variational and viscosity solutions of the evolutionary...

Variational and viscosity solutions of the evolutionaryHamilton-Jacobi equation

Valentine Roos

École Normale Supérieure de Lyon

New trends in Hamilton-Jacobi equationsJuly 1-6, 2019

Fudan University, Shanghai, China

Hamilton-Jacobi equation: dynamical point of view

§ C2 Hamiltonian function H : pt, q, pq P Rˆ Rd ˆ Rd Ñ R.§ Evolutionary Hamilton-Jacobi equation:

Btupt, qq ` Hpt, q, Bqupt, qqq “ 0, (HJ)

coupled with a Lipschitz initial condition up0, ¨q “ u0.

§ Hamiltonian action along a C1 path γ “ pq, pq : RÑ Rd ˆ Rd

Atspγq “

sppτq ¨ 9qpτq ´ Hpτ, qpτq, ppτqqdτ.

§ Hamiltonian system"

9q “ BpHpt, q, pq9p “ ´BqHpt, q, pq

ù Hamiltonian flow φts .

Hamilton-Jacobi equation: dynamical point of view

§ C2 Hamiltonian function H : pt, q, pq P Rˆ Rd ˆ Rd Ñ R.§ Evolutionary Hamilton-Jacobi equation:

Btupt, qq ` Hpt, q, Bqupt, qqq “ 0, (HJ)

coupled with a Lipschitz initial condition up0, ¨q “ u0.§ Hamiltonian action along a C1 path γ “ pq, pq : RÑ Rd ˆ Rd

Atspγq “

sppτq ¨ 9qpτq ´ Hpτ, qpτq, ppτqqdτ.

§ Hamiltonian system"

9q “ BpHpt, q, pq9p “ ´BqHpt, q, pq

ù Hamiltonian flow φts .

Method of characteristics for classical solutions

Let u be a C2 solution on r0,T s ˆ Rd . Then for all 0 ď t ď Tp

tp “ dqu0u

p0, q0qpt, qq

tp “ dqutu

and upt, qq “ up0, q0q `At0 pφ

τ0pq0, dq0u0qq.

Method of characteristics for classical solutions

Consequence: no C2 solution even for smooth data.p

tp “ dqu0u

p0, q0qpt, qq

After the characteristics cross, the method defines a multivaluedsolution.

Multivalued solution

A multivalued solution is a multivalued function defined on r0,T s ˆRd

with multigraph matching at each time the wavefront F t0u0 Ă Rd`1

F t0u0 :“

q, u0pq0q `At0 pφ

τ0pq0, dq0u0qq

ˇ φt0pq0, dq0u0q “ pq, ‹q(

F t0u0 “ grputq

φt0pgrpdu0qq “ grpdutq

FT0 u0

qφT0 pgrpdu0qq

The convex case: Lax-Oleinik semigroup

§ H is Tonelli (i.e. C2, superlinear and strictly convex in p)ðñ Lpt, q, vq “ sup

p ¨ v ´ Hpt, q, pq is Tonelli.

§ t ÞÑ qptq solves the Euler-Lagrange equationðñ for pptq “ BvL pt, qptq, 9qptqq,

t ÞÑ pqptq, pptqq solves the Hamitonian system.

The viscosity solution is given by the Lax-Oleinik semigroup

T ts upqq :“ inf

c : rs, ts Ñ Rd

cptq “ q

upcpsqq `

sL pτ, cpτq, 9cpτqq dτ.

Tonelli: the infimum is attained by a C2 solution of the Euler-Lagrangeequation.Consequence: Existence of backward characteristics for the viscositysolution, i.e. the viscosity solution is part of the multivalued solution.

T ts upqq :“ inf

c : rs, ts Ñ Rd

cptq “ q

upcpsqq `

T ts upqq :“ inf

c : rs, ts Ñ Rd

cptq “ q

upcpsqq `

T ts upqq :“ inf

c : rs, ts Ñ Rd

cptq “ q

upcpsqq `

Nonconvex case: the viscosity solution is not necessarily partof the multivalued solution

Chenciner, Aspects géométriques de l’études des chocs dans les loisde conservation Problèmes d’évolution non linéaires, Séminaire deNice, 1975.

Viterbo, Solutions of Hamilton-Jacobi equations and symplecticgeometry Séminaire sur les Équations aux Dérivées Partielles,1994-1995.

Bernardi, Cardin, On C 0-variational solutions for Hamilton-Jacobiequations Discrete Contin. Dyn. Syst., 2011.

Wei, Viscosity solution of the Hamilton-Jacobi equation by alimiting minimax method Nonlinearity, 2014.

In these examples, the wavefront has a unique continuous section witha shock denying the entropy condition. Hence the viscosity solutioncannot be part of the multivalued solution.

Viscosity solutions: an axiomatic characterisation

Alvarez, Guichard, Lions, Morel, Axioms and fundamentalequations of image processing Arch. Rational Mech. Anal., 1993.

Viscosity operator§ If }d2Hpt, q, pq} ď C , and pV t

s qsďt : LippRdq Ñ LippRdq is s.t.1. Consistency: if u is a C2 solution of HJ, then V t

s us “ ut ,2. Monotonicity: u ď v ñ V t

s u ď V ts v for s ď t,

3. Additivity: for c P R, V ts pc ` uq “ c ` V t

s u,4. Regularity: pt, qq ÞÑ V t

τ upqq is locally Lipschitz and q ÞÑ V tτ upqq

Lipschitz uniformly w.r.t. t P rτ,T s,5. Markov: V t

s “ V tτ ˝ V

τs for s ď τ ď t.

then pt, qq ÞÑ V ts upqq solves (HJ) in the viscosity sense, with

initial condition u at time s.

s “ V tτ ˝ V

Bernard, The Lax-Oleinik semi-group: a Hamiltonian point of viewProc. Roy. Soc. Edinburgh, 2012.

s “ V tτ ˝ V

initial condition u at time s.§ If furthermore }dHpt, q, pq} ď C p1` }p}q, then there exists suchan operator and it is unique.

s “ V tτ ˝ V

initial condition u at time s.§ If furthermore }dHpt, q, pq} ď C p1` }p}q, then there exists suchan operator and it is unique.Ishii, Uniqueness of unbounded viscosity solution ofHamilton-Jacobi equations Indiana Univ. Math. J., 1984.

Variational operator: requirements & first consequences

Aim: select a continuous section in the wavefront associated with theCauchy problem.

Variational operatorIt is a family of operators pRt

s qsďt : LippRdq Ñ LippRdq s.t.1. Variational property: if u is C1, for all s ă t, the graph of Rt

s u iscontained in the wavefront F t

s u.( ùñ Consistency)

2. Monotonicity: u ď v ñ Rts u ď Rt

s v when s ď t,3. Additivity: if c P R, Rt

s pc ` uq “ c ` Rts u.

§ 2.+3. ùñ }Rts u ´ Rt

s v}8 ď }u ´ v}8.§ 1.+2. ùñ if u is semiconcave, a variational operator gives forsmall time the minimal section of the wavefront, which is C0.

§ u C2 ùñ pt, qq ÞÑ Rt0upqq solves (HJ) a.e. on r0,`8q ˆ Rd .

§ global notion (no definition of "to be a variational solution of (HJ)at pt, qq")

§ a priori no Markov property.

§ u Lipschitz???ùñ pt, qq ÞÑ Rt

0upqq solves (HJ) a.e.

Variational operator: Chaperon-Sikorav method

Generating family of the geometric solutionFind a C1 function S t

0u0 : Rd ˆ Rk Ñ R n.d.quadratic at infinity s.t.

φt0pgr du0q “

pq, BqSt0u0pq, χqq, BχS

t0u0pq, χq “ 0

F t0u0 “

pq,S t0u0pq, χqq, BχS

t0u0pq, χq “ 0

BtSt0u0pq, χq “ ´Hpt, q, BqS

t0u0pq, χqq if BχS t

0u0pq, χq “ 0.

Remark: if pt, qq ÞÑ χpt, qq is C1 and satisfies BχS t0u0pq, χpt, qqq “ 0,

then pt, qq ÞÑ S t0u0pq, χpt, qqq solves Hamilton-Jacobi.

Crit. points of S t0u0pq, ¨q

( 1:1ÐÑ

Ham. traj. withˇ

γp0q P grpdu0q,γptq P T ‹qRd

with associated critical value corresponding to the Hamiltonian actionof the trajectory plus the initial cost u0pq0q.

φt0pgr du0q “

t0u0pq, χq “ 0

F t0u0 “

t0u0pq, χq “ 0

0u0pq, χq “ 0.

( 1:1ÐÑ

Ham. traj. withˇ

φt0pgr du0q “

t0u0pq, χq “ 0

F t0u0 “

t0u0pq, χq “ 0

0u0pq, χq “ 0.

( 1:1ÐÑ

Ham. traj. withˇ

Critical value selector σ (ex: minmax selector)It selects for any smooth and n.d.quadratic at infinity function on Rk acritical value, with

§ σpc ` f q “ c ` σpf q for c P R,§ if φ is a C1-diffeo., σpf ˝ φq “ σpf q,§ σpf ‘ Qq “ σpf q,§ σpf q ď σpgq when f ď g and f ´ g Lipschitz,§ if µ ÞÑ fµ is such that critical points and values of fµ do notdepend on µ, µ ÞÑ σpfµq is constant.

PropositionIf H is a C2 Hamiltonian satisfying

}d2Hpt, q, pq} ď C , }dHpt, q, pq} ď C p1` }p}q,

then Rt0u0pqq :“ σ pS t

0u0pq, ¨qq defines a variational operator.

Critical value selector σ (ex: minmax selector)It selects for any smooth and n.d.quadratic at infinity function on Rk acritical value, with

§ σpc ` f q “ c ` σpf q for c P R,§ if φ is a C1-diffeo., σpf ˝ φq “ σpf q,§ σpf ‘ Qq “ σpf q,§ σpf q ď σpgq when f ď g and f ´ g Lipschitz,§ if µ ÞÑ fµ is such that critical points and values of fµ do notdepend on µ, µ ÞÑ σpfµq is constant.

PropositionIf H is a C2 Hamiltonian satisfying

}d2Hpt, q, pq} ď C , }dHpt, q, pq} ď C p1` }p}q,

then Rt0u0pqq :“ σ pS t

0u0pq, ¨qq defines a variational operator.

References on variational solutions

Chaperon, Une idée du type "géodésiques brisées" pour lessystèmes hamiltoniens C. R. Acad. Sci., 1984.

Sikorav, Sur les immersions lagrangiennes dans un fibré cotangentadmettant une phase génératrice globale C. R. Acad. Sci., 1986.

Viterbo, Symplectic topology and Hamilton-Jacobi equations,2006.

Graph selector with more sophisticated symplectic tools

Floer, Morse theory for Lagrangian intersections J. DifferentialGeom., 1988.

Oh, Symplectic topology as the geometry of action functional J.Differential Geom., 1997.

Guillermou, Quantization of conic Lagrangian submanifolds ofcotangent bundles arXiv:1212.5818, 2012.

Chaperon, Une idée du type "géodésiques brisées" pour lessystèmes hamiltoniens C. R. Acad. Sci., 1984.

Sikorav, Sur les immersions lagrangiennes dans un fibré cotangentadmettant une phase génératrice globale C. R. Acad. Sci., 1986.

Viterbo, Symplectic topology and Hamilton-Jacobi equations,2006.

Graph selector with more sophisticated symplectic tools

Floer, Morse theory for Lagrangian intersections J. DifferentialGeom., 1988.

Oh, Symplectic topology as the geometry of action functional J.Differential Geom., 1997.

Guillermou, Quantization of conic Lagrangian submanifolds ofcotangent bundles arXiv:1212.5818, 2012.

Cardin, Viterbo, Commuting Hamiltonians and Hamilton-Jacobimulti-time equations Duke Math. J., 2008.

§ existence of variational solutions for (HJ) multi-time equations forcommuting Hamiltonians, whereas

Davini, Zavidovique, On the (non) existence of viscositysolutions of multi-time Hamilton-Jacobi equations J.Differential Equations, 2015.

§ Appendix C: existence of variational solutions for (multi-time) HJcontact equation

§ Appendix B: extension to the non-compact case for Hamiltoniansflow with finite propagation speed

Properties of the variational operator

Let u and v be Lipschitz functions and H (or K ) be a Hamiltonian s.t.

}d2Hpt, q, pq} ď C , }dHpt, q, pq} ď C p1` }p}q.

§ Rts u is Lipschitz and 1` LippRt

s uq ď eCpt´sqp1` Lippuqq,§ Rt

s u ď Rts v if u ď v ,

§ }Rts u ´ Rt

s v}8 ď }u ´ v}8,§ Rt

s,H ď Rts,K if H ě K ,

§ }Rts,Hu ´ Rt

s,Ku}8 ď pt ´ sq}H ´ K}8.

Consequence: extension to Lipschitz initial conditions and Hamiltonians(C0-variational solutions).

Remark: the non-expansion and monotonicity properties can belocalized with explicit bounds on the Hamiltonian trajectories.

§ }Rts u ´ Rt

s v}8 ď }u ´ v}8,§ Rt

§ }Rts,Hu ´ Rt

§ }Rts u ´ Rt

s v}8 ď }u ´ v}8,§ Rt

§ }Rts,Hu ´ Rt

§ }Rts u ´ Rt

s v}8 ď }u ´ v}8,§ Rt

§ }Rts,Hu ´ Rt

Convergence of the iterated variational operator

Conjectured by Chaperon and Viterbo:

Theorem (Wei, R.)If s ď t1 ď ¨ ¨ ¨ ď tN ď t is a N-step subdivision with maximal steptending to 0,

RttN˝ RtN

tN´1˝ ¨ ¨ ¨ ˝ Rt1

s upqq ÑNÑ8

V ts upqq ploc. uniform in s ď t, qq.

RttN˝ RtN

tN´1˝ ¨ ¨ ¨ ˝ Rt1

s upqq ÑNÑ8

Souganidis, Approximation schemes for viscosity solutions ofHamilton-Jacobi equations J. Differential Equations, 1985.

Wei, Viscosity solution of the Hamilton-Jacobi equation by alimiting minimax method Nonlinearity, 2014.

Castillo Colmenares, Geometric and viscosity solutions for theCauchy Problem of first Order arXiv:1611.10293, 2017.

Roos, Variational and viscosity operators for the evolutionaryHamilton–Jacobi equation Commun. Contemp. Math., 2017.

RttN˝ RtN

tN´1˝ ¨ ¨ ¨ ˝ Rt1

s upqq ÑNÑ8

Sketch of the proof:§ Local semi-group type Lipschitz estimates w.r.t. s, t, u, q,

§ Arzela-Ascoli extraction for ps, t, u, qq in some compact subsets,§ Diagonal extraction,§ Check that any accumulation point satisfies the 5 axioms of the(unique) viscosity operator.

RttN˝ RtN

tN´1˝ ¨ ¨ ¨ ˝ Rt1

s upqq ÑNÑ8

Sketch of the proof:§ Local semi-group type Lipschitz estimates w.r.t. s, t, u, q,§ Arzela-Ascoli extraction for ps, t, u, qq in some compact subsets,

§ Diagonal extraction,§ Check that any accumulation point satisfies the 5 axioms of the(unique) viscosity operator.

RttN˝ RtN

tN´1˝ ¨ ¨ ¨ ˝ Rt1

s upqq ÑNÑ8

Sketch of the proof:§ Local semi-group type Lipschitz estimates w.r.t. s, t, u, q,§ Arzela-Ascoli extraction for ps, t, u, qq in some compact subsets,§ Diagonal extraction,

§ Check that any accumulation point satisfies the 5 axioms of the(unique) viscosity operator.

RttN˝ RtN

tN´1˝ ¨ ¨ ¨ ˝ Rt1

s upqq ÑNÑ8

Sketch of the proof:§ Local semi-group type Lipschitz estimates w.r.t. s, t, u, q,§ Arzela-Ascoli extraction for ps, t, u, qq in some compact subsets,§ Diagonal extraction,§ Check that any accumulation point satisfies the 5 axioms of the(unique) viscosity operator.

Variational = viscosity?

Joukovskaia ’91: If H Tonelli, the Lax-Oleinik semigroup coincideswith the variational solution obtained by the Chaperon-Sikorav method.

Consequence: p ÞÑ Hpt, q, pq convex (concave) ùñ V “ R .

Lax-Hopf formula: Hppq, u convex (concave) ùñ V ts u “ Rt

Bernardi-Cardin ’09: the solutions coincide for convex-concaveHamiltonians if the initial condition has separated variables, i.e.Hpt, q, pq “ H1pt, q1, p1q ` H2pt, q2, p2q and upqq “ u1pq1q ` u2pq2qwith p1 ÞÑ Hpt, q1, p1q convex and p2 ÞÑ Hpt, q2, p2q concave.

Ñ does there exists u such that Rt0u ‰ V t

0u for Hpp1, p2q “ p21 ´ p2

Theorem (R. ’18)If Hppq is neither convex nor concave, there exists a smooth u suchthat Rt

0u ‰ V t0u.

For this initial condition, the graph of the viscosity solution is notcontained in the wavefront.

Joukovskaia ’91: If H Tonelli, the Lax-Oleinik semigroup coincideswith the variational solution obtained by the Chaperon-Sikorav method.Consequence: p ÞÑ Hpt, q, pq convex (concave) ùñ V “ R .

0u ‰ V t0u.

Joukovskaia ’91: If H Tonelli, the Lax-Oleinik semigroup coincideswith the variational solution obtained by the Chaperon-Sikorav method.Consequence: p ÞÑ Hpt, q, pq convex (concave) ðñ

HppqV “ R .

0u ‰ V t0u.

Semi-Lagrangian numerical scheme for nonconvexHamiltonians (j.w. in progress with H. Hivert)

Input:§ An integrable C2 1D Hamiltonian Hppq, entered as a function.

ù φt0pq, pq “`

q ` tH 1ppq, p˘

, At0pγq “ t

pH 1ppq ´ Hppq˘

§ Spatial step ∆x , time step ∆t.§ A semiconcave initial condition u with a finite number of shocks,processed as a list of 3-tuples pxi , ui , pi q ("Clarke 1-jet").

One step:§ ∆t Hamiltonian transform on each 3-tuple:pxi , ui , pi q Ñ pxi `∆tH 1ppi q, ui `∆t ppiH

1ppi q ´ Hppi qq , pi q.§ "Follow" and select the minimal section.§ Insert points of shocks with Clarke derivatives in the selected list.

Then iterate.

ù φt0pq, pq “`

q ` tH 1ppq, p˘

, At0pγq “ t

pH 1ppq ´ Hppq˘

Then iterate.

ù φt0pq, pq “`

q ` tH 1ppq, p˘

, At0pγq “ t

pH 1ppq ´ Hppq˘

Then iterate.

Semi-Lagrangian scheme: comparison with Lax-Friedrichs

-0.2 -0.1 0 0.1 0.2 0.3 0.4

Semi-Lagrangian scheme, x = 0.01, t = 0.025

Lax-Friedrichs scheme, = 0.2, x = 0.01, t = 0.0005

Lax-Friedrichs

Semi-Lagrangian

Initial data

Hppq “ p4 ´ p2 u0pqq “

q if q ă 0´3

4q ` q2{2 if q ą 0

Semi-Lagrangian scheme: precision test

-20 -10 0 10 20 30 40 50

T = 50, t = 0.05, x = 0.01

Initial data

Explicit variational solution

Explicit viscosity solution

Approximated variational solution

Approximated viscosity solution

37 38 39 40 41 42 43 44 45

Hppq “

p ` p2 if p ă 0p ´ p2 if p ą δ

u0pqq “

q if q ă 0´q ` q2{2 if q ą 0

Explicit example where the solutions differ

u0pqq “

q if q ă 0´q ` q2{2 if q ą 0

Hppq “

with H C2 on R and such that H2 cancels exactly once in p0, δq.

Characteristics for the variational solution

In the blue domain, the variational solution coincides with the classicalsolution fr for Hppq “ p ` p2 with smooth initial conditionf pqq “ ´q ` q2{2.

u0pqq “

q if q ă 0´q ` q2{2 if q ą 0

Hppq “

Oleinik’s guessIf xptq denotes the position of the presumed viscosity shock, then

x 1ptq “Hpp`ptqq ´ Hpp´ptqq

p`ptq ´ p´ptq“ H 1pp`ptqq

where p´ptq “ Bx fr pt, xptqq and p`ptq “ ψpp´ptqq.

ψppqp

u0pqq “

q if q ă 0´q ` q2{2 if q ą 0

Hppq “

Oleinik’s guessIf xptq denotes the position of the presumed viscosity shock, then

x 1ptq “Hpp`ptqq ´ Hpp´ptqq

p`ptq ´ p´ptq“ H 1pp`ptqq

where p´ptq “ Bx fr pt, xptqq and p`ptq “ ψpp´ptqq.

ψppqp

Characteristics for the viscosity solution

Characteristics for the viscosity solution are tangentially issued from theshock.

Characteristics for the variational solution

Characteristics for the viscosity solution are tangentially issued from theshock.

-20 -10 0 10 20 30 40 50

T = 50, t = 0.05, x = 0.01

Initial data

Explicit variational solution

Explicit viscosity solution

Approximated variational solution

Approximated viscosity solution

37 38 39 40 41 42 43 44 45

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