...) and 1-Laplacians (total variationflows) (Caselles et al). And more.

Book about the PME. J. L. Vázquez, "The Porous Medium Equation.Mathematical Theory", Oxford Univ. Press, 2007, xxii+624 pages.

About estimates and scaling. J. L. Vázquez, “Smoothing and Decay Estimates forNonlinear Parabolic Equations of Porous Medium Type”, Oxford Univ. Press,2006, 234 pages.About asymptotic behaviour. (Following Lyapunov and Boltzmann)J. L. Vázquez. Asymptotic behaviour for the Porous Medium Equation posed inthe whole space. Journal of Evolution Equations 3 (2003), 67–118.M. Bonforte, J. Dolbeault, G. Grillo, J. L. Vázquez, Sharp rates of decay ofsolutions to the nonlinear fast diffusion equation via functional inequalities,Proceedings Nat. Acad. Sciences, 107, no. 38 (2010) 16459–16464.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 7 / 53

References* Well-known work starting with Barenblatt (1952), Oleinik et al (1958), and thenAronson, Bénilan, Brezis, Crandall, Caffarelli, Diaz, Di Benedetto, Friedman,Kamin, Kenig, Dalhberg, Koch, Peletier, Pierre, Wolanski, JLV.* Some recent: Carrillo-Toscani-Markowich-Otto on entropies and gradient flow;Daskalopoulos-Hamilton-Lee-Vazquez on concavity. Many works on Fastdiffusion flows (like my work with Bonforte, Dolbeault and Grillo) and logarithmicdiffusion (Daskal., del Pino, V.), on p−Laplacian flows, with recent interest on∞-Laplacian (Aronsson, Crandall, Evans,...) and 1-Laplacians (total variationflows) (Caselles et al). And more.

Book about the PME. J. L. Vázquez, "The Porous Medium Equation.Mathematical Theory", Oxford Univ. Press, 2007, xxii+624 pages.

About estimates and scaling. J. L. Vázquez, “Smoothing and Decay Estimates forNonlinear Parabolic Equations of Porous Medium Type”, Oxford Univ. Press,2006, 234 pages.About asymptotic behaviour. (Following Lyapunov and Boltzmann)J. L. Vázquez. Asymptotic behaviour for the Porous Medium Equation posed inthe whole space. Journal of Evolution Equations 3 (2003), 67–118.M. Bonforte, J. Dolbeault, G. Grillo, J. L. Vázquez, Sharp rates of decay ofsolutions to the nonlinear fast diffusion equation via functional inequalities,Proceedings Nat. Acad. Sciences, 107, no. 38 (2010) 16459–16464.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 7 / 53

References* Well-known work starting with Barenblatt (1952), Oleinik et al (1958), and thenAronson, Bénilan, Brezis, Crandall, Caffarelli, Diaz, Di Benedetto, Friedman,Kamin, Kenig, Dalhberg, Koch, Peletier, Pierre, Wolanski, JLV.* Some recent: Carrillo-Toscani-Markowich-Otto on entropies and gradient flow;Daskalopoulos-Hamilton-Lee-Vazquez on concavity. Many works on Fastdiffusion flows (like my work with Bonforte, Dolbeault and Grillo) and logarithmicdiffusion (Daskal., del Pino, V.), on p−Laplacian flows, with recent interest on∞-Laplacian (Aronsson, Crandall, Evans,...) and 1-Laplacians (total variationflows) (Caselles et al). And more.

Book about the PME. J. L. Vázquez, "The Porous Medium Equation.Mathematical Theory", Oxford Univ. Press, 2007, xxii+624 pages.

About estimates and scaling. J. L. Vázquez, “Smoothing and Decay Estimates forNonlinear Parabolic Equations of Porous Medium Type”, Oxford Univ. Press,2006, 234 pages.About asymptotic behaviour. (Following Lyapunov and Boltzmann)J. L. Vázquez. Asymptotic behaviour for the Porous Medium Equation posed inthe whole space. Journal of Evolution Equations 3 (2003), 67–118.M. Bonforte, J. Dolbeault, G. Grillo, J. L. Vázquez, Sharp rates of decay ofsolutions to the nonlinear fast diffusion equation via functional inequalities,Proceedings Nat. Acad. Sciences, 107, no. 38 (2010) 16459–16464.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 7 / 53

References* Well-known work starting with Barenblatt (1952), Oleinik et al (1958), and thenAronson, Bénilan, Brezis, Crandall, Caffarelli, Diaz, Di Benedetto, Friedman,Kamin, Kenig, Dalhberg, Koch, Peletier, Pierre, Wolanski, JLV.* Some recent: Carrillo-Toscani-Markowich-Otto on entropies and gradient flow;Daskalopoulos-Hamilton-Lee-Vazquez on concavity. Many works on Fastdiffusion flows (like my work with Bonforte, Dolbeault and Grillo) and logarithmicdiffusion (Daskal., del Pino, V.), on p−Laplacian flows, with recent interest on∞-Laplacian (Aronsson, Crandall, Evans,...) and 1-Laplacians (total variationflows) (Caselles et al). And more.

Book about the PME. J. L. Vázquez, "The Porous Medium Equation.Mathematical Theory", Oxford Univ. Press, 2007, xxii+624 pages.

About estimates and scaling. J. L. Vázquez, “Smoothing and Decay Estimates forNonlinear Parabolic Equations of Porous Medium Type”, Oxford Univ. Press,2006, 234 pages.About asymptotic behaviour. (Following Lyapunov and Boltzmann)J. L. Vázquez. Asymptotic behaviour for the Porous Medium Equation posed inthe whole space. Journal of Evolution Equations 3 (2003), 67–118.M. Bonforte, J. Dolbeault, G. Grillo, J. L. Vázquez, Sharp rates of decay ofsolutions to the nonlinear fast diffusion equation via functional inequalities,Proceedings Nat. Acad. Sciences, 107, no. 38 (2010) 16459–16464.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 7 / 53

Outline

1 Introduction

2 Traditional porous medium

3 Nonlinear diffusion with nonlocal effectsPresentationThe modelOther models

4 TheoryMain estimatesTheory

5 Asymptotic behaviourAsymptotic behavior for the nonlocal PMERenormalized estimates

6 The second Fractional PME model

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 8 / 53

PME with fractional diffusion operatorsWe have been studying two kinds of models for flow in porous media in-cluding nonlocal (long-range) diffusion effects.The first model is based on Darcy’s law and the pressure is related to thedensity by an inverse fractional Laplacian operator. We prove existence ofsolutions that propagate with finite speed.The model has the very interesting property that mass preserving self-similar solutions can be found by solving an elliptic obstacle problem withfractional Laplacian for the pair pressure-density.We use entropy methods to show that the asymptotic behaviour is de-scribed after renormalization by these solutions which play the role of theBarenblatt profiles of the standard porous medium model.This is a joint ongoing project with Luis Caffarelli, Univ. Texas.Regularity is studied also with Fernando Soria, UAM. The hydrodynamiclimit with Sylvia Serfaty, ParisThe second model is more in the spirit of fractional Laplacian flows, butnonlinear. Contrary to PME flows, it has infinite speed of propagation andtypical power-like tails. Joint work with Arturo de Pablo, Fernando Quirósand Ana Rodriguez, Madrid. Latest works with Matteo Bonforte, BrunoVolzone, Diana Stan, Felix del Teso.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 9 / 53

PME with fractional diffusion operatorsWe have been studying two kinds of models for flow in porous media in-cluding nonlocal (long-range) diffusion effects.The first model is based on Darcy’s law and the pressure is related to thedensity by an inverse fractional Laplacian operator. We prove existence ofsolutions that propagate with finite speed.The model has the very interesting property that mass preserving self-similar solutions can be found by solving an elliptic obstacle problem withfractional Laplacian for the pair pressure-density.We use entropy methods to show that the asymptotic behaviour is de-scribed after renormalization by these solutions which play the role of theBarenblatt profiles of the standard porous medium model.This is a joint ongoing project with Luis Caffarelli, Univ. Texas.Regularity is studied also with Fernando Soria, UAM. The hydrodynamiclimit with Sylvia Serfaty, ParisThe second model is more in the spirit of fractional Laplacian flows, butnonlinear. Contrary to PME flows, it has infinite speed of propagation andtypical power-like tails. Joint work with Arturo de Pablo, Fernando Quirósand Ana Rodriguez, Madrid. Latest works with Matteo Bonforte, BrunoVolzone, Diana Stan, Felix del Teso.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 9 / 53

PME with fractional diffusion operatorsWe have been studying two kinds of models for flow in porous media in-cluding nonlocal (long-range) diffusion effects.The first model is based on Darcy’s law and the pressure is related to thedensity by an inverse fractional Laplacian operator. We prove existence ofsolutions that propagate with finite speed.The model has the very interesting property that mass preserving self-similar solutions can be found by solving an elliptic obstacle problem withfractional Laplacian for the pair pressure-density.We use entropy methods to show that the asymptotic behaviour is de-scribed after renormalization by these solutions which play the role of theBarenblatt profiles of the standard porous medium model.This is a joint ongoing project with Luis Caffarelli, Univ. Texas.Regularity is studied also with Fernando Soria, UAM. The hydrodynamiclimit with Sylvia Serfaty, ParisThe second model is more in the spirit of fractional Laplacian flows, butnonlinear. Contrary to PME flows, it has infinite speed of propagation andtypical power-like tails. Joint work with Arturo de Pablo, Fernando Quirósand Ana Rodriguez, Madrid. Latest works with Matteo Bonforte, BrunoVolzone, Diana Stan, Felix del Teso.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 9 / 53

PME with fractional diffusion operatorsWe have been studying two kinds of models for flow in porous media in-cluding nonlocal (long-range) diffusion effects.The first model is based on Darcy’s law and the pressure is related to thedensity by an inverse fractional Laplacian operator. We prove existence ofsolutions that propagate with finite speed.The model has the very interesting property that mass preserving self-similar solutions can be found by solving an elliptic obstacle problem withfractional Laplacian for the pair pressure-density.We use entropy methods to show that the asymptotic behaviour is de-scribed after renormalization by these solutions which play the role of theBarenblatt profiles of the standard porous medium model.This is a joint ongoing project with Luis Caffarelli, Univ. Texas.Regularity is studied also with Fernando Soria, UAM. The hydrodynamiclimit with Sylvia Serfaty, ParisThe second model is more in the spirit of fractional Laplacian flows, butnonlinear. Contrary to PME flows, it has infinite speed of propagation andtypical power-like tails. Joint work with Arturo de Pablo, Fernando Quirósand Ana Rodriguez, Madrid. Latest works with Matteo Bonforte, BrunoVolzone, Diana Stan, Felix del Teso.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 9 / 53

PME with fractional diffusion operatorsWe have been studying two kinds of models for flow in porous media in-cluding nonlocal (long-range) diffusion effects.The first model is based on Darcy’s law and the pressure is related to thedensity by an inverse fractional Laplacian operator. We prove existence ofsolutions that propagate with finite speed.The model has the very interesting property that mass preserving self-similar solutions can be found by solving an elliptic obstacle problem withfractional Laplacian for the pair pressure-density.We use entropy methods to show that the asymptotic behaviour is de-scribed after renormalization by these solutions which play the role of theBarenblatt profiles of the standard porous medium model.This is a joint ongoing project with Luis Caffarelli, Univ. Texas.Regularity is studied also with Fernando Soria, UAM. The hydrodynamiclimit with Sylvia Serfaty, ParisThe second model is more in the spirit of fractional Laplacian flows, butnonlinear. Contrary to PME flows, it has infinite speed of propagation andtypical power-like tails. Joint work with Arturo de Pablo, Fernando Quirósand Ana Rodriguez, Madrid. Latest works with Matteo Bonforte, BrunoVolzone, Diana Stan, Felix del Teso.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 9 / 53

PME with fractional diffusion operatorsWe have been studying two kinds of models for flow in porous media in-cluding nonlocal (long-range) diffusion effects.The first model is based on Darcy’s law and the pressure is related to thedensity by an inverse fractional Laplacian operator. We prove existence ofsolutions that propagate with finite speed.The model has the very interesting property that mass preserving self-similar solutions can be found by solving an elliptic obstacle problem withfractional Laplacian for the pair pressure-density.We use entropy methods to show that the asymptotic behaviour is de-scribed after renormalization by these solutions which play the role of theBarenblatt profiles of the standard porous medium model.This is a joint ongoing project with Luis Caffarelli, Univ. Texas.Regularity is studied also with Fernando Soria, UAM. The hydrodynamiclimit with Sylvia Serfaty, ParisThe second model is more in the spirit of fractional Laplacian flows, butnonlinear. Contrary to PME flows, it has infinite speed of propagation andtypical power-like tails. Joint work with Arturo de Pablo, Fernando Quirósand Ana Rodriguez, Madrid. Latest works with Matteo Bonforte, BrunoVolzone, Diana Stan, Felix del Teso.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 9 / 53

PME with fractional diffusion operatorsWe have been studying two kinds of models for flow in porous media in-cluding nonlocal (long-range) diffusion effects.The first model is based on Darcy’s law and the pressure is related to thedensity by an inverse fractional Laplacian operator. We prove existence ofsolutions that propagate with finite speed.The model has the very interesting property that mass preserving self-similar solutions can be found by solving an elliptic obstacle problem withfractional Laplacian for the pair pressure-density.We use entropy methods to show that the asymptotic behaviour is de-scribed after renormalization by these solutions which play the role of theBarenblatt profiles of the standard porous medium model.This is a joint ongoing project with Luis Caffarelli, Univ. Texas.Regularity is studied also with Fernando Soria, UAM. The hydrodynamiclimit with Sylvia Serfaty, ParisThe second model is more in the spirit of fractional Laplacian flows, butnonlinear. Contrary to PME flows, it has infinite speed of propagation andtypical power-like tails. Joint work with Arturo de Pablo, Fernando Quirósand Ana Rodriguez, Madrid. Latest works with Matteo Bonforte, BrunoVolzone, Diana Stan, Felix del Teso.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 9 / 53

Nonlocal diffusion model I

The model arises from the consideration of a continuum, say, a fluid, representedby a density distribution u(x , t) ≥ 0 that evolves with time following a velocity fieldv(x, t), according to the continuity equation

ut +∇ · (u v) = 0.

We assume next that v derives from a potential, v = −∇p, as happens in fluids inporous media according to Darcy’s law, an in that case p is the pressure. Butpotential velocity fields are found in many other instances, like Hele-Shaw cells,and other recent examples.

We still need a closure relation to relate u and p. In the case of gases in porousmedia, as modeled by Leibenzon and Muskat, the closure relation takes the formof a state law p = f (u), where f is a nondecreasing scalar function, which is linearwhen the flow is isothermal, and a power of u if it is adiabatic.The linear relationship happens also in the simplified description of waterinfiltration in an almost horizontal soil layer according to Boussinesq. In bothcases we get the standard porous medium equation, ut = c∆(u2).See PME Book for these and other applications (around 20!).

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 10 / 53

Nonlocal diffusion model I

The model arises from the consideration of a continuum, say, a fluid, representedby a density distribution u(x , t) ≥ 0 that evolves with time following a velocity fieldv(x, t), according to the continuity equation

ut +∇ · (u v) = 0.

We assume next that v derives from a potential, v = −∇p, as happens in fluids inporous media according to Darcy’s law, an in that case p is the pressure. Butpotential velocity fields are found in many other instances, like Hele-Shaw cells,and other recent examples.

We still need a closure relation to relate u and p. In the case of gases in porousmedia, as modeled by Leibenzon and Muskat, the closure relation takes the formof a state law p = f (u), where f is a nondecreasing scalar function, which is linearwhen the flow is isothermal, and a power of u if it is adiabatic.The linear relationship happens also in the simplified description of waterinfiltration in an almost horizontal soil layer according to Boussinesq. In bothcases we get the standard porous medium equation, ut = c∆(u2).See PME Book for these and other applications (around 20!).

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 10 / 53

Nonlocal diffusion model I

The model arises from the consideration of a continuum, say, a fluid, representedby a density distribution u(x , t) ≥ 0 that evolves with time following a velocity fieldv(x, t), according to the continuity equation

ut +∇ · (u v) = 0.

We assume next that v derives from a potential, v = −∇p, as happens in fluids inporous media according to Darcy’s law, an in that case p is the pressure. Butpotential velocity fields are found in many other instances, like Hele-Shaw cells,and other recent examples.

We still need a closure relation to relate u and p. In the case of gases in porousmedia, as modeled by Leibenzon and Muskat, the closure relation takes the formof a state law p = f (u), where f is a nondecreasing scalar function, which is linearwhen the flow is isothermal, and a power of u if it is adiabatic.The linear relationship happens also in the simplified description of waterinfiltration in an almost horizontal soil layer according to Boussinesq. In bothcases we get the standard porous medium equation, ut = c∆(u2).See PME Book for these and other applications (around 20!).

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 10 / 53

Nonlocal diffusion model I

The model arises from the consideration of a continuum, say, a fluid, representedby a density distribution u(x , t) ≥ 0 that evolves with time following a velocity fieldv(x, t), according to the continuity equation

ut +∇ · (u v) = 0.

We assume next that v derives from a potential, v = −∇p, as happens in fluids inporous media according to Darcy’s law, an in that case p is the pressure. Butpotential velocity fields are found in many other instances, like Hele-Shaw cells,and other recent examples.

We still need a closure relation to relate u and p. In the case of gases in porousmedia, as modeled by Leibenzon and Muskat, the closure relation takes the formof a state law p = f (u), where f is a nondecreasing scalar function, which is linearwhen the flow is isothermal, and a power of u if it is adiabatic.The linear relationship happens also in the simplified description of waterinfiltration in an almost horizontal soil layer according to Boussinesq. In bothcases we get the standard porous medium equation, ut = c∆(u2).See PME Book for these and other applications (around 20!).

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 10 / 53

Nonlocal diffusion model I. The problemThe diffusion model with nonlocal effects we propose here uses a closure relationof the form p = K(u), where K is a linear integral operator, which we assume inpractice to be the inverse of a fractional Laplacian. Hence, p es related to uthrough a fractional potential operator, K = (−∆)−s, 0 < s < 1, with kernel

k(x , y) = c|x − y |−(n−2s)

(i.e., a Riesz operator). We have (−∆)sp = u.

The diffusion model with nonlocal effects is thus given by the system

(1) ut = ∇ · (u∇p), p = K(u).

where u is a function of the variables (x , t) to be thought of as a density orconcentration, and therefore nonnegative, while p is the pressure, which is relatedto u via a linear operator K.

The problem is posed for x ∈ Rn, n ≥ 1, and t > 0, and we give initial conditions

(2) u(x , 0) = u0(x), x ∈ Rn,

where u0 is a nonnegative, bounded and integrable function in Rn.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 11 / 53

Nonlocal diffusion model I. The problemThe diffusion model with nonlocal effects we propose here uses a closure relationof the form p = K(u), where K is a linear integral operator, which we assume inpractice to be the inverse of a fractional Laplacian. Hence, p es related to uthrough a fractional potential operator, K = (−∆)−s, 0 < s < 1, with kernel

k(x , y) = c|x − y |−(n−2s)

(i.e., a Riesz operator). We have (−∆)sp = u.

The diffusion model with nonlocal effects is thus given by the system

(1) ut = ∇ · (u∇p), p = K(u).

where u is a function of the variables (x , t) to be thought of as a density orconcentration, and therefore nonnegative, while p is the pressure, which is relatedto u via a linear operator K.

The problem is posed for x ∈ Rn, n ≥ 1, and t > 0, and we give initial conditions

(2) u(x , 0) = u0(x), x ∈ Rn,

where u0 is a nonnegative, bounded and integrable function in Rn.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 11 / 53

Nonlocal diffusion model I. The problemThe diffusion model with nonlocal effects we propose here uses a closure relationof the form p = K(u), where K is a linear integral operator, which we assume inpractice to be the inverse of a fractional Laplacian. Hence, p es related to uthrough a fractional potential operator, K = (−∆)−s, 0 < s < 1, with kernel

k(x , y) = c|x − y |−(n−2s)

(i.e., a Riesz operator). We have (−∆)sp = u.

The diffusion model with nonlocal effects is thus given by the system

(1) ut = ∇ · (u∇p), p = K(u).

where u is a function of the variables (x , t) to be thought of as a density orconcentration, and therefore nonnegative, while p is the pressure, which is relatedto u via a linear operator K.

The problem is posed for x ∈ Rn, n ≥ 1, and t > 0, and we give initial conditions

(2) u(x , 0) = u0(x), x ∈ Rn,

where u0 is a nonnegative, bounded and integrable function in Rn.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 11 / 53

The fractional Laplacian operator

Different formulas for fractional Laplacian operator.We assume that the space variable x ∈ Rn, and the fractional exponentis 0 < s < 1. First, pseudo differential operator given by the Fourier transform:

(−∆)su(ξ) = |ξ|2su(ξ)

Singular integral operator:

(−∆)su(x) = Cn,s

∫Rn

u(x)− u(y)

|x − y |n+2s dy

With this definition, it is the inverse of the Riesz integral operator (−∆)−su. Thisone has kernel C1|x − y |n−2s, which is not integrable.Take the random walk for Lévy processes:

un+1j =

∑k

Pjk unk

where Pik denotes the transition function which has a . tail (i.e, power decay withthe distance |i − k |). In the limit you get an operator A as the infinitesimalgenerator of a Levy process: if Xt is the isotropic α-stable Lévy process we have

Au(x) = limh→0

E(u(x)− u(x + Xh))

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 12 / 53

The fractional Laplacian operator

Different formulas for fractional Laplacian operator.We assume that the space variable x ∈ Rn, and the fractional exponentis 0 < s < 1. First, pseudo differential operator given by the Fourier transform:

(−∆)su(ξ) = |ξ|2su(ξ)

Singular integral operator:

(−∆)su(x) = Cn,s

∫Rn

u(x)− u(y)

|x − y |n+2s dy

With this definition, it is the inverse of the Riesz integral operator (−∆)−su. Thisone has kernel C1|x − y |n−2s, which is not integrable.Take the random walk for Lévy processes:

un+1j =

∑k

Pjk unk

where Pik denotes the transition function which has a . tail (i.e, power decay withthe distance |i − k |). In the limit you get an operator A as the infinitesimalgenerator of a Levy process: if Xt is the isotropic α-stable Lévy process we have

Au(x) = limh→0

E(u(x)− u(x + Xh))

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 12 / 53

The fractional Laplacian operator

Different formulas for fractional Laplacian operator.We assume that the space variable x ∈ Rn, and the fractional exponentis 0 < s < 1. First, pseudo differential operator given by the Fourier transform:

(−∆)su(ξ) = |ξ|2su(ξ)

Singular integral operator:

(−∆)su(x) = Cn,s

∫Rn

u(x)− u(y)

|x − y |n+2s dy

With this definition, it is the inverse of the Riesz integral operator (−∆)−su. Thisone has kernel C1|x − y |n−2s, which is not integrable.Take the random walk for Lévy processes:

un+1j =

∑k

Pjk unk

where Pik denotes the transition function which has a . tail (i.e, power decay withthe distance |i − k |). In the limit you get an operator A as the infinitesimalgenerator of a Levy process: if Xt is the isotropic α-stable Lévy process we have

Au(x) = limh→0

E(u(x)− u(x + Xh))

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 12 / 53

The fractional Laplacian operator II

The α-harmonic extension: Find first the solution of the (n + 1) problem

∇ · (y1−α∇v) = 0 (x , y) ∈ Rn × R+; v(x , 0) = u(x), x ∈ Rn.

Then, putting α = 2s we have

(−∆)su(x) = −Cα limy→0

y1−α ∂v∂y

When s = 1/2 i.e. α = 1, the extended function v is harmonic (in n + 1 variables)and the operator is the Dirichlet-to-Neumann map on the base space x ∈ Rn. Itwas proposed in PDEs by Caffarelli and Silvestre.

Remark. In Rn all these versions are equivalent. In a bounded domain we have tore-examine all of them. Three main alternatives are studied in probability andPDEs, corresponding to different options about what happens to particles at theboundary or what is the domain of the functionals.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 13 / 53

The fractional Laplacian operator II

The α-harmonic extension: Find first the solution of the (n + 1) problem

∇ · (y1−α∇v) = 0 (x , y) ∈ Rn × R+; v(x , 0) = u(x), x ∈ Rn.

Then, putting α = 2s we have

(−∆)su(x) = −Cα limy→0

y1−α ∂v∂y

When s = 1/2 i.e. α = 1, the extended function v is harmonic (in n + 1 variables)and the operator is the Dirichlet-to-Neumann map on the base space x ∈ Rn. Itwas proposed in PDEs by Caffarelli and Silvestre.

Remark. In Rn all these versions are equivalent. In a bounded domain we have tore-examine all of them. Three main alternatives are studied in probability andPDEs, corresponding to different options about what happens to particles at theboundary or what is the domain of the functionals.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 13 / 53

Extreme cases

If we take s = 0, K = the identity operator, we get the standard porous mediumequation, whose behaviour is well-known, see references later.

In the other end of the s interval, when s = 1 and we take K = −∆ we get

(3) ut = ∇u · ∇p − u2, −∆p = u.

In one dimension this leads to ut = ux px − u2, pxx = −u. In terms ofv = −px =

∫u dx we have

vt = upx + c(t) = −vx v + c(t),

For c = 0 this is the Burgers equation vt + vvx = 0 which generates shocks infinite time but only if we allow for u to have two signs.

HYDRODYNAMIC LIMIT. The case s = 1 in several dimensions is more interestingbecause it does not reduce to a simple Burgers equation.

ut = ∇ · (u∇p) = ∇u · ∇p − u2; , p = (−∆)−1u ,

Applications in superconductivity and superfluidity, see below.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 14 / 53

Extreme cases

If we take s = 0, K = the identity operator, we get the standard porous mediumequation, whose behaviour is well-known, see references later.

In the other end of the s interval, when s = 1 and we take K = −∆ we get

(3) ut = ∇u · ∇p − u2, −∆p = u.

In one dimension this leads to ut = ux px − u2, pxx = −u. In terms ofv = −px =

∫u dx we have

vt = upx + c(t) = −vx v + c(t),

For c = 0 this is the Burgers equation vt + vvx = 0 which generates shocks infinite time but only if we allow for u to have two signs.

HYDRODYNAMIC LIMIT. The case s = 1 in several dimensions is more interestingbecause it does not reduce to a simple Burgers equation.

ut = ∇ · (u∇p) = ∇u · ∇p − u2; , p = (−∆)−1u ,

Applications in superconductivity and superfluidity, see below.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 14 / 53

Extreme cases

If we take s = 0, K = the identity operator, we get the standard porous mediumequation, whose behaviour is well-known, see references later.

In the other end of the s interval, when s = 1 and we take K = −∆ we get

(3) ut = ∇u · ∇p − u2, −∆p = u.

In one dimension this leads to ut = ux px − u2, pxx = −u. In terms ofv = −px =

∫u dx we have

vt = upx + c(t) = −vx v + c(t),

For c = 0 this is the Burgers equation vt + vvx = 0 which generates shocks infinite time but only if we allow for u to have two signs.

HYDRODYNAMIC LIMIT. The case s = 1 in several dimensions is more interestingbecause it does not reduce to a simple Burgers equation.

ut = ∇ · (u∇p) = ∇u · ∇p − u2; , p = (−∆)−1u ,

Applications in superconductivity and superfluidity, see below.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 14 / 53

Nonlocal diffusion model

The interest in using fractional Laplacians in modeling diffusive processes has awide literature, especially when one wants to model long-range diffusiveinteraction, and this interest has been activated by the recent progress in themathematical theory as represented in [AC]1, [CSS]2. See also L. Silvestre‘s workstarting with his thesis, or work by X. Cabré and coauthors.

A variant of the proposed model was studied by Lions and Mas-Gallic [LMG] 3

They study the regularization of the velocity field in the standard porous mediumequation by means of a convolution kernel to get a system like ours, with adifference, namely that they assume the kernel to be smooth and integrable.Since the kernel of the fractional operator (−∆)s is k(x , y) = |x − y |n−2s, we areaway from that case, but it may serve as a regularization step below.

1[AC] I. Athanasopoulos and L. A. Caffarelli. Optimal regularity of lower dimensionalobstacle problems. Steklov, 2004.

2[CSS] L. A. Caffarelli, S. Salsa, and L. Silvestre. Regularity estimates for thesolution and the free boundary to the obstacle problem for the fractional LaplacianInvent. Math. 171 (2008).

3[LMG] P. L. Lions and S. Mas-Gallic. Une méthode particulaire déterministe pourdes équations diffusives non linéaires, C.R. Acad. Sci. Paris t. 332 (série 1) (2001)369–376.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 15 / 53

Nonlocal diffusion model

The interest in using fractional Laplacians in modeling diffusive processes has awide literature, especially when one wants to model long-range diffusiveinteraction, and this interest has been activated by the recent progress in themathematical theory as represented in [AC]1, [CSS]2. See also L. Silvestre‘s workstarting with his thesis, or work by X. Cabré and coauthors.

A variant of the proposed model was studied by Lions and Mas-Gallic [LMG] 3

They study the regularization of the velocity field in the standard porous mediumequation by means of a convolution kernel to get a system like ours, with adifference, namely that they assume the kernel to be smooth and integrable.Since the kernel of the fractional operator (−∆)s is k(x , y) = |x − y |n−2s, we areaway from that case, but it may serve as a regularization step below.

1[AC] I. Athanasopoulos and L. A. Caffarelli. Optimal regularity of lower dimensionalobstacle problems. Steklov, 2004.

2[CSS] L. A. Caffarelli, S. Salsa, and L. Silvestre. Regularity estimates for thesolution and the free boundary to the obstacle problem for the fractional LaplacianInvent. Math. 171 (2008).

3[LMG] P. L. Lions and S. Mas-Gallic. Une méthode particulaire déterministe pourdes équations diffusives non linéaires, C.R. Acad. Sci. Paris t. 332 (série 1) (2001)369–376.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 15 / 53

Nonlocal diffusion model. ApplicationsModeling dislocation dynamics as a continuum.• A. K. Head. Dislocation group dynamics II. Similarity solutions of the continuumapproximation. Phil. Mag. 26 (1972), 65–72.• P. Biler, G. Karch, and R. Monneau. Nonlinear diffusion of dislocation densityand self-similar solutions. Comm. Math. Phys., (2009).This is a one-dimensional model. By integration in x they introduce viscositysolutions a la Crandall-Evans-LIons. Uniqueness holds.Equations of the more general form ut = ∇ · (σ(u)∇Lu) have appeared recentlyin a number of applications in particle physics. Thus, Giacomin and Lebowitz (J.Stat. Phys. (1997)) consider a lattice gas with general short-range interactionsand a Kac potential, and passing to the limit, the macroscopic density profileρ(r , t) satisfies the equation

∂ρ

∂t= ∇ ·

[σs(ρ)∇δF (ρ)

δρ

]See also (GL2) and the review paper (GLP). The model is used to study phasesegregation in (GLM, 2000).More generally, it could be assumed that K is an operator of integral type definedby convolution on all of Rn, with the assumptions that is positive and symmetric.The fact the K is a homogeneous operator of degree 2s, 0 < s < 1, will beimportant in the proofs. An interesting variant would be K = (−∆ + cI)−s. We arenot exploring such extensions.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 16 / 53

Nonlocal diffusion model. ApplicationsModeling dislocation dynamics as a continuum.• A. K. Head. Dislocation group dynamics II. Similarity solutions of the continuumapproximation. Phil. Mag. 26 (1972), 65–72.• P. Biler, G. Karch, and R. Monneau. Nonlinear diffusion of dislocation densityand self-similar solutions. Comm. Math. Phys., (2009).This is a one-dimensional model. By integration in x they introduce viscositysolutions a la Crandall-Evans-LIons. Uniqueness holds.Equations of the more general form ut = ∇ · (σ(u)∇Lu) have appeared recentlyin a number of applications in particle physics. Thus, Giacomin and Lebowitz (J.Stat. Phys. (1997)) consider a lattice gas with general short-range interactionsand a Kac potential, and passing to the limit, the macroscopic density profileρ(r , t) satisfies the equation

∂ρ

∂t= ∇ ·

[σs(ρ)∇δF (ρ)

δρ

]See also (GL2) and the review paper (GLP). The model is used to study phasesegregation in (GLM, 2000).More generally, it could be assumed that K is an operator of integral type definedby convolution on all of Rn, with the assumptions that is positive and symmetric.The fact the K is a homogeneous operator of degree 2s, 0 < s < 1, will beimportant in the proofs. An interesting variant would be K = (−∆ + cI)−s. We arenot exploring such extensions.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 16 / 53

Nonlocal diffusion model. ApplicationsModeling dislocation dynamics as a continuum.• A. K. Head. Dislocation group dynamics II. Similarity solutions of the continuumapproximation. Phil. Mag. 26 (1972), 65–72.• P. Biler, G. Karch, and R. Monneau. Nonlinear diffusion of dislocation densityand self-similar solutions. Comm. Math. Phys., (2009).This is a one-dimensional model. By integration in x they introduce viscositysolutions a la Crandall-Evans-LIons. Uniqueness holds.Equations of the more general form ut = ∇ · (σ(u)∇Lu) have appeared recentlyin a number of applications in particle physics. Thus, Giacomin and Lebowitz (J.Stat. Phys. (1997)) consider a lattice gas with general short-range interactionsand a Kac potential, and passing to the limit, the macroscopic density profileρ(r , t) satisfies the equation

∂ρ

∂t= ∇ ·

[σs(ρ)∇δF (ρ)

δρ

]See also (GL2) and the review paper (GLP). The model is used to study phasesegregation in (GLM, 2000).More generally, it could be assumed that K is an operator of integral type definedby convolution on all of Rn, with the assumptions that is positive and symmetric.The fact the K is a homogeneous operator of degree 2s, 0 < s < 1, will beimportant in the proofs. An interesting variant would be K = (−∆ + cI)−s. We arenot exploring such extensions.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 16 / 53

Hydrodynamic caseThis equation was first studied in R2 by Lin-Zhang ( DCDS (2000)). There,existence is proven by a vortex point approximation. In dimension 2, the equationis directly related to the Chapman-Rubinstein-Schatzman mean-field model ofsuperconductivity (1996) . Also to Weinan E’s model of superfluidity (Phys. Rev.B (1994)). which would correspond rather to the equation

(4) ut = ∇ · (|u|∇p)

which coincides with previous equation when u ≥ 0. That model is a mean-fieldmodel for the motion of vortices in a superconductor in the Ginzburg-Landautheory. There, u represents the local vortex-density, and p represents the inducedmagnetic field in the sample. In this application the equation is really posed in abounded domain and for signed “vorticity measures" u (which can be measuresinstead of functions).Ambrosio and Serfaty4 studied the model in a bounded two-dimensional domain(taking into account the possibility of flux of vortices through the boundary), via agradient flow approach. It is equivalent to our case s = 1 but posed in a boundeddomain.The interpolation we study has better properties than s = 1 but is different inmany properties from s = 0. Work with Serfaty to appear in Calc. Var. PDEs.

4L. Ambrosio, S. Serfaty. A gradient flow approach to an evolution problem arising insuperconductivity, Comm. Pure Appl. Math. 61 (2008).

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 17 / 53

Hydrodynamic caseThis equation was first studied in R2 by Lin-Zhang ( DCDS (2000)). There,existence is proven by a vortex point approximation. In dimension 2, the equationis directly related to the Chapman-Rubinstein-Schatzman mean-field model ofsuperconductivity (1996) . Also to Weinan E’s model of superfluidity (Phys. Rev.B (1994)). which would correspond rather to the equation

(4) ut = ∇ · (|u|∇p)

which coincides with previous equation when u ≥ 0. That model is a mean-fieldmodel for the motion of vortices in a superconductor in the Ginzburg-Landautheory. There, u represents the local vortex-density, and p represents the inducedmagnetic field in the sample. In this application the equation is really posed in abounded domain and for signed “vorticity measures" u (which can be measuresinstead of functions).Ambrosio and Serfaty4 studied the model in a bounded two-dimensional domain(taking into account the possibility of flux of vortices through the boundary), via agradient flow approach. It is equivalent to our case s = 1 but posed in a boundeddomain.The interpolation we study has better properties than s = 1 but is different inmany properties from s = 0. Work with Serfaty to appear in Calc. Var. PDEs.

4L. Ambrosio, S. Serfaty. A gradient flow approach to an evolution problem arising insuperconductivity, Comm. Pure Appl. Math. 61 (2008).

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 17 / 53

Outline

1 Introduction

2 Traditional porous medium

3 Nonlinear diffusion with nonlocal effectsPresentationThe modelOther models

4 TheoryMain estimatesTheory

5 Asymptotic behaviourAsymptotic behavior for the nonlocal PMERenormalized estimates

6 The second Fractional PME model

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 18 / 53

Results of our first project.

Existence of weak energy solutions and property of finite propagationL. Caffarelli and J. L. Vázquez, Nonlinear porous medium flow withfractional potential pressure, Arch. Rational Mech. Anal. 202 (2011),537–565.

Existence of self-similar profiles, renormalized Fokker-Planck equationand entropy-based proof of stabilizationL. Caffarelli and J. L. Vazquez, Asymptotic behaviour of a porous mediumequation with fractional diffusion, DCDS-A 29, no. 4 (2011), 1393–1404.

Regularity in three levels: L1 → L2, L2 → L∞, and bounded implies Cα

L. Caffarelli, F. Soria, and J. L. Vazquez, Regularity of porous mediumequation with fractional diffusion, J. Eur. Math. Soc., 15, 5 (2013),1701–1746.S. Serfaty, and J. L. Vazquez, Hydrodynamic Limit of Nonlinear. Diffusionwith Fractional Laplacian Operators, Calc. Var. PDEs 526, online;arXiv:1205.6322v1 [math.AP], 2012.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 19 / 53

Results of our first project.

Existence of weak energy solutions and property of finite propagationL. Caffarelli and J. L. Vázquez, Nonlinear porous medium flow withfractional potential pressure, Arch. Rational Mech. Anal. 202 (2011),537–565.

Existence of self-similar profiles, renormalized Fokker-Planck equationand entropy-based proof of stabilizationL. Caffarelli and J. L. Vazquez, Asymptotic behaviour of a porous mediumequation with fractional diffusion, DCDS-A 29, no. 4 (2011), 1393–1404.

Regularity in three levels: L1 → L2, L2 → L∞, and bounded implies Cα

L. Caffarelli, F. Soria, and J. L. Vazquez, Regularity of porous mediumequation with fractional diffusion, J. Eur. Math. Soc., 15, 5 (2013),1701–1746.S. Serfaty, and J. L. Vazquez, Hydrodynamic Limit of Nonlinear. Diffusionwith Fractional Laplacian Operators, Calc. Var. PDEs 526, online;arXiv:1205.6322v1 [math.AP], 2012.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 19 / 53

Results of our first project.

Existence of weak energy solutions and property of finite propagationL. Caffarelli and J. L. Vázquez, Nonlinear porous medium flow withfractional potential pressure, Arch. Rational Mech. Anal. 202 (2011),537–565.

Existence of self-similar profiles, renormalized Fokker-Planck equationand entropy-based proof of stabilizationL. Caffarelli and J. L. Vazquez, Asymptotic behaviour of a porous mediumequation with fractional diffusion, DCDS-A 29, no. 4 (2011), 1393–1404.

Regularity in three levels: L1 → L2, L2 → L∞, and bounded implies Cα

L. Caffarelli, F. Soria, and J. L. Vazquez, Regularity of porous mediumequation with fractional diffusion, J. Eur. Math. Soc., 15, 5 (2013),1701–1746.S. Serfaty, and J. L. Vazquez, Hydrodynamic Limit of Nonlinear. Diffusionwith Fractional Laplacian Operators, Calc. Var. PDEs 526, online;arXiv:1205.6322v1 [math.AP], 2012.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 19 / 53

Results of our first project.

Existence of weak energy solutions and property of finite propagationL. Caffarelli and J. L. Vázquez, Nonlinear porous medium flow withfractional potential pressure, Arch. Rational Mech. Anal. 202 (2011),537–565.

Existence of self-similar profiles, renormalized Fokker-Planck equationand entropy-based proof of stabilizationL. Caffarelli and J. L. Vazquez, Asymptotic behaviour of a porous mediumequation with fractional diffusion, DCDS-A 29, no. 4 (2011), 1393–1404.

Regularity in three levels: L1 → L2, L2 → L∞, and bounded implies Cα

L. Caffarelli, F. Soria, and J. L. Vazquez, Regularity of porous mediumequation with fractional diffusion, J. Eur. Math. Soc., 15, 5 (2013),1701–1746.S. Serfaty, and J. L. Vazquez, Hydrodynamic Limit of Nonlinear. Diffusionwith Fractional Laplacian Operators, Calc. Var. PDEs 526, online;arXiv:1205.6322v1 [math.AP], 2012.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 19 / 53

Results of our first project.

Existence of weak energy solutions and property of finite propagationL. Caffarelli and J. L. Vázquez, Nonlinear porous medium flow withfractional potential pressure, Arch. Rational Mech. Anal. 202 (2011),537–565.

Existence of self-similar profiles, renormalized Fokker-Planck equationand entropy-based proof of stabilizationL. Caffarelli and J. L. Vazquez, Asymptotic behaviour of a porous mediumequation with fractional diffusion, DCDS-A 29, no. 4 (2011), 1393–1404.

Regularity in three levels: L1 → L2, L2 → L∞, and bounded implies Cα

L. Caffarelli, F. Soria, and J. L. Vazquez, Regularity of porous mediumequation with fractional diffusion, J. Eur. Math. Soc., 15, 5 (2013),1701–1746.S. Serfaty, and J. L. Vazquez, Hydrodynamic Limit of Nonlinear. Diffusionwith Fractional Laplacian Operators, Calc. Var. PDEs 526, online;arXiv:1205.6322v1 [math.AP], 2012.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 19 / 53

Results of our first project.

Existence of weak energy solutions and property of finite propagationL. Caffarelli and J. L. Vázquez, Nonlinear porous medium flow withfractional potential pressure, Arch. Rational Mech. Anal. 202 (2011),537–565.

Existence of self-similar profiles, renormalized Fokker-Planck equationand entropy-based proof of stabilizationL. Caffarelli and J. L. Vazquez, Asymptotic behaviour of a porous mediumequation with fractional diffusion, DCDS-A 29, no. 4 (2011), 1393–1404.

Regularity in three levels: L1 → L2, L2 → L∞, and bounded implies Cα

L. Caffarelli, F. Soria, and J. L. Vazquez, Regularity of porous mediumequation with fractional diffusion, J. Eur. Math. Soc., 15, 5 (2013),1701–1746.S. Serfaty, and J. L. Vazquez, Hydrodynamic Limit of Nonlinear. Diffusionwith Fractional Laplacian Operators, Calc. Var. PDEs 526, online;arXiv:1205.6322v1 [math.AP], 2012.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 19 / 53

Results of our first project.

Existence of weak energy solutions and property of finite propagationL. Caffarelli and J. L. Vázquez, Nonlinear porous medium flow withfractional potential pressure, Arch. Rational Mech. Anal. 202 (2011),537–565.

Existence of self-similar profiles, renormalized Fokker-Planck equationand entropy-based proof of stabilizationL. Caffarelli and J. L. Vazquez, Asymptotic behaviour of a porous mediumequation with fractional diffusion, DCDS-A 29, no. 4 (2011), 1393–1404.

Regularity in three levels: L1 → L2, L2 → L∞, and bounded implies Cα

L. Caffarelli, F. Soria, and J. L. Vazquez, Regularity of porous mediumequation with fractional diffusion, J. Eur. Math. Soc., 15, 5 (2013),1701–1746.S. Serfaty, and J. L. Vazquez, Hydrodynamic Limit of Nonlinear. Diffusionwith Fractional Laplacian Operators, Calc. Var. PDEs 526, online;arXiv:1205.6322v1 [math.AP], 2012.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 19 / 53

Main estimates5

The equation is ∂tu = ∇ · (u∇K (u)), posed in the whole space Rn.We consider K = (−∆)−s for some 0 < s < 1 acting on Schwartz class functionsdefined in the whole space. It is a positive essentially self-adjoint operator.We let H = K 1/2 = (−∆)−s/2.We do formal calculations, assuming that u ≥ 0 satisfies the required smoothnessand integrability assumptions. This is to be justified later by approximation.

Conservation of mass

(5)ddt

∫u(x , t) dx = 0.

First energy estimate:

(6)ddt

∫u(x , t) log u(x , t) dx = −

∫|∇Hu|2 dx .

Second energy estimate

(7)ddt

∫|Hu(x , t)|2 dx = −2

∫u|∇Ku|2 dx .

5Luis Caffarelli and J. L. Vázquez, Nonlinear porous medium flow with fractionalpotential pressure. arXiv:1001.0410v1 [math.AP].

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 20 / 53

Main estimates II

Conservation of positivity: u0 ≥ 0 implies that u(t) ≥ 0 for all times.

L∞ estimate. We prove that the L∞ norm does not increase in time.Proof. At a point of maximum of u at time t = t0, say x = 0, we have

ut = ∇u · ∇P + u ∆K (u).

The first term is zero, and for the second we have −∆K = L where L = (−∆)q

with q = 1− s so that

∆Ku(0) = −Lu(0) = −∫

u(0)− u(y)

|y |n+2(1−s) dy ≤ 0.

This concludes the proof.

We did not find a clean comparison theorem, a form of the usual maximumprinciple is not proved.

Lp bounds are conserved

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 21 / 53

Finite propagation. Solutions with compactsupport

One of the most important features of the porous medium equation and otherrelated degenerate parabolic equations is the property of finite propagation,whereby compactly supported initial data u0(x) give rise to solutions u(x , t) thathave the same property for all positive times, i.e., the support of u(·, t) iscontained in a ball BR(t)(0) for all t > 0.

One possible proof in the case of the PME is by constructing explicit weaksolutions with that property and then using the comparison principle, that holdsfor that equation.

Since we do not have such a general principle here, we have to devise acomparison method with a suitable family of “strict supersolutions”, in factexcessive supersolutions. The technique has to be adapted to the peculiar formof the integral kernels involved in operator K.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 22 / 53

Solutions with compact supportWe begin with n = 1. We assume that our solution u(x , t) ≥ 0 has bounded initialdata u0(x) = u(x , t0) ≤ M with compact support and is such that

u0 is below the parabola a(x − b)2, a, b > 0.

with graphs strictly separated. We may assume that u0 is located under the leftbranch of the parabola. We take as comparison function

U(x , t) = a(Ct − (x − b))2,

and argue at the first point and time where u(x , t) touches U the left branch ofthe parabola from below.

Lemma

Assume that u is a bounded solution of equation (1) with K = (−∆)−s and 0 < s < 1.Assume that u has compact support for all t > t0 and is C2 for u > 0. If C is largeenough, then there is no contact from below between u and U at a point where u > 0.Actually, if s < 1/2 we can take C = C0(n, s)M(1/2)+sa(1/2)−s.

Conclusion is true for n = 1 with restriction on s, 0 < s < 1/2

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 23 / 53

Positivity. Instantaneous BoundednessSolutions that are positive at t = 0 remain positive for all times. Butpositivity at one point can be lost. Example of two bumps,the large onepushing the smaller one

Solutions are bounded in terms of data in Lp, 1 ≤ p ≤ ∞.Use (the de Giorgi or the Moser) iteration technique on theCaffarelli-Silvestre extension as in Caffarelli-Vasseur.Or use energy estimates based on the properties of the quadratic andbilinear forms associated to the fractional operator, and then the iterationtechnique

Theorem Let u be a weak solution the IVP for the FPME with datau0 ∈ L1(Rn) ∩ L∞(Rn), as constructed before. Then, there exists apositive constant C such that for every t > 0

(8) supx∈Rn|u(x , t)| ≤ C t−α‖u0‖γL1(Rn)

with α = n/(n + 2− 2s), γ = (2− 2s)/((n + 2− 2s). The constant Cdepends only on n and s.This theorem allows to extend the theory to data u0 ∈ L1(Rn), u0 ≥ 0,with global existence of bounded weak solutions.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 24 / 53

Positivity. Instantaneous BoundednessSolutions that are positive at t = 0 remain positive for all times. Butpositivity at one point can be lost. Example of two bumps,the large onepushing the smaller one

Solutions are bounded in terms of data in Lp, 1 ≤ p ≤ ∞.Use (the de Giorgi or the Moser) iteration technique on theCaffarelli-Silvestre extension as in Caffarelli-Vasseur.Or use energy estimates based on the properties of the quadratic andbilinear forms associated to the fractional operator, and then the iterationtechnique

Theorem Let u be a weak solution the IVP for the FPME with datau0 ∈ L1(Rn) ∩ L∞(Rn), as constructed before. Then, there exists apositive constant C such that for every t > 0

(8) supx∈Rn|u(x , t)| ≤ C t−α‖u0‖γL1(Rn)

with α = n/(n + 2− 2s), γ = (2− 2s)/((n + 2− 2s). The constant Cdepends only on n and s.This theorem allows to extend the theory to data u0 ∈ L1(Rn), u0 ≥ 0,with global existence of bounded weak solutions.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 24 / 53

Positivity. Instantaneous BoundednessSolutions that are positive at t = 0 remain positive for all times. Butpositivity at one point can be lost. Example of two bumps,the large onepushing the smaller one

Solutions are bounded in terms of data in Lp, 1 ≤ p ≤ ∞.Use (the de Giorgi or the Moser) iteration technique on theCaffarelli-Silvestre extension as in Caffarelli-Vasseur.Or use energy estimates based on the properties of the quadratic andbilinear forms associated to the fractional operator, and then the iterationtechnique

Theorem Let u be a weak solution the IVP for the FPME with datau0 ∈ L1(Rn) ∩ L∞(Rn), as constructed before. Then, there exists apositive constant C such that for every t > 0

(8) supx∈Rn|u(x , t)| ≤ C t−α‖u0‖γL1(Rn)

with α = n/(n + 2− 2s), γ = (2− 2s)/((n + 2− 2s). The constant Cdepends only on n and s.This theorem allows to extend the theory to data u0 ∈ L1(Rn), u0 ≥ 0,with global existence of bounded weak solutions.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 24 / 53

Energy and bilinear formsEnergy solutions: The basis of the boundedness analysis is a propertythat goes beyond the definition of weak solution. We will review theformulas with attention to the constants that appear since this is not donein [CSV]. The general energy property is as follows: for any F smoothand such that f = F ′ is bounded and nonnegative, we have for every0 ≤ t1 ≤ t2 ≤ T ,∫

F (u(t2)) dx −∫

F (u(t1)) dx = −∫ t2

t1

∫∇[f (u)]u∇p dx dt =

−∫ t2

t1

∫∇h(u)∇(−∆)−su dx dt

where h is a function satisfying h′(u) = u f ′(u). We can write the lastintegral as a bilinear form∫

∇h(u)∇(−∆)−su dx = Bs(h(u),u)

This bilinear form Bs is defined on the Sobolev space W 1,2(Rn) by

(9) Bs(v ,w) = Cn,s

∫∫∇v(x)

1|x − y |n−2s∇w(y) dx dy .

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 25 / 53

Energy and bilinear formsEnergy solutions: The basis of the boundedness analysis is a propertythat goes beyond the definition of weak solution. We will review theformulas with attention to the constants that appear since this is not donein [CSV]. The general energy property is as follows: for any F smoothand such that f = F ′ is bounded and nonnegative, we have for every0 ≤ t1 ≤ t2 ≤ T ,∫

F (u(t2)) dx −∫

F (u(t1)) dx = −∫ t2

t1

∫∇[f (u)]u∇p dx dt =

−∫ t2

t1

∫∇h(u)∇(−∆)−su dx dt

where h is a function satisfying h′(u) = u f ′(u). We can write the lastintegral as a bilinear form∫

∇h(u)∇(−∆)−su dx = Bs(h(u),u)

This bilinear form Bs is defined on the Sobolev space W 1,2(Rn) by

(9) Bs(v ,w) = Cn,s

∫∫∇v(x)

1|x − y |n−2s∇w(y) dx dy .

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 25 / 53

Energy and bilinear forms II

This bilinear form Bs is defined on the Sobolev space W 1,2(Rn) by

Bs(v ,w) = Cn,s∫∫∇v(x) 1

|x−y|n−2s∇w(y) dx dy =∫∫N−s(x , y)∇v(x)∇w(y) dx dy

where N−s(x , y) = Cn,s|x − y |−(n−2s) is the kernel of operator (−∆)−s.After some integrations by parts we also have(10)

Bs(v ,w) = Cn,1−s

∫∫(v(x)− v(y))

1|x − y |n+2(1−s) (w(x)− w(y)) dx dy

since −∆N−s = N1−s.It is known (Stein) that Bs(u,u) is an equivalent norm for the fractionalSobolev space W 1−s,2(Rn).We will need below that Cn,1−s ∼ Kn(1− s) as s → 1, for some constantKn depending only on n.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 26 / 53

Energy and bilinear forms II

This bilinear form Bs is defined on the Sobolev space W 1,2(Rn) by

Bs(v ,w) = Cn,s∫∫∇v(x) 1

|x−y|n−2s∇w(y) dx dy =∫∫N−s(x , y)∇v(x)∇w(y) dx dy

where N−s(x , y) = Cn,s|x − y |−(n−2s) is the kernel of operator (−∆)−s.After some integrations by parts we also have(10)

Bs(v ,w) = Cn,1−s

∫∫(v(x)− v(y))

1|x − y |n+2(1−s) (w(x)− w(y)) dx dy

since −∆N−s = N1−s.It is known (Stein) that Bs(u,u) is an equivalent norm for the fractionalSobolev space W 1−s,2(Rn).We will need below that Cn,1−s ∼ Kn(1− s) as s → 1, for some constantKn depending only on n.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 26 / 53

Energy and bilinear forms II

This bilinear form Bs is defined on the Sobolev space W 1,2(Rn) by

Bs(v ,w) = Cn,s∫∫∇v(x) 1

|x−y|n−2s∇w(y) dx dy =∫∫N−s(x , y)∇v(x)∇w(y) dx dy

where N−s(x , y) = Cn,s|x − y |−(n−2s) is the kernel of operator (−∆)−s.After some integrations by parts we also have(10)

Bs(v ,w) = Cn,1−s

∫∫(v(x)− v(y))

1|x − y |n+2(1−s) (w(x)− w(y)) dx dy

since −∆N−s = N1−s.It is known (Stein) that Bs(u,u) is an equivalent norm for the fractionalSobolev space W 1−s,2(Rn).We will need below that Cn,1−s ∼ Kn(1− s) as s → 1, for some constantKn depending only on n.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 26 / 53

Continuity

Bounded weak solutions u are Cα if 0 < s < 1.

• Case 0 < s < 1/2, which means diffusion almost as in the standardcase.Use variant of Luis local regularity theory for elliptic and parabolic PDE’s.The basic idea follows the energy analysis in Caffarelli-Chan-Vasseur.The nonlinearity is tackled by using a sophisticated test function on theweak formulation. The energy analysis is quite complex.The crucial point is to get a local version of the energy inequalities thatcan be iterated. It involves a delicate manipulation of the kernels ofbilinear forms associated to the fractional operators (Harmonic Analysis),with amounts to knowing well the Hs spaces

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 27 / 53

Continuity

Bounded weak solutions u are Cα if 0 < s < 1.

• Case 0 < s < 1/2, which means diffusion almost as in the standardcase.Use variant of Luis local regularity theory for elliptic and parabolic PDE’s.The basic idea follows the energy analysis in Caffarelli-Chan-Vasseur.The nonlinearity is tackled by using a sophisticated test function on theweak formulation. The energy analysis is quite complex.The crucial point is to get a local version of the energy inequalities thatcan be iterated. It involves a delicate manipulation of the kernels ofbilinear forms associated to the fractional operators (Harmonic Analysis),with amounts to knowing well the Hs spaces

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 27 / 53

Continuity II

Case 1/2 < s < 1, which means diffusion close to hydrodunamics.Here the bad integral is tackled by means of drift techniques. It amountsin practice to make a change of variables to a frame that moves rigidlywith one characteristic line of the flow

Case s = 1/2, which means middle of the way, he good case for theCaffarelli-Silvestre extension. This is the case with the very difficultanalysis, due to the existence of a bad term in the energy inequalities onwhich the iteration is based.

This is the topic for a 1 hour talk on hard analysis of singular integrals.

See details in paper CSV, arxiv 2012; JEMS 2013. The case s = 1/2 isstill in process.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 28 / 53

Continuity II

Case 1/2 < s < 1, which means diffusion close to hydrodunamics.Here the bad integral is tackled by means of drift techniques. It amountsin practice to make a change of variables to a frame that moves rigidlywith one characteristic line of the flow

Case s = 1/2, which means middle of the way, he good case for theCaffarelli-Silvestre extension. This is the case with the very difficultanalysis, due to the existence of a bad term in the energy inequalities onwhich the iteration is based.

This is the topic for a 1 hour talk on hard analysis of singular integrals.

See details in paper CSV, arxiv 2012; JEMS 2013. The case s = 1/2 isstill in process.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 28 / 53

Continuity II

Case 1/2 < s < 1, which means diffusion close to hydrodunamics.Here the bad integral is tackled by means of drift techniques. It amountsin practice to make a change of variables to a frame that moves rigidlywith one characteristic line of the flow

Case s = 1/2, which means middle of the way, he good case for theCaffarelli-Silvestre extension. This is the case with the very difficultanalysis, due to the existence of a bad term in the energy inequalities onwhich the iteration is based.

This is the topic for a 1 hour talk on hard analysis of singular integrals.

See details in paper CSV, arxiv 2012; JEMS 2013. The case s = 1/2 isstill in process.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 28 / 53

Outline

1 Introduction

2 Traditional porous medium

3 Nonlinear diffusion with nonlocal effectsPresentationThe modelOther models

4 TheoryMain estimatesTheory

5 Asymptotic behaviourAsymptotic behavior for the nonlocal PMERenormalized estimates

6 The second Fractional PME model

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 29 / 53

Barenblatt profiles and Asymptotics

These profiles are the alternative to the Gaussian profiles.They are source solutions. Source means that u(x , t)→ M δ(x) as t → 0.usExplicit formulas (1950, 52):

B(x , t ;M) = t−αF(x/tβ), F(ξ) =(

C − kξ2)1/(m−1)

+

α = n2+n(m−1)

β = 12+n(m−1) < 1/2

Height u = Ct−α

Free boundary at distance |x | = ctβ

Scaling law; anomalous diffusion versus Brownian motion

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 30 / 53

Barenblatt profiles and Asymptotics

These profiles are the alternative to the Gaussian profiles.They are source solutions. Source means that u(x , t)→ M δ(x) as t → 0.usExplicit formulas (1950, 52):

B(x , t ;M) = t−αF(x/tβ), F(ξ) =(

C − kξ2)1/(m−1)

+

α = n2+n(m−1)

β = 12+n(m−1) < 1/2

Height u = Ct−α

Free boundary at distance |x | = ctβ

Scaling law; anomalous diffusion versus Brownian motion

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 30 / 53

Asymptotic behaviour INonlinear Central Limit Theorem

Choice of domain: Rn. Choice of data: u0(x) ∈ L1(Rn). We can write

ut = ∆(|u|m−1u) + f

Let us put f ∈ L1x,t . Let M =

∫u0(x) dx +

∫∫f dxdt .

Asymptotic Theorem [Kamin and Friedman, 1980; V. 2001] Let B(x , t ; M) bethe Barenblatt with the asymptotic mass M; let f = 0 (or small at infinity).Then u converges to B after renormalization

tα|u(x , t)− B(x , t)| → 0 ,

and for every p ≥ 1 we have

‖u(t)− B(t)‖p = o(t−α/p′), p′ = p/(p − 1).

Result is valid for p = 1 with only restriction that f ∈ L1.

Notes: α and β = α/n = 1/(2 + n(m − 1)) are the zooming exponents asin B(x , t). Starting result by FK takes u0 ≥ 0, compact support and f = 0.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 31 / 53

Asymptotic behaviour INonlinear Central Limit Theorem

Choice of domain: Rn. Choice of data: u0(x) ∈ L1(Rn). We can write

ut = ∆(|u|m−1u) + f

Let us put f ∈ L1x,t . Let M =

∫u0(x) dx +

∫∫f dxdt .

Asymptotic Theorem [Kamin and Friedman, 1980; V. 2001] Let B(x , t ; M) bethe Barenblatt with the asymptotic mass M; let f = 0 (or small at infinity).Then u converges to B after renormalization

tα|u(x , t)− B(x , t)| → 0 ,

and for every p ≥ 1 we have

‖u(t)− B(t)‖p = o(t−α/p′), p′ = p/(p − 1).

Result is valid for p = 1 with only restriction that f ∈ L1.

Notes: α and β = α/n = 1/(2 + n(m − 1)) are the zooming exponents asin B(x , t). Starting result by FK takes u0 ≥ 0, compact support and f = 0.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 31 / 53

Asymptotic behaviour INonlinear Central Limit Theorem

Choice of domain: Rn. Choice of data: u0(x) ∈ L1(Rn). We can write

ut = ∆(|u|m−1u) + f

Let us put f ∈ L1x,t . Let M =

∫u0(x) dx +

∫∫f dxdt .

Asymptotic Theorem [Kamin and Friedman, 1980; V. 2001] Let B(x , t ; M) bethe Barenblatt with the asymptotic mass M; let f = 0 (or small at infinity).Then u converges to B after renormalization

tα|u(x , t)− B(x , t)| → 0 ,

and for every p ≥ 1 we have

‖u(t)− B(t)‖p = o(t−α/p′), p′ = p/(p − 1).

Result is valid for p = 1 with only restriction that f ∈ L1.

Notes: α and β = α/n = 1/(2 + n(m − 1)) are the zooming exponents asin B(x , t). Starting result by FK takes u0 ≥ 0, compact support and f = 0.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 31 / 53

Calculations of the entropy ratesWe rescale the function as u(x , t) = r(t)n ρ(y r(t), s)where r(t) is the Barenblatt radius at t + 1, and “new time" iss = log(1 + t). Equation becomes

ρs = div (ρ(∇ρm−1 +c2∇y2)).

Then define the entropyE(u)(t) =

∫(

1mρm +

c2ρy2) dy

The minimum of entropy is identified as the Barenblatt profile.Calculate

dEds

= −∫ρ|∇ρm−1 + cy |2 dy = −D

Moreover, dDds

= −R, R ∼ λD.

We conclude exponential decay of D in new time s, which is potential inreal time t. It follows that E decays to a minimum E∞ > 0 and we provethat this is the level of the Barenblatt solution.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 32 / 53

Calculations of the entropy ratesWe rescale the function as u(x , t) = r(t)n ρ(y r(t), s)where r(t) is the Barenblatt radius at t + 1, and “new time" iss = log(1 + t). Equation becomes

ρs = div (ρ(∇ρm−1 +c2∇y2)).

Then define the entropyE(u)(t) =

∫(

1mρm +

c2ρy2) dy

The minimum of entropy is identified as the Barenblatt profile.Calculate

dEds

= −∫ρ|∇ρm−1 + cy |2 dy = −D

Moreover, dDds

= −R, R ∼ λD.

We conclude exponential decay of D in new time s, which is potential inreal time t. It follows that E decays to a minimum E∞ > 0 and we provethat this is the level of the Barenblatt solution.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 32 / 53

Calculations of the entropy ratesWe rescale the function as u(x , t) = r(t)n ρ(y r(t), s)where r(t) is the Barenblatt radius at t + 1, and “new time" iss = log(1 + t). Equation becomes

ρs = div (ρ(∇ρm−1 +c2∇y2)).

Then define the entropyE(u)(t) =

∫(

1mρm +

c2ρy2) dy

The minimum of entropy is identified as the Barenblatt profile.Calculate

dEds

= −∫ρ|∇ρm−1 + cy |2 dy = −D

Moreover, dDds

= −R, R ∼ λD.

We conclude exponential decay of D in new time s, which is potential inreal time t. It follows that E decays to a minimum E∞ > 0 and we provethat this is the level of the Barenblatt solution.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 32 / 53

Asymptotic behavior

for the nonlocal PME 6

6Asymptotic behavior of a porous medium equation with fractional diffusion,Luis Caffarelli, Juan Luis Vázquez, Discrete Cont. Dynam. Systems, special issue 2011.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 33 / 53

Rescaling for the NL-PMEWe now begin the study of the large time behavior.Inspired by the asymptotics of the standard porous medium equation, we definethe renormalized flow through the transformation

(11) u(x , t) = t−αv(x/tβ , τ)

with new time τ = log(1 + t). We also put y = x/tβ as rescaled space variable.In order to cancel the factors including t explicitly, we get the condition on theexponents

(12) α + (2− 2s)β = 1

We use the homogeneity of K in the form

(13) (Ku)(x , t) = t−α+2sβ(Kv)(y , τ).

If we also want conservation of (finite) mass, then we must put α = nβ, and wearrive at the the precise value of the exponents:

β = 1/(n + 2− 2s), α = n/(n + 2− 2s).

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 34 / 53

Rescaling for the NL-PMEWe now begin the study of the large time behavior.Inspired by the asymptotics of the standard porous medium equation, we definethe renormalized flow through the transformation

(11) u(x , t) = t−αv(x/tβ , τ)

with new time τ = log(1 + t). We also put y = x/tβ as rescaled space variable.In order to cancel the factors including t explicitly, we get the condition on theexponents

(12) α + (2− 2s)β = 1

We use the homogeneity of K in the form

(13) (Ku)(x , t) = t−α+2sβ(Kv)(y , τ).

If we also want conservation of (finite) mass, then we must put α = nβ, and wearrive at the the precise value of the exponents:

β = 1/(n + 2− 2s), α = n/(n + 2− 2s).

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 34 / 53

Rescaling for the NL-PMEWe now begin the study of the large time behavior.Inspired by the asymptotics of the standard porous medium equation, we definethe renormalized flow through the transformation

(11) u(x , t) = t−αv(x/tβ , τ)

with new time τ = log(1 + t). We also put y = x/tβ as rescaled space variable.In order to cancel the factors including t explicitly, we get the condition on theexponents

(12) α + (2− 2s)β = 1

We use the homogeneity of K in the form

(13) (Ku)(x , t) = t−α+2sβ(Kv)(y , τ).

If we also want conservation of (finite) mass, then we must put α = nβ, and wearrive at the the precise value of the exponents:

β = 1/(n + 2− 2s), α = n/(n + 2− 2s).

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 34 / 53

Renormalized flowWe thus arrive at the nonlinear, nonlocal Fokker-Plank equation

(14) vτ = ∇y · (v (∇y K (v) + βy))

This formula implies a transformation for the pressure of the form(15)

p(u)(x , t) = t−σp(v)(x/tβ , τ), with σ = α− 2sβ = 1− 2β =n − 2s

n + 2− 2s< 1.

Stationary renormalized solutions. They are the solutions U(y) of

(16) ∇y · (U∇y (P + a|y |2)) = 0, P = K (U).

where a = β/2, and β defined just above. Since we are looking for asymptoticprofiles of the standard solutions of the NL-PME we also want U ≥ 0 andintegrable. The simplest possibility is integrating once and getting the radialversion

(17) U∇y (P + a|y |2)) = 0, P = K (U), U ≥ 0.

The first equation gives an alternative choice that reminds of the complementaryformulation of the obstacle problems.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 35 / 53

Renormalized flowWe thus arrive at the nonlinear, nonlocal Fokker-Plank equation

(14) vτ = ∇y · (v (∇y K (v) + βy))

This formula implies a transformation for the pressure of the form(15)

p(u)(x , t) = t−σp(v)(x/tβ , τ), with σ = α− 2sβ = 1− 2β =n − 2s

n + 2− 2s< 1.

Stationary renormalized solutions. They are the solutions U(y) of

(16) ∇y · (U∇y (P + a|y |2)) = 0, P = K (U).

where a = β/2, and β defined just above. Since we are looking for asymptoticprofiles of the standard solutions of the NL-PME we also want U ≥ 0 andintegrable. The simplest possibility is integrating once and getting the radialversion

(17) U∇y (P + a|y |2)) = 0, P = K (U), U ≥ 0.

The first equation gives an alternative choice that reminds of the complementaryformulation of the obstacle problems.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 35 / 53

Renormalized flowWe thus arrive at the nonlinear, nonlocal Fokker-Plank equation

(14) vτ = ∇y · (v (∇y K (v) + βy))

This formula implies a transformation for the pressure of the form(15)

p(u)(x , t) = t−σp(v)(x/tβ , τ), with σ = α− 2sβ = 1− 2β =n − 2s

n + 2− 2s< 1.

Stationary renormalized solutions. They are the solutions U(y) of

(16) ∇y · (U∇y (P + a|y |2)) = 0, P = K (U).

where a = β/2, and β defined just above. Since we are looking for asymptoticprofiles of the standard solutions of the NL-PME we also want U ≥ 0 andintegrable. The simplest possibility is integrating once and getting the radialversion

(17) U∇y (P + a|y |2)) = 0, P = K (U), U ≥ 0.

The first equation gives an alternative choice that reminds of the complementaryformulation of the obstacle problems.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 35 / 53

Obstacle problem

Indeed, if we solve the obstacle problem with fractional Laplacian we will obtain aunique solution P(y) of the problem:

(18) P ≥ Φ, U = (−∆)sP ≥ 0;either P = Φ or U = 0.

with 0 < s < 1. Here we have to choose as obstacle

Φ = C − a |y |2,

where C is any positive constant and a = β/2. For uniqueness we also need thecondition P → 0 as |y | → ∞.

The theory is developed in A-C-S, C-S-S, the solution is unique and belongs tothe space H−s with pressure in Hs. The solutions have P ∈ C1,s and U ∈ C1−s.

Note that for C ≤ 0 the solution is trivial, P = 0, U = 0, hence we choose C > 0.We also note the pressure is defined but for a constant, so that we may takewithout loss of generality C = 0 and take as pressure P = P − C instead of P.But then P → 0 implies that P → −C as |y | → ∞, so we get a one parameterfamily of stationary profiles that we denote UC(y).

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 36 / 53

Obstacle problem

Indeed, if we solve the obstacle problem with fractional Laplacian we will obtain aunique solution P(y) of the problem:

(18) P ≥ Φ, U = (−∆)sP ≥ 0;either P = Φ or U = 0.

with 0 < s < 1. Here we have to choose as obstacle

Φ = C − a |y |2,

where C is any positive constant and a = β/2. For uniqueness we also need thecondition P → 0 as |y | → ∞.

The theory is developed in A-C-S, C-S-S, the solution is unique and belongs tothe space H−s with pressure in Hs. The solutions have P ∈ C1,s and U ∈ C1−s.

Note that for C ≤ 0 the solution is trivial, P = 0, U = 0, hence we choose C > 0.We also note the pressure is defined but for a constant, so that we may takewithout loss of generality C = 0 and take as pressure P = P − C instead of P.But then P → 0 implies that P → −C as |y | → ∞, so we get a one parameterfamily of stationary profiles that we denote UC(y).

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 36 / 53

Obstacle problem

Indeed, if we solve the obstacle problem with fractional Laplacian we will obtain aunique solution P(y) of the problem:

(18) P ≥ Φ, U = (−∆)sP ≥ 0;either P = Φ or U = 0.

with 0 < s < 1. Here we have to choose as obstacle

Φ = C − a |y |2,

where C is any positive constant and a = β/2. For uniqueness we also need thecondition P → 0 as |y | → ∞.

The theory is developed in A-C-S, C-S-S, the solution is unique and belongs tothe space H−s with pressure in Hs. The solutions have P ∈ C1,s and U ∈ C1−s.

Note that for C ≤ 0 the solution is trivial, P = 0, U = 0, hence we choose C > 0.We also note the pressure is defined but for a constant, so that we may takewithout loss of generality C = 0 and take as pressure P = P − C instead of P.But then P → 0 implies that P → −C as |y | → ∞, so we get a one parameterfamily of stationary profiles that we denote UC(y).

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 36 / 53

Barenblatt solutions of new type.

The above solution of this obstacle problem produces a correct weak solution ofthe NL-PM equation with initial data a Dirac delta for the density u, i.e., it is thesource-type or Barenblatt solution for this problem, which is a profile U ≥ 0.It is positive in the contact set of the obstacle problem, C = {|y | ≤ R(C)},and is zero outside, hence it has compact support.

It is clear that R is smaller than the intersection of the parabola Φ with the axisR1 = (C/a)1/2.

On the other hand, the pressure P(|y |) is always positive and decays to zero as|y | → ∞ according to fractional potential theory, cf Landau, Stein.The rate is P = O(|y |2s−n).

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 37 / 53

Barenblatt solutions of new type.

The above solution of this obstacle problem produces a correct weak solution ofthe NL-PM equation with initial data a Dirac delta for the density u, i.e., it is thesource-type or Barenblatt solution for this problem, which is a profile U ≥ 0.It is positive in the contact set of the obstacle problem, C = {|y | ≤ R(C)},and is zero outside, hence it has compact support.

It is clear that R is smaller than the intersection of the parabola Φ with the axisR1 = (C/a)1/2.

On the other hand, the pressure P(|y |) is always positive and decays to zero as|y | → ∞ according to fractional potential theory, cf Landau, Stein.The rate is P = O(|y |2s−n).

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 37 / 53

Barenblatt solutions of new type.

The above solution of this obstacle problem produces a correct weak solution ofthe NL-PM equation with initial data a Dirac delta for the density u, i.e., it is thesource-type or Barenblatt solution for this problem, which is a profile U ≥ 0.It is positive in the contact set of the obstacle problem, C = {|y | ≤ R(C)},and is zero outside, hence it has compact support.

It is clear that R is smaller than the intersection of the parabola Φ with the axisR1 = (C/a)1/2.

On the other hand, the pressure P(|y |) is always positive and decays to zero as|y | → ∞ according to fractional potential theory, cf Landau, Stein.The rate is P = O(|y |2s−n).

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 37 / 53

The solution of the obstacle problem with parabolic obstacle

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 38 / 53

Estimates for the renormalized problem.Entropy dissipation.

Our main problem is now to prove that these profiles are attractors for therenormalized flow.We review the estimates of Main Estimates Section above in order toadapt them to the renormalized problem. çThere is no problem is reproving mass conservation or positivity.First energy estimate becomes (recall that H = K 1/2)

(19)

ddτ

∫v(y , τ) log v(y , τ) dy

= −∫|∇Hv |2 dy − β

∫∇v · y

= −∫|∇Hv |2 dy + α

∫v .

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 39 / 53

Estimates for the renormalized problem.Entropy dissipation.

Our main problem is now to prove that these profiles are attractors for therenormalized flow.We review the estimates of Main Estimates Section above in order toadapt them to the renormalized problem. çThere is no problem is reproving mass conservation or positivity.First energy estimate becomes (recall that H = K 1/2)

(19)

ddτ

∫v(y , τ) log v(y , τ) dy

= −∫|∇Hv |2 dy − β

∫∇v · y

= −∫|∇Hv |2 dy + α

∫v .

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 39 / 53

Estimates for the renormalized problem. Entropydissipation.

However, the second energy estimate has an essential change. We need todefine the entropy of the renormalized flow as

(20) E(v(τ)) :=12

∫Rn

(v K (v) + βy2v) dy

The entropy contains two terms. The first is

E1(v(τ)) :=

∫Rn

v K (v) dy =

∫Rn|Hv |2 dy , H = K 1/2

hence positive. The second is the moment E2(v(τ)) = M2(v(τ)) :=∫

y2v dy ,also positive. By differentiation we get

(21)d

dτE(v) = −I(v), I(v) :=

∫ ∣∣∣∣∇(Kv +β

2y2)

∣∣∣∣2 vdy .

This means that whenever the initial entropy is finite, then E(v(τ)) is uniformlybounded for all τ > 0, I(v) is integrable in (0,∞) and

E(v(τ)) +

∫∫ ∣∣∣∣∇(Kv +β

2y2)

∣∣∣∣2 vdy dt ≤ E(v0).

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 40 / 53

Estimates for the renormalized problem. Entropydissipation.

However, the second energy estimate has an essential change. We need todefine the entropy of the renormalized flow as

(20) E(v(τ)) :=12

∫Rn

(v K (v) + βy2v) dy

The entropy contains two terms. The first is

E1(v(τ)) :=

∫Rn

v K (v) dy =

∫Rn|Hv |2 dy , H = K 1/2

hence positive. The second is the moment E2(v(τ)) = M2(v(τ)) :=∫

y2v dy ,also positive. By differentiation we get

(21)d

dτE(v) = −I(v), I(v) :=

∫ ∣∣∣∣∇(Kv +β

2y2)

∣∣∣∣2 vdy .

This means that whenever the initial entropy is finite, then E(v(τ)) is uniformlybounded for all τ > 0, I(v) is integrable in (0,∞) and

E(v(τ)) +

∫∫ ∣∣∣∣∇(Kv +β

2y2)

∣∣∣∣2 vdy dt ≤ E(v0).

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 40 / 53

Estimates for the renormalized problem. Entropydissipation.

However, the second energy estimate has an essential change. We need todefine the entropy of the renormalized flow as

(20) E(v(τ)) :=12

∫Rn

(v K (v) + βy2v) dy

The entropy contains two terms. The first is

E1(v(τ)) :=

∫Rn

v K (v) dy =

∫Rn|Hv |2 dy , H = K 1/2

hence positive. The second is the moment E2(v(τ)) = M2(v(τ)) :=∫

y2v dy ,also positive. By differentiation we get

(21)d

dτE(v) = −I(v), I(v) :=

∫ ∣∣∣∣∇(Kv +β

2y2)

∣∣∣∣2 vdy .

This means that whenever the initial entropy is finite, then E(v(τ)) is uniformlybounded for all τ > 0, I(v) is integrable in (0,∞) and

E(v(τ)) +

∫∫ ∣∣∣∣∇(Kv +β

2y2)

∣∣∣∣2 vdy dt ≤ E(v0).

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 40 / 53

Convergence.

The standard idea is to let t →∞ in the renormalized flow. Since the entropygoes down there is a limit

E∗ = limt→∞E(t) ≥ 0.

Since u is bounded in L1x unif. in t , and also ux2 is bounded in L1

x unif. in t , andmoreover |∇H(u)| ∈ L2

x unif in t , we have that u(t) is a compact family that thereis a subsequence tj →∞ that converges in L1

x and almost everywhere to a limitu∗ ≥ 0. The mass of u∗ is the same mass of u. One consequence is that the liminf of the component E2(u(tj )) is equal or larger that M2(u∗).

We also have H(u) ∈ L2x uniformly in t . The boundedness of ∇H(u) in L2

x impliesthe compactness of H(u) in space, so that it converges along a subsequence tov∗ . This allows to pass to the limit in E1(u(tj )) and obtain a correct limit. We havev∗ = H(u∗).

Since the mass of u for large x is small On the other hand, also

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 41 / 53

Convergence.

The standard idea is to let t →∞ in the renormalized flow. Since the entropygoes down there is a limit

E∗ = limt→∞E(t) ≥ 0.

Since u is bounded in L1x unif. in t , and also ux2 is bounded in L1

x unif. in t , andmoreover |∇H(u)| ∈ L2

x unif in t , we have that u(t) is a compact family that thereis a subsequence tj →∞ that converges in L1

x and almost everywhere to a limitu∗ ≥ 0. The mass of u∗ is the same mass of u. One consequence is that the liminf of the component E2(u(tj )) is equal or larger that M2(u∗).

We also have H(u) ∈ L2x uniformly in t . The boundedness of ∇H(u) in L2

x impliesthe compactness of H(u) in space, so that it converges along a subsequence tov∗ . This allows to pass to the limit in E1(u(tj )) and obtain a correct limit. We havev∗ = H(u∗).

Since the mass of u for large x is small On the other hand, also

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 41 / 53

Convergence.

The standard idea is to let t →∞ in the renormalized flow. Since the entropygoes down there is a limit

E∗ = limt→∞E(t) ≥ 0.

Since u is bounded in L1x unif. in t , and also ux2 is bounded in L1

x unif. in t , andmoreover |∇H(u)| ∈ L2

x unif in t , we have that u(t) is a compact family that thereis a subsequence tj →∞ that converges in L1

x and almost everywhere to a limitu∗ ≥ 0. The mass of u∗ is the same mass of u. One consequence is that the liminf of the component E2(u(tj )) is equal or larger that M2(u∗).

We also have H(u) ∈ L2x uniformly in t . The boundedness of ∇H(u) in L2

x impliesthe compactness of H(u) in space, so that it converges along a subsequence tov∗ . This allows to pass to the limit in E1(u(tj )) and obtain a correct limit. We havev∗ = H(u∗).

Since the mass of u for large x is small On the other hand, also

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 41 / 53

Convergence

Now we get the consequence that for every h > 0 fixed∫ tj+h

tj

∫ ∣∣∣∣∇(Ku +β

2x2)

∣∣∣∣2 udx dt → 0.

This implies that if w(x , t) = Ku + β2 x2 and wh(x , t) = w(x , t + h), then uh|∇wh|2

converges to zero as h→∞ in L1(Rn × (0,T ). Then wh converges to a constantin space wherever u is not zero, and that constant must be Ku∗ + β

2 x2 along thesaid subsequence, hence constant also in time

This means that the limit is a solution of the Barenblatt obstacle problem.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 42 / 53

Convergence

Now we get the consequence that for every h > 0 fixed∫ tj+h

tj

∫ ∣∣∣∣∇(Ku +β

2x2)

∣∣∣∣2 udx dt → 0.

This implies that if w(x , t) = Ku + β2 x2 and wh(x , t) = w(x , t + h), then uh|∇wh|2

converges to zero as h→∞ in L1(Rn × (0,T ). Then wh converges to a constantin space wherever u is not zero, and that constant must be Ku∗ + β

2 x2 along thesaid subsequence, hence constant also in time

This means that the limit is a solution of the Barenblatt obstacle problem.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 42 / 53

Recent workBiler, Imbert and Karch. In a note just submitted to CRAS (Barenblattprofiles for a nonlocal porous medium equation) the authors study themore general equation

ut = ∇ · (uΛα−1um), 0 < α < 2and obtain our type of Barenblatt solutions for every m > 1 with a verynice added information, they happen to be explicit of the form

u(x , t) = Ct−µ(R2 − x2t−2ν))α/2(m−1)+

it uses an important identity by Getoor, (−∆)α/2(1− y2)α/2+ = K , valid

inside the support. Observe the boundary behavior.The work with Serfaty (2012) shows that the limit s → 1 of the solutions ofthe fractional diffusion problem produces the solutions of thehydrodynamic limit

ut = ∇ · (u∇p), −∆p = u .This happens in all dimensions n ≥ 2. The corresponding self-similarsolutions take the form

U(x , t) = t−1H(x/t1/n)and H is the characteristic function of a ball BR(0). This means in Physicsand expanding vortex patch. This type of solutions is NOT continuous.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 43 / 53

Recent workBiler, Imbert and Karch. In a note just submitted to CRAS (Barenblattprofiles for a nonlocal porous medium equation) the authors study themore general equation

ut = ∇ · (uΛα−1um), 0 < α < 2and obtain our type of Barenblatt solutions for every m > 1 with a verynice added information, they happen to be explicit of the form

u(x , t) = Ct−µ(R2 − x2t−2ν))α/2(m−1)+

it uses an important identity by Getoor, (−∆)α/2(1− y2)α/2+ = K , valid

inside the support. Observe the boundary behavior.The work with Serfaty (2012) shows that the limit s → 1 of the solutions ofthe fractional diffusion problem produces the solutions of thehydrodynamic limit

ut = ∇ · (u∇p), −∆p = u .This happens in all dimensions n ≥ 2. The corresponding self-similarsolutions take the form

U(x , t) = t−1H(x/t1/n)and H is the characteristic function of a ball BR(0). This means in Physicsand expanding vortex patch. This type of solutions is NOT continuous.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 43 / 53

Recent workBiler, Imbert and Karch. In a note just submitted to CRAS (Barenblattprofiles for a nonlocal porous medium equation) the authors study themore general equation

ut = ∇ · (uΛα−1um), 0 < α < 2and obtain our type of Barenblatt solutions for every m > 1 with a verynice added information, they happen to be explicit of the form

u(x , t) = Ct−µ(R2 − x2t−2ν))α/2(m−1)+

it uses an important identity by Getoor, (−∆)α/2(1− y2)α/2+ = K , valid

inside the support. Observe the boundary behavior.The work with Serfaty (2012) shows that the limit s → 1 of the solutions ofthe fractional diffusion problem produces the solutions of thehydrodynamic limit

ut = ∇ · (u∇p), −∆p = u .This happens in all dimensions n ≥ 2. The corresponding self-similarsolutions take the form

U(x , t) = t−1H(x/t1/n)and H is the characteristic function of a ball BR(0). This means in Physicsand expanding vortex patch. This type of solutions is NOT continuous.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 43 / 53

Recent workBiler, Imbert and Karch. In a note just submitted to CRAS (Barenblattprofiles for a nonlocal porous medium equation) the authors study themore general equation

ut = ∇ · (uΛα−1um), 0 < α < 2and obtain our type of Barenblatt solutions for every m > 1 with a verynice added information, they happen to be explicit of the form

u(x , t) = Ct−µ(R2 − x2t−2ν))α/2(m−1)+

it uses an important identity by Getoor, (−∆)α/2(1− y2)α/2+ = K , valid

inside the support. Observe the boundary behavior.The work with Serfaty (2012) shows that the limit s → 1 of the solutions ofthe fractional diffusion problem produces the solutions of thehydrodynamic limit

ut = ∇ · (u∇p), −∆p = u .This happens in all dimensions n ≥ 2. The corresponding self-similarsolutions take the form

U(x , t) = t−1H(x/t1/n)and H is the characteristic function of a ball BR(0). This means in Physicsand expanding vortex patch. This type of solutions is NOT continuous.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 43 / 53

Recent workBiler, Imbert and Karch. In a note just submitted to CRAS (Barenblattprofiles for a nonlocal porous medium equation) the authors study themore general equation

ut = ∇ · (uΛα−1um), 0 < α < 2and obtain our type of Barenblatt solutions for every m > 1 with a verynice added information, they happen to be explicit of the form

u(x , t) = Ct−µ(R2 − x2t−2ν))α/2(m−1)+

it uses an important identity by Getoor, (−∆)α/2(1− y2)α/2+ = K , valid

inside the support. Observe the boundary behavior.The work with Serfaty (2012) shows that the limit s → 1 of the solutions ofthe fractional diffusion problem produces the solutions of thehydrodynamic limit

ut = ∇ · (u∇p), −∆p = u .This happens in all dimensions n ≥ 2. The corresponding self-similarsolutions take the form

U(x , t) = t−1H(x/t1/n)and H is the characteristic function of a ball BR(0). This means in Physicsand expanding vortex patch. This type of solutions is NOT continuous.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 43 / 53

Work to DoStudy the regularity of the solutions

Study the equation and regularity of the free boundary

Study fine asymptotic behavior, and extend to other classes of data

Study these nonlocal problems in bounded domains

Decide conditions of uniqueness

Decide conditions of comparison

Write a performing numerical code

Consider different nonlocal nonlinear diffusion problems

Discuss the Stochastic Particle Models in the literature that involvelong-range effects and anomalous diffusion parameters.

∗ SURVEY Juan Luis Vázquez, Nonlinear Diffusion with FractionalLaplacian Operators. “Nonlinear partial differential equations: the AbelSymposium 2010”, Holden, Helge & Karlsen, Kenneth H. eds., Springer,2012. Pp. 271–298. Based on talk given at Academy Sciences, Oslo, 1Oct 2011.See JLV’s webpage, preprints.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 44 / 53

Work to DoStudy the regularity of the solutions

Study the equation and regularity of the free boundary

Study fine asymptotic behavior, and extend to other classes of data

Study these nonlocal problems in bounded domains

Decide conditions of uniqueness

Decide conditions of comparison

Write a performing numerical code

Consider different nonlocal nonlinear diffusion problems

Discuss the Stochastic Particle Models in the literature that involvelong-range effects and anomalous diffusion parameters.

∗ SURVEY Juan Luis Vázquez, Nonlinear Diffusion with FractionalLaplacian Operators. “Nonlinear partial differential equations: the AbelSymposium 2010”, Holden, Helge & Karlsen, Kenneth H. eds., Springer,2012. Pp. 271–298. Based on talk given at Academy Sciences, Oslo, 1Oct 2011.See JLV’s webpage, preprints.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 44 / 53

Work to DoStudy the regularity of the solutions

Study the equation and regularity of the free boundary

Study fine asymptotic behavior, and extend to other classes of data

Study these nonlocal problems in bounded domains

Decide conditions of uniqueness

Decide conditions of comparison

Write a performing numerical code

Consider different nonlocal nonlinear diffusion problems

Discuss the Stochastic Particle Models in the literature that involvelong-range effects and anomalous diffusion parameters.

∗ SURVEY Juan Luis Vázquez, Nonlinear Diffusion with FractionalLaplacian Operators. “Nonlinear partial differential equations: the AbelSymposium 2010”, Holden, Helge & Karlsen, Kenneth H. eds., Springer,2012. Pp. 271–298. Based on talk given at Academy Sciences, Oslo, 1Oct 2011.See JLV’s webpage, preprints.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 44 / 53

Work to DoStudy the regularity of the solutions

Study the equation and regularity of the free boundary

Study fine asymptotic behavior, and extend to other classes of data

Study these nonlocal problems in bounded domains

Decide conditions of uniqueness

Decide conditions of comparison

Write a performing numerical code

Consider different nonlocal nonlinear diffusion problems

Discuss the Stochastic Particle Models in the literature that involvelong-range effects and anomalous diffusion parameters.

∗ SURVEY Juan Luis Vázquez, Nonlinear Diffusion with FractionalLaplacian Operators. “Nonlinear partial differential equations: the AbelSymposium 2010”, Holden, Helge & Karlsen, Kenneth H. eds., Springer,2012. Pp. 271–298. Based on talk given at Academy Sciences, Oslo, 1Oct 2011.See JLV’s webpage, preprints.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 44 / 53

Work to DoStudy the regularity of the solutions

Study the equation and regularity of the free boundary

Study fine asymptotic behavior, and extend to other classes of data

Study these nonlocal problems in bounded domains

Decide conditions of uniqueness

Decide conditions of comparison

Write a performing numerical code

Consider different nonlocal nonlinear diffusion problems

Discuss the Stochastic Particle Models in the literature that involvelong-range effects and anomalous diffusion parameters.

∗ SURVEY Juan Luis Vázquez, Nonlinear Diffusion with FractionalLaplacian Operators. “Nonlinear partial differential equations: the AbelSymposium 2010”, Holden, Helge & Karlsen, Kenneth H. eds., Springer,2012. Pp. 271–298. Based on talk given at Academy Sciences, Oslo, 1Oct 2011.See JLV’s webpage, preprints.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 44 / 53

Work to DoStudy the regularity of the solutions

Study the equation and regularity of the free boundary

Study fine asymptotic behavior, and extend to other classes of data

Study these nonlocal problems in bounded domains

Decide conditions of uniqueness

Decide conditions of comparison

Write a performing numerical code

Consider different nonlocal nonlinear diffusion problems

Discuss the Stochastic Particle Models in the literature that involvelong-range effects and anomalous diffusion parameters.

∗ SURVEY Juan Luis Vázquez, Nonlinear Diffusion with FractionalLaplacian Operators. “Nonlinear partial differential equations: the AbelSymposium 2010”, Holden, Helge & Karlsen, Kenneth H. eds., Springer,2012. Pp. 271–298. Based on talk given at Academy Sciences, Oslo, 1Oct 2011.See JLV’s webpage, preprints.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 44 / 53

Work to DoStudy the regularity of the solutions

Study the equation and regularity of the free boundary

Study fine asymptotic behavior, and extend to other classes of data

Study these nonlocal problems in bounded domains

Decide conditions of uniqueness

Decide conditions of comparison

Write a performing numerical code

Consider different nonlocal nonlinear diffusion problems

Discuss the Stochastic Particle Models in the literature that involvelong-range effects and anomalous diffusion parameters.

∗ SURVEY Juan Luis Vázquez, Nonlinear Diffusion with FractionalLaplacian Operators. “Nonlinear partial differential equations: the AbelSymposium 2010”, Holden, Helge & Karlsen, Kenneth H. eds., Springer,2012. Pp. 271–298. Based on talk given at Academy Sciences, Oslo, 1Oct 2011.See JLV’s webpage, preprints.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 44 / 53

Outline

1 Introduction

2 Traditional porous medium

3 Nonlinear diffusion with nonlocal effectsPresentationThe modelOther models

4 TheoryMain estimatesTheory

5 Asymptotic behaviourAsymptotic behavior for the nonlocal PMERenormalized estimates

6 The second Fractional PME model

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 45 / 53

Alternative model For Porous Medium Flows

An alternative natural equation is

∂tu + (−∆)sum = 0.

This model arises from stochastic differential equations when modelingfor instance heat conduction with anomalous properties and oneintroduces jump processes into the modeling. The physical situationlooks still a bit confusing to me.Applied References: Stefano Olla. Milton Jara and collab., : • Jara, M.,Komorowski, T., Olla, S., Limit theorems for additive functionals of a Markovchain. http://arxiv.org/abs/0809.0177; M. Jara, C. Landim, S. Sethuraman,Nonequilibrium fluctuations for a tagged particle in mean-zero one-dimensionalzero-range processes, Probab. Theory Relat. Fields 145 (2009), 565–590.

Mathematics:• Athanasopoulos, I.; Caffarelli, L. A. Continuity of thetemperature in boundary heat control problems, Adv. Math. 224 (2010), no. 1,293–315. • Arturo de Pablo, Fernando Quirós, Ana Rodríguez, and Juan LuisVázquez, A fractional porous medium equation, Advances in Math. 226 (2011)1378–1409. A general fractional porous medium equation. Comm. Pure Appl.Math. 65 (2012), no. 9, 1242–1284.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 46 / 53

The second Fractional PME model. Short reportProblem is ut + (−∆)s/2(|u|m−1u) = 0 for x ∈ Rn, 0 < m <∞, 0 < s < 2,

with initial data in L1(Rn).The model represents another type of nonlinear interpolation, this time between

ut −∆(|u|m−1u) = 0 and ut + |u|m−1u = 0

A complete analysis of the Cauchy problem done byA. de Pablo, F. Quirós, AnaRodríguez, and J.L.V. A general fractional porous medium equation, Comm. PureApplied Math, 2012.Semigroup of weak energy solutions is constructed, smoothing effect works, Cα

regularity (if m is not near 0), infinite propagation for all m and s⇒ no compactsupport.

Asymptotic behavior is given by new self-similar solutions, see JLV, Barenblattsolutions and asymptotic behavior for a nonlinear fractional heat equation ofporous medium type. arXiv:1205.6332v1 [math.AP], May 2012. JEMS, to appear.

U(x , t) = t−nβF (x t−β)

with β = 1/(n(m − 1) + σ). F is only known explicitly for cases like m = 1, σ = 1.Its decay at infinity characterizes the fat tail of the fractional diffusion process.

Can you guess what is the decay ?

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 47 / 53

The second Fractional PME model. Short reportProblem is ut + (−∆)s/2(|u|m−1u) = 0 for x ∈ Rn, 0 < m <∞, 0 < s < 2,

with initial data in L1(Rn).The model represents another type of nonlinear interpolation, this time between

ut −∆(|u|m−1u) = 0 and ut + |u|m−1u = 0

A complete analysis of the Cauchy problem done byA. de Pablo, F. Quirós, AnaRodríguez, and J.L.V. A general fractional porous medium equation, Comm. PureApplied Math, 2012.Semigroup of weak energy solutions is constructed, smoothing effect works, Cα

regularity (if m is not near 0), infinite propagation for all m and s⇒ no compactsupport.

Asymptotic behavior is given by new self-similar solutions, see JLV, Barenblattsolutions and asymptotic behavior for a nonlinear fractional heat equation ofporous medium type. arXiv:1205.6332v1 [math.AP], May 2012. JEMS, to appear.

U(x , t) = t−nβF (x t−β)

with β = 1/(n(m − 1) + σ). F is only known explicitly for cases like m = 1, σ = 1.Its decay at infinity characterizes the fat tail of the fractional diffusion process.

Can you guess what is the decay ?

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 47 / 53

The second Fractional PME model. Short reportProblem is ut + (−∆)s/2(|u|m−1u) = 0 for x ∈ Rn, 0 < m <∞, 0 < s < 2,

with initial data in L1(Rn).The model represents another type of nonlinear interpolation, this time between

ut −∆(|u|m−1u) = 0 and ut + |u|m−1u = 0

A complete analysis of the Cauchy problem done byA. de Pablo, F. Quirós, AnaRodríguez, and J.L.V. A general fractional porous medium equation, Comm. PureApplied Math, 2012.Semigroup of weak energy solutions is constructed, smoothing effect works, Cα

regularity (if m is not near 0), infinite propagation for all m and s⇒ no compactsupport.

Asymptotic behavior is given by new self-similar solutions, see JLV, Barenblattsolutions and asymptotic behavior for a nonlinear fractional heat equation ofporous medium type. arXiv:1205.6332v1 [math.AP], May 2012. JEMS, to appear.

U(x , t) = t−nβF (x t−β)

with β = 1/(n(m − 1) + σ). F is only known explicitly for cases like m = 1, σ = 1.Its decay at infinity characterizes the fat tail of the fractional diffusion process.

Can you guess what is the decay ?

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 47 / 53

The second Fractional PME model. Short reportProblem is ut + (−∆)s/2(|u|m−1u) = 0 for x ∈ Rn, 0 < m <∞, 0 < s < 2,

with initial data in L1(Rn).The model represents another type of nonlinear interpolation, this time between

ut −∆(|u|m−1u) = 0 and ut + |u|m−1u = 0

A complete analysis of the Cauchy problem done byA. de Pablo, F. Quirós, AnaRodríguez, and J.L.V. A general fractional porous medium equation, Comm. PureApplied Math, 2012.Semigroup of weak energy solutions is constructed, smoothing effect works, Cα

regularity (if m is not near 0), infinite propagation for all m and s⇒ no compactsupport.

Asymptotic behavior is given by new self-similar solutions, see JLV, Barenblattsolutions and asymptotic behavior for a nonlinear fractional heat equation ofporous medium type. arXiv:1205.6332v1 [math.AP], May 2012. JEMS, to appear.

U(x , t) = t−nβF (x t−β)

with β = 1/(n(m − 1) + σ). F is only known explicitly for cases like m = 1, σ = 1.Its decay at infinity characterizes the fat tail of the fractional diffusion process.

Can you guess what is the decay ?

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 47 / 53

The second Fractional PME model. Short reportProblem is ut + (−∆)s/2(|u|m−1u) = 0 for x ∈ Rn, 0 < m <∞, 0 < s < 2,

with initial data in L1(Rn).The model represents another type of nonlinear interpolation, this time between

ut −∆(|u|m−1u) = 0 and ut + |u|m−1u = 0

A complete analysis of the Cauchy problem done byA. de Pablo, F. Quirós, AnaRodríguez, and J.L.V. A general fractional porous medium equation, Comm. PureApplied Math, 2012.Semigroup of weak energy solutions is constructed, smoothing effect works, Cα

regularity (if m is not near 0), infinite propagation for all m and s⇒ no compactsupport.

Asymptotic behavior is given by new self-similar solutions, see JLV, Barenblattsolutions and asymptotic behavior for a nonlinear fractional heat equation ofporous medium type. arXiv:1205.6332v1 [math.AP], May 2012. JEMS, to appear.

U(x , t) = t−nβF (x t−β)

with β = 1/(n(m − 1) + σ). F is only known explicitly for cases like m = 1, σ = 1.Its decay at infinity characterizes the fat tail of the fractional diffusion process.

Can you guess what is the decay ?

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 47 / 53

The second Fractional PME model. Short reportProblem is ut + (−∆)s/2(|u|m−1u) = 0 for x ∈ Rn, 0 < m <∞, 0 < s < 2,

with initial data in L1(Rn).The model represents another type of nonlinear interpolation, this time between

ut −∆(|u|m−1u) = 0 and ut + |u|m−1u = 0

A complete analysis of the Cauchy problem done byA. de Pablo, F. Quirós, AnaRodríguez, and J.L.V. A general fractional porous medium equation, Comm. PureApplied Math, 2012.Semigroup of weak energy solutions is constructed, smoothing effect works, Cα

regularity (if m is not near 0), infinite propagation for all m and s⇒ no compactsupport.

Asymptotic behavior is given by new self-similar solutions, see JLV, Barenblattsolutions and asymptotic behavior for a nonlinear fractional heat equation ofporous medium type. arXiv:1205.6332v1 [math.AP], May 2012. JEMS, to appear.

U(x , t) = t−nβF (x t−β)

with β = 1/(n(m − 1) + σ). F is only known explicitly for cases like m = 1, σ = 1.Its decay at infinity characterizes the fat tail of the fractional diffusion process.

Can you guess what is the decay ?

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 47 / 53

The second Fractional PME model. Short reportProblem is ut + (−∆)s/2(|u|m−1u) = 0 for x ∈ Rn, 0 < m <∞, 0 < s < 2,

with initial data in L1(Rn).The model represents another type of nonlinear interpolation, this time between

ut −∆(|u|m−1u) = 0 and ut + |u|m−1u = 0

A complete analysis of the Cauchy problem done byA. de Pablo, F. Quirós, AnaRodríguez, and J.L.V. A general fractional porous medium equation, Comm. PureApplied Math, 2012.Semigroup of weak energy solutions is constructed, smoothing effect works, Cα

regularity (if m is not near 0), infinite propagation for all m and s⇒ no compactsupport.

Asymptotic behavior is given by new self-similar solutions, see JLV, Barenblattsolutions and asymptotic behavior for a nonlinear fractional heat equation ofporous medium type. arXiv:1205.6332v1 [math.AP], May 2012. JEMS, to appear.

U(x , t) = t−nβF (x t−β)

with β = 1/(n(m − 1) + σ). F is only known explicitly for cases like m = 1, σ = 1.Its decay at infinity characterizes the fat tail of the fractional diffusion process.

Can you guess what is the decay ?

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 47 / 53

The second Fractional PME model. Short reportProblem is ut + (−∆)s/2(|u|m−1u) = 0 for x ∈ Rn, 0 < m <∞, 0 < s < 2,

with initial data in L1(Rn).The model represents another type of nonlinear interpolation, this time between

ut −∆(|u|m−1u) = 0 and ut + |u|m−1u = 0

A complete analysis of the Cauchy problem done byA. de Pablo, F. Quirós, AnaRodríguez, and J.L.V. A general fractional porous medium equation, Comm. PureApplied Math, 2012.Semigroup of weak energy solutions is constructed, smoothing effect works, Cα

regularity (if m is not near 0), infinite propagation for all m and s⇒ no compactsupport.

Asymptotic behavior is given by new self-similar solutions, see JLV, Barenblattsolutions and asymptotic behavior for a nonlinear fractional heat equation ofporous medium type. arXiv:1205.6332v1 [math.AP], May 2012. JEMS, to appear.

U(x , t) = t−nβF (x t−β)

with β = 1/(n(m − 1) + σ). F is only known explicitly for cases like m = 1, σ = 1.Its decay at infinity characterizes the fat tail of the fractional diffusion process.

Can you guess what is the decay ?

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 47 / 53

The second Fractional PME model. Short reportProblem is ut + (−∆)s/2(|u|m−1u) = 0 for x ∈ Rn, 0 < m <∞, 0 < s < 2,

with initial data in L1(Rn).The model represents another type of nonlinear interpolation, this time between

ut −∆(|u|m−1u) = 0 and ut + |u|m−1u = 0

A complete analysis of the Cauchy problem done byA. de Pablo, F. Quirós, AnaRodríguez, and J.L.V. A general fractional porous medium equation, Comm. PureApplied Math, 2012.Semigroup of weak energy solutions is constructed, smoothing effect works, Cα

regularity (if m is not near 0), infinite propagation for all m and s⇒ no compactsupport.

Asymptotic behavior is given by new self-similar solutions, see JLV, Barenblattsolutions and asymptotic behavior for a nonlinear fractional heat equation ofporous medium type. arXiv:1205.6332v1 [math.AP], May 2012. JEMS, to appear.

U(x , t) = t−nβF (x t−β)

with β = 1/(n(m − 1) + σ). F is only known explicitly for cases like m = 1, σ = 1.Its decay at infinity characterizes the fat tail of the fractional diffusion process.

Can you guess what is the decay ?

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 47 / 53

Existence of Self-similar solutions

Theorem For every choice of parameters s ∈ (0,1) andm > mc = max{(N − 2s)/N,0}, and every M > 0, equation (1) admits aunique fundamental solution; it is a nonnegative and continuous weaksolution for t > 0 and takes the initial data M δ(x) as a trace in the senseof Radon measures. Such solution has the self-similar form

(22) u∗M(x , t) = t−αf (|x | t−β)

for suitable α and β that can be calculated in terms of N and s in adimensional way, precisely

(23) α =N

N(m − 1) + 2s, β =

1N(m − 1) + 2s

.

The profile function f (r), r ≥ 0, is a bounded and Hölder continuousfunction, it is positive everywhere, it is monotone and goes to zero atinfinity in a precise way.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 48 / 53

Space Decay and Attractivity

Figure: Plot of decay rates of the self-similar profiles. Here, N = 3 and s = 0.8.

Theorem on asymptotic convergence. The above Barenblatt solutionsare the attractors for every other solution with integrable and nonnegativeinitial data .

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 49 / 53

Latest. Quantitative positivityThe reference here is : Quantitative Local and Global A Priori Estimatesfor Fractional Nonlinear Diffusion Equations, by Matteo Bonforte andJ.L.V., preprint 2012; Advances in Math, 2013.

Theorem on Local lower bounds Let u be a weak solution to theCauchy Problem, corresponding to u0 ∈ L1(Rd ). Assume thatd/(d − 2s) := m∗ < m < 1, so that ϑ := 1/[2s − d(1−m)] > 0. Thenthere exists a time

(24) t∗ := C∗ R1ϑ

0 ‖u0‖1−mL1(BR0 )

such that

(25) infx∈BR0/2

u(t , x) ≥ K1 R− 2s

1−m0 t

11−m if 0 ≤ t ≤ t∗ ,

and

(26) infx∈BR0/2

u(t , x) ≥ K2

‖u0‖2sϑL1(BR0 )

tdθ if t ≥ t∗ .

The positive constants C∗,K1,K2 depend only on m, s and d.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 50 / 53

Quantitative positivityHere is the theorem for m > 1:

Theorem on Local lower bounds Let u be a weak solution to theCauchy Problem , corresponding to u0 ∈ L1(Rd ). and let m > 1. We putϑ := 1/[2s − d(1−m)] > 0. Then there exists a time

(27) t∗ := C R1ϑ ‖u0‖−(m−1)

L1(BR)

such that for every t ≥ t∗ we have the lower bound

(28) infx∈BR/2

u(t , x) ≥ K‖u0‖2sϑ

L1(BR)

tdθ

valid for all R > 0. The positive constants C and K depend only on m, sand d ≥ 3, and not on R.Remark. This result can be written alternatively as saying that thereexists a universal constant C1 = C1(d , s,m) such for all solutions in theabove class we have

(29)∫

BR(0)u0(x) dx ≤ C1

(R1/ϑ(m−1)t−1/(m−1) + u(0, t)1/2sϑtd/2s

).

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 51 / 53

Semi-Harnack estimatesThe last estimate reminds of the Aronson-Caffarelli estimate for initialtrace-lower bounds in the standard PME. It is a form of lower Harnackinequality.We have been able to obtain such half Harnack for the standard FDE inthe papers* Global positivity estimates and Harnack inequalities for the fast diffusionequation, J. Funct. Anal. 240, no.2, (2006) 399–428. Qualitative (in thesense of the parabolic Harnack inequalities of Moser, 1963).

* Positivity, local smoothing, and Harnack inequalities for very fastdiffusion equations, Advances in Math. 223 (2010), 529–578.

Against some prejudice due to the nonlocal character of the diffusion, weare able to obtain them here for fractional PME/FDE using the techniqueof weighted integrals.Such "dimensional estimates" are popular in heat equations flows onmanifolds after the work of Li and Yau, and they can be applied to porousmedium on manifolds (Lu, Ni, Vazq., Villani), and should work forfractional diffusion on manifolds.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 52 / 53

Semi-Harnack estimatesThe last estimate reminds of the Aronson-Caffarelli estimate for initialtrace-lower bounds in the standard PME. It is a form of lower Harnackinequality.We have been able to obtain such half Harnack for the standard FDE inthe papers* Global positivity estimates and Harnack inequalities for the fast diffusionequation, J. Funct. Anal. 240, no.2, (2006) 399–428. Qualitative (in thesense of the parabolic Harnack inequalities of Moser, 1963).

* Positivity, local smoothing, and Harnack inequalities for very fastdiffusion equations, Advances in Math. 223 (2010), 529–578.

Against some prejudice due to the nonlocal character of the diffusion, weare able to obtain them here for fractional PME/FDE using the techniqueof weighted integrals.Such "dimensional estimates" are popular in heat equations flows onmanifolds after the work of Li and Yau, and they can be applied to porousmedium on manifolds (Lu, Ni, Vazq., Villani), and should work forfractional diffusion on manifolds.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 52 / 53

Semi-Harnack estimatesThe last estimate reminds of the Aronson-Caffarelli estimate for initialtrace-lower bounds in the standard PME. It is a form of lower Harnackinequality.We have been able to obtain such half Harnack for the standard FDE inthe papers* Global positivity estimates and Harnack inequalities for the fast diffusionequation, J. Funct. Anal. 240, no.2, (2006) 399–428. Qualitative (in thesense of the parabolic Harnack inequalities of Moser, 1963).

* Positivity, local smoothing, and Harnack inequalities for very fastdiffusion equations, Advances in Math. 223 (2010), 529–578.

Against some prejudice due to the nonlocal character of the diffusion, weare able to obtain them here for fractional PME/FDE using the techniqueof weighted integrals.Such "dimensional estimates" are popular in heat equations flows onmanifolds after the work of Li and Yau, and they can be applied to porousmedium on manifolds (Lu, Ni, Vazq., Villani), and should work forfractional diffusion on manifolds.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 52 / 53

Semi-Harnack estimatesThe last estimate reminds of the Aronson-Caffarelli estimate for initialtrace-lower bounds in the standard PME. It is a form of lower Harnackinequality.We have been able to obtain such half Harnack for the standard FDE inthe papers* Global positivity estimates and Harnack inequalities for the fast diffusionequation, J. Funct. Anal. 240, no.2, (2006) 399–428. Qualitative (in thesense of the parabolic Harnack inequalities of Moser, 1963).

* Positivity, local smoothing, and Harnack inequalities for very fastdiffusion equations, Advances in Math. 223 (2010), 529–578.

Against some prejudice due to the nonlocal character of the diffusion, weare able to obtain them here for fractional PME/FDE using the techniqueof weighted integrals.Such "dimensional estimates" are popular in heat equations flows onmanifolds after the work of Li and Yau, and they can be applied to porousmedium on manifolds (Lu, Ni, Vazq., Villani), and should work forfractional diffusion on manifolds.

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 52 / 53

There is much more going on on both models,but let us stop here.

End

Thank you♠ ♠ ♠

Gracias, eskarrik asko

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 53 / 53

There is much more going on on both models,but let us stop here.

End

Thank you♠ ♠ ♠

Gracias, eskarrik asko

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 53 / 53

There is much more going on on both models,but let us stop here.

End

Thank you♠ ♠ ♠

Gracias, eskarrik asko

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 53 / 53

There is much more going on on both models,but let us stop here.

End

Thank you♠ ♠ ♠

Gracias, eskarrik asko

Juan Luis Vázquez (Univ. Autónoma de Madrid) Nonlinear Fractional Diffusion 53 / 53