vii wpde vii workshop on partial differential...
TRANSCRIPT
Rio de Janeiro - Brazil, August 26-29, 2008
2008
VII WPDE
VII Workshop on
Partial Differential Equations
VII Workshop on Partial Differential Equations
August 26 - 29, 2008
Rio de Janeiro, Brazil
We hope that this new edition of the event will fulfill our expectations. Namely, to congregateresearchers in the area of Partial Differential Equations viewing the presentation and discussionsof their domains of interests, as well as, presenting to the graduate students some of the recentlines of research developed in the Brazilian and foreign institutions.
We thank the support from the Brazilian agencies Fundacao de Amparo a Pesquisa doEstado do Rio de Janeiro (FAPERJ), Coordenacao de Aperfeicoamento de Pessoal de NıvelSuperior (CAPES), Conselho Nacional de Desenvolvimento Cientıfico Tecnologico (CNPq) andthe support from Universidade Federal do Rio de Janeiro (UFRJ), Banco do Brasil, LaboratorioNacional de Computacao Cientıfica (LNCC), Fundacao Universitaria Jose Bonifacio (FUJB),Ministerio da Ciencia e Tecnologia and Instituto do Milenio (IM-AGIMB).
Scientific Committee
Gustavo Ponce (USA)Felipe Linares (Brazil)Mauro Fabrizio (Italy)
Roberto Triggiani (USA)Djairo Figueiredo (Brazil)
Vilmos Komornik (France)Mauricio Sepulveda (Chile)Reinhard Racke (Germany)
Organizing Committee
Wladimir Neves (UFRJ, Brazil)Ademir F. Pazoto (UFRJ, Brazil)Marcelo Cavalcanti (UEM, Brazil)
Mauro de Lima Santos (UFPA, Brazil)Santina de Fatima Arantes (LNCC, Brazil)
Jaime E. Munoz Rivera (LNCC-UFRJ, Brazil)
Latin American Committee
Jaime Ortega (Chile)Roxana Lopez (Peru)
Octavio Vera Villagran (Chile)Gustavo Perla Menzala (Brazil)
ii
Rio de Janeiro - Brazil, August 26-29, 2008
A B S T R A C T S
VII WPDE
VII Workshop on
Partial Differential Equations
iii
Contents
Luis Aguirre Castillo
A general reduction principle of stability and its application to systems ofpartial differential equations. 1
Fatiha Alabau-Boussouira
Energy decay rates for evolution equations with boundary memory feedbacks. 1
Djairo Figueiredo
Nonlinear elliptic systems with exponential non-linearities. 1
Dilberto da Silva Almeida Junior
Asymptotic behavior in thermoelasticity for Timoshenko beam. 2
Felipe Linares
On Zakharov-Rubenchick system. 2
Margareth da Silva Alves
Exponential decay in a thermoelastic mixture of solids. 3
Piero D’Ancona
Almost optimal local well posedness for the Mazwell-Dirac system. 3
Santina de Fatima Arantes
Numerical analysis of the exponential and polynomial decay for an hyperbolicparabolic system. 4
Fabio Souto Azevedo
Asymptotic analysis of radiative flows. 5
Pedro D. Damazio
The equations of motion for the Oldroyd model of viscoelastic fluids. 6
Andres Avila
Numerical solutions of a singularly nonlinear Schrodinger equation. 7
Victor Emilio Carrera Barrantes
Exponential decay of wave equation with a viscoelastic boundary conditionand source term. 8
Valeria N. Domingos Cavalcanti
Uniform stabilization of the wave equation on compact manifolds and locallydistributed damping - A sharp result. 8
Ruy Coimbra Charao
The semilinear system of elastic waves in Rn with nonabsorption and a cri-
tical damping potential. 9
iv
Patricio Cumsille Atala
Detecting a rigid body moving inside of an ideal fluid via boundary measure-ments. 10
Emmanuelle Crepeau
New results on the exact controllability of the Korteweg-de Vries equation. 11
Uri Ascher
Surprising computations. 12
Mauro Fabrizio
Minimal state in viscoelasticity and applications to PDEs. 12
Giovany Malcher Figueiredo
On a p-Kirchhoff equation via Krasnoselskii’s Genus. 13
Gustavo Perla Menzala
Anisotropic Maxwell equations: decay properties. 13
Pedro Gamboa
Polynomial decay to thermoelastic plates with memory. 14
Maurizio Grasselli
Hyperbolic Cahn-Hilliard equations. 15
Aıssa Guesmia
On convexity and decay rates of Timoshenko-type systems. 16
Bernadette Miara
Strongly heteregeneous piezoelectric composites. 16
Thierry Horsin
Controllability of some partial differential equations in Lagrangian descri-ption. 17
Masaki Kurokiba
Stationary solutions to the drift-diffusion model in the whole space Rn. 17
Irena Lasiecka
Finite dimensional and smooth attractors arising in wave dynamics with ge-ometrically restricted dissipation and nonlinear critical exponents. 18
Sergey Lurie
Variational model of nonholonomic 4D-media and its application to the cou-ple problem of the thermo- and visco-elasticity. 19
Salim A. Messaoudi
A stability result in a memory-type Timoshenko system. 20
v
Cleverson R. da Luz
Decay properties of an anisotropic coupled model in electromagnetism/elas-ticity. 21
Sorin Micu
Numerical approximation of the boundary control of the wave equation. 22
Gustavo Ponce
Persistence properties and unique continuation results for canonical disper-sive equations. 23
Reinhard Racke
Asymptotic behavior in nonlinear hyperbolic thermoelasticity. 23
Rubia G. Nascimento
Existence and multiplicity results for elliptic equation of p-Kirchhoff typewith discontinuous nonlinearities. 24
Vilmos Komornik
Explicit multidimensional Ingham-Beurling type estimates. 24
Maria Grazia Naso
On the vibrations of an extensible thermoelastic beam. 25
Luz de Teresa
Controllability results for 1-d coupled degenerate parabolic equations. 25
Andre Novotny
Topological derivative in multi-scale linear elasticity models. 26
Michael Renardy
Short wave stability for inviscid shear flow. 28
Lionel Rosier
Exact controllability of the nonlinear Schrodinger equation. 28
Octavio Vera Villagran
Gain of regularity for the KP-I equation. 28
Paulo X. Pamplona
Stability of linear porous-thermo-elasticity system. 29
Hugo D. Fernandez Sare
On the stability of Mindlin-Timoshenko plates. 30
Roberto Triggiani
Uniform stabilization of a coupled parabolic-hyperbolic fluid-structure intera-ction model with dissipation at the interface. 30
vi
Eugenio Cabanillas Lapa
Uniform decay for the wave equation of Kirchhoff type with viscoelastic bo-undary condition and source term. 31
vii
A general reduction principle of stability and itsapplication to systems of PDE’s
Luis Aguirre Castillo
Universidad Autonoma Metropolitana-IztapalapaDepto. de Matematicas
San Rafael Atlixco 186, Vicentina, C.P. 09340, Mexico, D.F., MexicoE-mail: [email protected]
A general principle is announced which permits the reduction of the problem of stabilityof a composed system to the corresponding problem for its subsystems. This general result isapplied to systems of partial differential equations with emphasis on proving global asymptoticstability, where conventional techniques yield only local results.
Energy decay rates for evolution equations
with boundary memory feedbacks
Fatiha Alabau-Boussouira
Universite Paul Verlaine-MetzLaboratoire de Mathematiques et Applications, UMR 7122Batiment A, Ile du Saulcy F-57045 Metz Cedex 1, France
E-mail: [email protected]
We study stabilization properties of visco-elastic materials. For such materials, the feedbackoperator acts as a convolution operator in time of space second order derivatives of the solutionwith a time-dependent kernel. We present some of the recent advances for memory dampingson the boundary.
Nonlinear elliptic systems with exponentialnon-linearities
Djairo Figueiredo
Universidade Estadual de Campinas - UNICAMPIMECC
13081-970 Campinas - SP, BrasilE-mail: [email protected]
We’ll discuss elliptic systems in two dimensions with non-linearity criticism of the typeTrudinger-Moser.
1
Asymptotic behavior in thermoelasticity for Timoshenko beam
Dilberto da Silva Almeida Junior
Universidade Federal do Para, UFPAFaculdade de Matematica
Rua Augusto Corra Street, 01, 66075-110, Belem - PA, BrasilE-mail: [email protected]
For some dissipative systems of Timoshenko, the property of exponential decay is constrainedto a particular relationship between the coefficients of the system ([1],[2]).
In this work, we study the mathematical issues of a Timoshenko system in linear thermoe-lasticity, whose thermal coupling was only realized in the shear S = kGA(ϕx + ψ)[3], and itresults in a system of partial differential equations where the effect of the coupling exists in allthe equations. The property of exponential decay occurs if, and only if, the speeds associatesto the propagations of the waves are equal; this is the property we prove numerically using anumerical scheme in the context of the finite-difference space semidiscretization.
References
[1] A. Soufyane: Stabilisation de la poutre de Timoshenko. C. R. Acad. Sci. Paris, Ser. I, 328(1999), 731–734.
[2] Reinhard Racke and J. E. Muoz Rivera: Mildly dissipative nonlinear Timoshenko systems— global existence and exponential stability. Journal of Mathematical Analysis and Appli-cations. Volume 276, Numero 1, 248–278 (2002).
[3] S. P. Timoshenko: On the correction for shear of the differential equation for transversevibrations of prismatic bars. Philosophical Magazine, 6(1921), 744–746.
∗ Joint work with Jaime E. Munoz Rivera.
On Zakharov-Rubenchick system
Felipe Linares
IMPAEstrada Dona Castorina, 110
22460-320 Rio de Janeiro - RJ, BrasilE-mail: [email protected]
In this talk we will discuss recent results regarding the one-dimensional initial value prob-lem associated to the Zakharov-Rubenchick system. The results include local and global well-posedness for rough data. This is a joint work with Carlos Matheus. We will also mentionsome problems concerning the 2D and 3D cases.
2
Exponential decay in a thermoelastic mixtureof solids
Margareth da Silva Alves
Universidade Federal de Vicosa, UFVDepartamento de Matematica36570-000 Vicosa - MG, Brasil
E-mail: [email protected]
In this work we investigate the asymptotic behaviour of solutions to the initial boundary valueproblem for a one-dimensional mixture of thermoelastic solids. Our main result is to establisha necessary and sufficient condition over the coefficients of the system to get the exponentialstability of the corresponding semigroup. We also prove the impossibility of time localization ofsolutions.
∗ Joint work with Jaime E. Munoz Rivera and R. Quintanilla.
Almost optimal local well posedness for theMazwell-Dirac system
Piero D’Ancona
Universita di Roma ”La Sapienza”Dipartimento di Matematica
Piazzale Aldo Moro, 2 - I-00185, ItalyE-mail: [email protected]
In a joint work with Damiano Foschi and Sigmund Selberg, we recently closed a long-standingconjecture concerning the MD system, namely the local well posedness in Hs × Hs−1/2 (spinorfield x electromagnetic field) for all s > 0. Recall that L2 × H−1/2 is the scale invariant spacefor the system. Main features of this result are:
1) we work in the Lorentz gauge, which was previously considered a ”bad” gauge for MD andMKG but in fact turned out to be the correct one;
2) previous results on MD did not uncover the full null structure of the system. To get it, it isnecessary to keep into account the full algebraic structure of the system and embed it in suitabletri- and quadrilinear estimates;
3) we follow the standard iteration method in suitable wave-Sobolev spaces, however new refinedestimates involving angular decompositions are necessary to close the iteration.
3
Numerical analysis of the exponentialand polynomial decay for an hyperbolic
parabolic system
Santina de Fatima Arantes
Laboratorio Nacional de Computacao Cientıfica - LNCCAvenida Getulio Vargas, 333 - Quitandinha
25651-075 Petropolis - RJ, BrasilE-mail: [email protected]
We consider the one dimensional hyperbolic parabolic system related to thermoelastic modelsdepending on a parameter α in a dimensionless form
utt + ρuxxxx − µBαθ = 0 in (0, L) × (0, T )
θt − κθxx + σBαut = 0 in (0, L) × (0, T )
with the initial and boundary conditions
u(x, 0) = u0(x), ut(x, 0) = u1(x), θ(x, 0) = θ0(x) in (0, L)
u(0, t) = u(L, t) = uxx(0, t) = uxx(L, t) = θ(0, t) = θ(L, t) = 0 in (0, T )
where T > 0 fixed is given and when α = 0, Bαw = w and when α = 1, Bαw = −wxx.
We show that for 0 ≤ α < 1/2, the system is not exponentially stable, but the solution decayspolynomially to zero with rates of decay that can be improved by improving the regularity ofthe initial data, and for 1/2 ≤ α ≤ 1, the system is exponentially stable. Finally, we getanalyticity if and only if α = 1. We present the graphics of energy E(t) and of solution u(x, t)of problem, when α = 0 and α = 1. We use semi-discrete finite element method to get ournumerical solution. To the computational results, we use the implemented code in Fortran 90.The graphics we did using Maple.
References
[1] Z. Y. Liu and M. Renardy, A Note on the Equations of a Thermoelastic Plate, Appl. Math.Lett. 8(3) (1995), pp. 1-6.
[2] Z. Liu and S. Zheng, Exponential stability of the Kirchhoff plate with thermal or viscoelasticdamping, Quart. Appl. Math. 53(1997), pp. 551-564.
[3] T. J. R. Hughes, The finite Element Method: Linear Static and Dynamic Finite ElementAnalysis, Dover Publications, INC. Mineola, New York, 2000.
∗ Joint work with Margareth da Silva Alves and Jaime E. Munoz Rivera.
4
Asymptotic analysis of radiative flows
Fabio Souto Azevedo
Universidade Federal do Rio Grande do Sul, UFRGSInstituto de Matematica
91509-900 Porto Alegre - RS, BrasilE-mail: [email protected]
The evolution of the temperature θ of fluid in the presence of radiative heat transfer, thermalconduction and convection can be modelled by the following simplified equation, see [1]:
Ltθ :=∂θ
∂t+ v · ∇θ −∇ · (k0∇θ) =
κ′
ε2
∫
S2
∫∞
ν0
(I − B(ν, θ)) dν, x ∈ D
ε∂θ
∂η= h(θb − θ) + απ
∫ ν0
0[B(ν, θb) − B(ν, θ)] dν, x ∈ ∂D
The radiative intensity is governed by the steady-state Boltzmann transport equation in anisotropic medium with semitransparent boundary. The coupled system is nonlinear and definedin a six-dimensional space ((x, t) ∈ R
3 × (0,∞), Ω ∈ S2), which often results in high costnumerical calculations. A less expensive approach consists in approximating the total fluxintroducing a simpler equation for I, which does not depend on the direction Ω. Indeed, variousefforts have been made in the literature to establish such approximations. For instance, [2]presents different orders approximation when σ = 0. According to this paper a second orderapproximation as ε → 0 can be achieved by I ≃ 4πΦ, where Φ solves:
(− ε2
κ′λ′ + 1
)Φ = B(ν, θ), x ∈ D
εb∂
∂ηΦ + Φ = B(ν, θb), x ∈ ∂D
Our main interest is to carry out an asymptotic expansion of the problem. Via the method ofsingular perturbations, we introduce stretched variables near the boundary and near the initialdata, indeed, we are able to establish that:
θa = θi + (θo + θτ ) ξ(x)
=(θ(0) + εθ
(1)i + ε2θ
(2)i
)+
(εθ(1)
o + ε2θ(2)o + εθ(1)
τ + ε2θ(2)τ
)ξ(x)
where 0 ≤ ξ(x) ≤ 1 is a smooth cutoff function vanish far from ∂D. The subindex i refersto the inner approximation while o and τ refers to the approximation near ∂D and t = 0
References
[1] Frank, M.; Seaid, M.; Klar, A.; Rinnam, R.; Thommes, G. (2004), A comparison of approx-imate models for radiation in gas turbines, Progress in Computational Fluid Dynamics, 4,191-197.
[2] Larsen, E. W.; Thommes, G.; Klas, A.; Seaid, M.; Gotz T. (2002), Simplified pn approxi-mation to the equations of radiative heat transfer on glass, J. Comp. Phys., 183, 652-675.
[3] Pao, C.; (1992) Nonlinear Parabolic and Elliptic Equations, New York: Plenum Press.
5
[4] Thompson, M.; Segatto, C.; Vilhena, M.T.; (2004) Existence Theory for the Solution ofa Stationary Nonlinear Conductive-Radiative Heat Transfer Problem in Three DimensionTransport Theory and Statistical Physics 33: 563-576.
The equations of motion for the
Oldroyd model of viscoelastic fluids
Pedro D. Damazio
Universidade Federal do Parana - UFPRDepartamento de Matematica
81531-990 Curitiba - PR, BrasilEmail: [email protected]
The equations of motion of the Oldroyd fluids can be described most naturally by the fol-lowing integro-differential equation
∂u
∂t+ u · ∇u − µ∆u −
∫ t
0β(t − τ)∆u(x, τ) dτ + ∇p = f(x, t), x ∈ Ω, t > 0,
and incompressibility condition
∇ · u = 0, x ∈ Ω, t > 0,
with initial and boundary conditions
u(x, 0) = u0 in Ω, u = 0, on ∂Ω, t ≥ 0.
Here, Ω is a bounded domain in two dimensional Euclidean space R2 with boundary ∂Ω, µ =2κλ−1 > 0 and the kernel β(t) = γ exp(−δt), where γ = λ−1(ν − κλ−1) and δ = λ−1.
In this work we show some a priori bounds for the solutions of such problem, under realis-tically assumed conditions on the initial data. Besides, we have proposed a time discretizationscheme based on linearized modification of the backward Euler method and analyzed the errorestimates for the approximations.
∗ Joint work with A. K. Pani and J. Y. Yuan.
6
Numerical solutions of a singularly nonlinearSchrodinger equation
Andres Avila
Universidad de La FronteraDepartamento de Ingenierıa Matematica
Temuco, ChileEmail: [email protected]
Our model is given by the equation
−ǫ2∆u + V (x)u = K(x)|u|p−1u in Ω,
with Dirichlet condition, 1 < p < N+2N−2 , V and K stricty positive and bounded functions. We
look for positive solutions with minimun energy, that is, such that
I(u) := ǫ21
2
∫
Ω‖∇u(x)‖2dx +
∫
ΩV (x)u2dx − 1
p + 1
∫
ΩK(x)|u|p+1dx
reaches its global minimum, which we will call groud state. For K ≡ 1, Rabinowitz [2] showedthat for ǫ small, the solutions concentrates about the local minimum of V . Moreover, Wang andZeng [3] proved, for K 6= 1, that both functions compete for the location of the maximum of theground state.
In this talk, we will modify the SIA algortihm developed by Chen et als. [1] to studythe location of the maximum. In particular, we will solve the linearization by two methodsspecially developed for singular lineal problems: multigrid approach and hp-FEM method. Weare interested when ǫ → 0 how the ǫ parameter affects the parameters of the methods. In thefirst case, we will focus on the structure of the mesh (V -cycles or W -cycles) and in the secondcase the relation with the step h and the polynomial degree p. We will show some numericalresults and some estimates of the numerical errors.
References
[1] G. Chen, J. Zhou and W.M. Ni, Algorithms and Visualization for Solutions of NonlinearElliptic Equations, Int. Jour. Bifurcation and Chaos 10 (2000), no. 7, 1565–1612.
[2] P. Rabinowitz, On a class of nonlinear Schrdinger equations, Z. angew Math Phys, Vol. 43(1992), 270– 291.
[3] X. Wang & B. Zeng, On concentration of positive bound states of nonlinear Schrodingerequations with competing potential functions, SIAM J. Math. Anal. Vol 28 (1997), 633-655.
∗ Joint work with Gino Montecinos and Miguel Navarro.
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Exponential decay of wave equation with aviscoelastic boundary condition and source term
Victor Emilio Carrera Barrantes
Universidad Nacional del CallaoAv. Juan Pablo II S/N - Bellavista, Callao - Peru
In this paper we concerned with the stability of solutions for the wave equa- tion with a vis-coelastic Boundary condition and source term by using the po- tential well method, the multipliertechnique and unique continuation theorem for the wave equation with variable coefficient.
∗ Joint work with F. Leon and J. Bernui.
Uniform stabilization of the wave equation
on compact manifolds and locally
distributed damping - A sharp result
Valeria N. Domingos Cavalcanti
Universidade Estadual de Maringa - UEMDepartamento de Matematica
87020-900 Maringa - PR, BrasilE-mail: [email protected]
Let (M,g) be an n-dimensional (n ≥ 2) compact Riemannian manifold with or withoutboundary where g denotes a Riemannian metric of class C∞. This paper is concerned with thestudy of the wave equation on (M,g) with locally distributed damping, described by
utt − ∆gu + a(x) g(ut) = 0 on M × ]0,∞[ , u = 0 on ∂M×]0,∞[,
where ∂M represents the boundary of M and the last condition is dropped when M is boundary-less. First of all we prove that for ǫ > 0, there exist an open subset V ⊂ M and a smooth functionf : M → R such that meas(V ) ≥ meas(M) − ǫ, Hessf ≈ g on V and infx∈V |∇f(x)| > 0.
Finally we prove that if a(x) ≥ a0 > 0 on an open subset M∗ ⊂ M that contains M\Vand if g is a monotonic increasing function such that k|s| ≤ |g(s)| ≤ K|s| for all |s| ≥ 1, thenuniform and optimal decay rates of the energy hold.
∗ Joint work with M. M. Cavalcanti, R. Fukuoka and J. A. Soriano.
8
The semilinear system of elastic waves in Rn
with nonabsorption and a critical
damping potential
Ruy Coimbra Charao
Universidade Federal de Santa Catarina, UFSCDepartamento de Matematica
88040-900 Florianopolis - SC, BrasilE-mail: [email protected]
We study the following initial value problem associated to the system of elasticity with anonabsoption nonlinear term and a critical potential type of damping
utt − a2u − (b2 − a2)∇divu + V (x)ut = |u|p−1u , (t, x) ∈ R
+ × Rn
u(0, x) = ǫu0(x), ut(0, x) = ǫu1(x), x ∈ Rn (1)
with ǫ > 0 and the power type of nonlinearity p satisfying
1 + 4n−1 < p < +∞ (n = 2)
1 + 4n−1 < p <
n + 2
n − 2(n ≥ 3) (2)
and the Lame coefficients a > 0, b > 0 satisfy 0 < a2 < b2. We suppose that the potentialV (x) ∈ L∞(Rn) in the equation (1) satisfies: there exists a constant C0 > 0 such that
V (x) ≥ C0
1 + |x| . (3)
Associated with the power nonlinearity p we consider the number F (n, p) given by
F (n, p) =n(p − 1) − p − 3
p − 1, n ≥ 2. (4)
Then, the hypotheses (2) on p imply that
0 < F (n, p) < 1.
According to the global existence and the asymptotic behavior to the problem (1) we havethe following result
Theorem 0.1 Let V ∈ C1(Rn) satisfying (3) and initial data (u0 , u1) ∈ (H1)n × (L2)n withcompact support in BR(0) , R > 0. Assume the power nonlinearity p also satisfies the additionalcondition
1 − C0
b< F (n, p) , if 0 < C0 ≤ b,
holds. The number b is the finite speed of propagation and C0 is defined in 3. Consider anumber δ satisfying
1 − C0
b < δ < F (n, p) if 0 < C0 ≤ b ,0 ≤ δ < F (n, p) if C0 > b.
(5)
9
where F (n, p) is defined in (4.) Then, there exists ǫ1 > 0 such that for ǫ ∈ (0, ǫ1) the problem(1) has a unique global solution
u ∈ C([0,∞), (H1(Rn))n) ∩ C1([0,∞), (L2(Rn))n)
satisfyingEu(t) ≤ C(1 + t)−(1−δ) , t > 0
with C > 0 constant depending on δ, R, p, ||V ||∞, u0 and u1.
where Eu(t) is defined by
Eu(t) =1
2
∫
Rn
[|ut|2 + a2|∇u|2 + (b2 − a2)(divu)2
]dx.
∗ Joint work with Ryo Ikehata.
Detecting a rigid body moving inside of an
ideal fluid via boundary measurements
Patricio Cumsille Atala
Universidad del Bıo-BıoDepartamento de Ciencias Basicas
Casilla 447, Chillan, ChileEmail: [email protected]
This conference deals with the following inverse problem: the detection of a rigid bodymoving inside of an ideal fluid. We wish to determine the location of the rigid body (i.e. itsinstantaneous position and velocity) at a certain time from the measurement of the velocity ofthe fluid on one part of the external boundary of the domain containing the fluid-solid system.Since there is no a unique continuation property for perfect fluids, we assume that the fluidis flowing in potential regime, i.e. its velocity is the gradient of a scalar function. In thatcase, we show that when the obstacle is a ball, its position and velocity of its center of masscan be identified from a single boundary measurement. Linear stability estimates will also beestablished by using shape differentiation techniques. Finally, based upon shape optimizationtechniques, we are studying the numerical reconstruction of the position and of the velocity of arigid disk by means of the minimization of a certain cost functional that takes into account thedynamics of the fluid-solid system, and whose optimal solution is exactly the position and thevelocity of the rigid disk. We will show some numerical experiences performing this optimizationprocess.
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New results on the exact controllabilityof the Korteweg-de Vries equation
Emmanuelle Crepeau
Universite de Versailles Saint-Quentin en Yvelines78035 Versailles, France
E-mail: [email protected]
Let T > 0, L > 0 be fixed. Let us consider the following control system for the Korteweg-deVries (KdV) equation
yt + yx + yxxx + yyx = 0, x ∈ (0, L), t ∈ (0, T ),y(t, 0) = 0, y(t, L) = 0, yx(t, L) = u(t), t ∈ (0, T ),
(1)
where the state is y(t, ·) : [0, L] → R and the control is u(t) ∈ R. The local controllability ofthis problem has already been studied by L. Rosier in [2]. He proved that if L is not in a critical
set of lengths, L /∈ N :=
2π
√k2+kl+l2
3 ; k, l ∈ N ∗
, then the linearized problem around the
origin is exactly controllable and the nonlinear problem is locally exactly controllable. Namelyfor every T > 0, L /∈ N , for every y0, yT in L2(0, L) sufficiently small, one can find a control uthat drives the solution of (1) from y0 to yT in time T .
In this talk, we study the local controllability for the critical lengths, L ∈ N . The followingresult is obtained by performing a power series expansion of the solution and studying thecascade linear system resulting of this expansion.
Theorem 0.2 [1] For every L > 0, there exists T (L) ≥ 0 such that for every T > T (L),thereexists r > 0 such that for any states y0, yT ∈ L2(0, L) with ‖y0‖L2(0,L) < r and ‖yT ‖L2(0,L) < r,there exists a control u such that the solution y of (1) satisfies y(0, ·) = y0 and y(T, ·) = yT .
References
[1] E. Cerpa, E. Crepeau, Boundary controlability for the non linear Korteweg-de Vries equationon any critical domain, annales de l’IHP, to be published, 2008.
[2] L. Rosier. Exact boundary controllability for the Korteweg-de Vries equation on a boundeddomain. ESAIM Control Optim. Calc. Var, 2, 1997, pp. 33–55.
∗ Joint work with Eduardo Cerpa.
11
Surprising computations
Uri Ascher
University of British ColumbiaComputer Science Department
Vancouver, CanadaE-mail: [email protected]
Computer simulations for differential equations (DEs) often require complex numerical meth-ods. It is important and often difficult to devise efficient methods for such purposes and to provetheir properties. The resulting computations usually produce expected results, at least qualita-tively, which in itself does not diminish the importance of the numerical methods.
Occasionally, however, one comes across a (correct) computation that yields surprising re-sults. In the process of writing a textbook on numerical methods for time dependent DEs I haveencountered some such, and this talk describes several instances including solving Hamiltoniansystems, KdV and NLS, and applying WENO methods for nonlinear conservation laws. Whatcan be qualified as “surprising” is of course a subjective matter, nonetheless the combined ef-fect of this talk hopefully sheds light on using marginally stable methods for solving marginallystable problems.
Minimal state in viscoelasticity and applications
to PDEs
Mauro Fabrizio
Universita di BolognaDipartimento di Matematica
Piazza di Porta S. Donato 5, 40126 Bologna, ItalyE-mail: [email protected]
We show the impact on the initial-boundary value problem of the use of a new notion ofstate based on the stress-response. Comparisons are made between this new approach and thetraditional one, which is based on the identification of histories as states. We shall refer to astress-response definition of state as the minimal state. Material with memory and with elasticrelaxation are discussed. Finally, we show how the evolution of a linear viscoelastic systemcan be described through a strongly continuous semigroup of linear contraction operators on aappropriate Hilbert space. The family of all solutions of the evolutionary system, obtained byvarying the initial data in such a space, is shown to have exponentially decaying energy.
12
On a p-Kirchhoff equation via Krasnoselskii’sGenus
Giovany Malcher Figueiredo
Universidade Federal do Para - UFPAFaculdade de Matematica
66.075-110 Belem - PA, BrasilE-mail: [email protected]
In this work will use the Genus theory, introduced by Krasnolselskii, to show a result ofexistence and multiplicity of solutions of the p-Kirchhoff equation
−[M
(∫
Ω|∇u|p dx
)]p−1
∆pu = f(x, u) in Ω, u = 0 on ∂Ω
where Ω is a bounded smooth domain of RN , 1 < p < N , and M and f are continuous functions.
2000 Mathematics Subject Classification : 45M20, 35J25, 34B18, 34C11, 34K12.
Keywords: Genus theory, nonlocal problems, Kirchhoff equation.
Anisotropic Maxwell equations: decay properties
Gustavo Perla Menzala
Laboratorio Nacional de Computacao Cientıfica - LNCCAvenida Getulio Vargas, 333 - Quitandinha
25651-075 Petropolis - RJ, Brasiland
Universidade Federal do Rio de Janeiro - UFRJInstituto de Matematica
Cidade Universitaria - Ilha do Fundao21941-972 Rio de Janeiro - RJ, Brasil
E-mail: [email protected] and [email protected]
We consider an open set in 3-d filled with an anisotropic electromagnetic medium and analyzethe asymptotic behavior as time approaches infinity of the total energy associated with the cor-responding Maxwell equations. We also consider a coupled anisotropic system Maxwell/Elasticwaves.
We treat either exterior domains or bounded domains with interior or boundary dissipations.Examples include the presence of electric conductivity or the classical Silver-Muller conditionwhich arises as a first order approximation to the so-called ”transparent” boundary condition.
∗ Joint work with Cleverson R. da Luz.
13
Polynomial decay to thermoelastic plateswith memory
Pedro Gamboa
Universidade Federal do Rio de Janeiro - UFRJInstituto de Matematica
Cidade Universitaria, Ilha do Fundao, P. O. Box 6853021945-970 Rio de Janeiro - RJ, Brasil
E-mail: [email protected]
Let Ω be an open bounded set of Rn with smooth boundary Γ. Here we consider the
transverse oscillations of a thermoelastic plate configurates over Ω. Denoting by u and θ thevertical deflection and the temperature of the plate, the model that defines the oscillations ofthe plate are given by
utt + γ∆2u + m∆θ = 0 in Ω×]0, +∞[
cθt −∫
∞
0κ(s)∆θ(t − s)ds − m∆ut = 0 in Ω×]0, +∞[.
We consider Dirichlet or clamped boundary conditions for u and Dirichlet boundary conditionfor θ that is
u = ∆u = θ = 0 in ∂Ω×]0, +∞[
or
u =∂u
∂ν= θ = 0 in ∂Ω×]0, +∞[
and initial value
u(0) = u0, u′(0) = u1, θ(0) = θ0, in Ω
θ(−s) = ψ(s) in Ω×]0, +∞[
where u0, u1, θ0 and ψ are given functions and κ : [0.+∞[→ R; is the memory kernel. Denotingby µ(s) = −κ′(s) the hypotheses we impose to µ are the following
µ ∈ C1(R+) ∩ H1(R+), µ(s) ≥ 0 µ′(s) ≤ 0 ∀s ∈ R+ µ′(s) + σµ(s) ≤ 0
In that article was proved that the system is not exponentially stable, and by using the La Salleprinciple the authors showed that the solution in general goes to zero but with out giving anyrate of decay. The main result of this paper is to show that there exists a polynomial rate ofdecay of the solution that can be improved by improving the regularity of the initial data.
14
Hyperbolic Cahn-Hilliard equations
Maurizio Grasselli
Politecnico di MilanoDipartimento di Matematica “F. Brioschi”
Via E. Bonardi, 9 - 20133 Milano, ItalyE-mail: [email protected]/utenti/grasselli
A modified Cahn-Hilliard equation has been proposed by P. Galenko et al. to model phaseseparation in certain binary systems like, for instance, glasses (cf. [2-5]). Supposing that thematerial occupies a bounded domain Ω ⊂ RN , N = 1, 2, 3, and denoting by ρ the concentrationof one species of atoms, the equation can be written in the following form
εχtt + γχt − ∆(−κ∆u + φ′(u)) = g,
where ε, γ and κ are positive constant, g is a given function and f is a nonconvex potential.Setting ε = 0 we recover the well-known Cahn-Hilliard equation. However, if ε > 0, the aboveequation has hyperbolic features. In the one-dimensional case, the mathematical analysis hasalready been carried out in a number of papers (see, e.g., [1, 6-8]). Here we intend to focus onthe more challenging cases N = 2 and N = 3.
References
[1] A. Debussche, A singular perturbation of the Cahn-Hilliard equation, Asymptotic Anal., 4
(1991), 161–185.
[2] P. Galenko, D. Jou, Diffuse-interface model for rapid phase transformations in nonequilib-rium systems, Phys. Rev. E, 71 (2005), 046125 (13 pages).
[3] P. Galenko, V. Lebedev, Analysis of the dispersion relation in spinodal decomposition of abinary system, Philos. Mag. Lett., 87 (2007), 821–827.
[4] P. Galenko, V. Lebedev, Local nonequilibrium effect on spinodal decomposition in a binarysystem, Int. J. Thermodyn., 11 (2008), 21–28.
[5] P. Galenko, V. Lebedev, Nonequilibrium effects in spinodal decomposition of a binary sys-tem, Phys. Lett. A, 372 (2008), 985–989.
[6] S. Gatti, M. Grasselli, A. Miranville, V. Pata, On the hyperbolic relaxation of the one-di-mensional Cahn-Hilliard equation, J. Math. Anal. Appl., 312 (2005), 230–247.
[7] S. Zheng, A.J. Milani, Global attractors for singular perturbations of the Cahn-Hilliardequations, J. Differential Equations, 209 (2005), 101–139.
[8] S. Zheng, A.J. Milani, Exponential attractors and inertial manifolds for singular perturba-tions of the Cahn-Hilliard equations, Nonlinear Anal., 57 (2004), 843–877.
15
On convexity and decay rates of Timoshenkotype systems
Aıssa Guesmia
Universite Paul Verlaine-MetzLaboratoire de Mathematiques et Applications, UMR 7122Batiment A, Ile du Saulcy F-57045 Metz Cedex 1, France
E-mail: [email protected]
In this work we consider the following Timoshenko system
ρ1ϕtt − k1(ϕx + ψ)x = 0,
ρ2ψtt − k2ψxx +
∫ t
0g(t − τ)(a(x)ψx(τ))xdτ + k1(ϕx + ψ) + b(x)h(ψt) = 0,
in (0, L) × R+, with Dirichlet boundary conditions and initial data where a, b, g, h are specific
functions and ρ1, ρ2, k1, k2, L are positive constants. We consider the case of equal speeds ofpropagation: k1
ρ1= k2
ρ2, and we establish general stability estimate using the multiplier method
and some properties of convex functions. Without imposing any growth condition at the originon h, we show that the energy of the system is bounded above by a quantity, depending on gand h, which tends to zero (as time goes to infinity). Our estimate allows us to consider largeclass of functions h with general growth at the origin. We give some illustrating examples toillustrate how to derive from our general estimate the polynomial, exponential or logarithmicdecay. The results of this paper improve and generalize some existing results in the literatureand generate some interesting open problems.
Keywords: general decay estimate, nonlinear damping, relaxation function, Timoshenko, vis-coelastic, convexity.
AMS Classification: 35B35, 35L20, 35L70.
∗ Joint work with Salim Messaoudi.
Strongly heteregeneous piezoelectric composites
Bernadette Miara
ESIEE93160 Noisy-le-Grand, France
E-mail: [email protected]
We consider a piezoelectric composite whose material properties (described by the tensorsof elasticity, of dielectricity and the coupling tensor) present strong heterogeneities. In thecase of a periodic distribution of the small inclusions in the composite we show that the limithomogenized material present band-gaps, i.e., frequency intervals in which there is no acousticwaves propagation.
16
Controllability of some partial differentialequations in Lagrangian description
Thierry Horsin
Universite de Versailles-St QuentinLaboratoire de Mathematiques Appliquees
45, Avenue des Etats-Unis, Batiment Fermat78030 Versailles Cedex, FranceE-mail: [email protected]
We discuss some controllability results of certain partial differential in a “Lagrangian de-scription”. We give some partial positive results for the viscous Burger’s equation, the heatequation, and the Euler equations (joint results with O. Glass). In the case of the Euler equa-tions we present some results that depend on the dimension of the space in which the problemis considered.
Stationary solutions to the drift-diffusion
model in the whole space Rn
Masaki Kurokiba
Fukuoka UniversityDepartment of Applied Mathematics
814-0180 Fukuoka, [email protected]
We are interested in the drift-diffusion model arising in semiconductor device simulation andplasma physics. In the simplest case, this model is described by the system of equations:
(P )
ut − ∆u + ∇ · (u∇ψ) = 0,vt − ∆v −∇ · (v∇ψ) = 0,−∆ψ = −(u − v) + g(x),
where u = u(x, t) and v = v(x, t) denote the electron and the hole densities, respectively, in thesemiconductor, while ψ = ψ(x, t) is the electric potential and g = g(x) is the impurity dopingprofile. Mathematical study of this equation has been extensively development.The main concernof those results is devoted to the initial boundary problem for (P ). Since some similar equationalso appear in the othe context as Nernst-Plank equation in astronomy, Keller-Segel model inchemotaxis, it is also interesting to consider the problem for (P ) in the whole space. In thisstudy we consider the stationary problem for the drift-diffusion model (P ) in the whole spaceR
n.
∗ Joint work with Shuichi Kawashima and Ryo Kobayashi.
17
Finite dimensional and smooth attractors arisingin wave dynamics with geometrically restricted
dissipation and nonlinear critical exponents
Irena Lasiecka
University of VirginiaDepartment of Applied Mathematics
Charlottesville, VA 22901, [email protected]
Recent developments in the area of long time behavior of nonlinear hyperbolic flows willbe presented. These include issues such as existence of global attractors, their smoothness anddimensionality see [1], [2] below. One of the difficulties arising in the analysis of attractorsfor hyperbolic flows is the lack of smoothing mechanism generated by the free dynamics (un-like parabolic flows). We shall consider wave dynamics with a nonlinear dissipation which issupported only on a portion of the boundary and with nonlinear sources of critical Sobolevexponents. Non-compactness of the sources, along with severely restricted boundary support ofthe dissipation causes new phenomena to arise. This is particularly at the level of exhibitingboth regularity and finite dimensionality of attractors. We will show that the dynamical system,governed by a 3-d “critical” wave equation defined on a bounded, convex and simply connecteddomain. The wave dynamics is subject to geometrically restricted nonlinear dissipation that issupported on a portion of the boundary. It will be shown that the system posseses an ultimatesmooth long time behavior which is also finite dimensional. The proof of this result is basedon the folllowing ingredients: (i) special Carlemans estimates handling criticality of potentialenergy, (ii) topological methods handling criticality of kinetic energy and (iii) special geometric(nonradial) multipliers handling the restricted support on the boundary.
References
[1] I. Chueshov and I. Lasiecka, “Long time behavior of second order evolutions with nonlineardissipation” - Memoires of AMS, 2008.
[2] I. Chueskov, I. Lasiecka and D. Toundykov , “Long-term dynamics of semilinear waveequation with nonlinear localized interior damping and a source term of critical exponent”,Discrete, Continuopus Dynamic Systems, vol 20, pp 459-609 , 2008.
18
Variational model of nonholonomic 4D-mediaand its application to the couple problem
of the thermo- and visco-elasticity
Sergey Lurie
Dorodnicyn Computer Centre RASMoscow, Russia
E-mail: [email protected]
The variational formalism of formulation of non-holonomic media models is offered. This ap-proach is based on the formal generalization of continued mechanics models of deformable bodieson the 4D-space of events with a four-dimensional vector of the generalized displacements.Thecoordinate system is determined by the spatial coordinates x1, x2, x3 and the fourth coordinatex4, x4 = ivt, i =
√−1, v - is the normalization factor, t - is the normalized time. Time of process
is included in the generalized 4D system of coordinates. The generalized displacement 4D-vectorRi(i = 1, 2, 3) was defined as well: Rj = rj + ivRNj , rk = Rj(δik −NkNj), rjNj = 0. The firstthree components of the generalized displacement vector ri = Rj(δij −NiNj) are the traditionalcomponents of the three dimensional, the fourth component is equal to the projection of thedisplacement 4D-vector onto the coordinate x4 axis.4D-strain tensor is defined as decomposition into the ”3D-tensor”of spatial strains, the ”3D-vector” of time displacements (the fourth column vector i = 1, 2, 3, j = 4, i 6= 4), and thefourth row vector i = 4, j = 1, 2, 3, j 6= 4), which we treat as the generalized velocity, and the”scalar” S (the last diagonal element in the matrix of 4D-strains, i = 4, j = 4), which we treatas the entropy density.To describe models of nonholonomic media, we use a variational approach based on the for-mulation of the kinematic constraints in the medium under study and on the construction ofthe corresponding variational forms using the Lagrange multipliers (a ”kinematic” variationprinciple). Generalized Cauchy relations determining generalized 4D-strains from generalized4D-displacements were considered as the kinematic constraints.The constitutive equations are determined with the help of the nonintegrability conditions forthe possible work of internal forces. We used symbolic notation to analyze the general structureof the equations and received the following common variational form for irreversible processesthe volume of the event space V and over the hypersurface F :
δW ∗ =
∫[1
2Cab(QbδQa − QaδQb)]dV −
∮[1
2Ddf (qfδqd − qdδqf )]dF
It was proved that for irreversible processes the physical parameters Cab, Ddf in the constitu-tive equations are nonsymmetric in the multiindexes, while for reversible processes the similarparameters Cab, Ddf (the tensor of elastic moduli) are of course symmetric with respect to per-mutations of the multiindexes.There are proved two fundamental statements: 1. The proposed model contains the First Lawof Thermodynamics as a special case. 2. In the framework of the proposed model, irreversibleprocesses run with positive dissipation, and the Second Law of Thermodynamics holds. It fol-lows from the nonintegrability conditions without additional propositions.Within the framework of specific model the generalized equation of heat conductivity is received,the treatments of Fourier hypothesis (for heat flow) and Duamel-Neumann hypothesis are given.It was shown that in the general case, an analog of the Duhamel-Neumann equation relates the
19
pressure p, temperature T , volume variation Θ and has the form of a relaxation-creep differentialequation: α1p+α2p = β1Θ+β2Θ+ψ1T +ψ2T , where α1, α2, β1, β2, ψ1, ψ2, are certain physicalconstants of the model.Us result the general concept for the components of the variational equation corresponding tothe conservative and nonconservative processes were received and a model of coupled dynamicthermoelasticity with dissipation taken into account was elaborated. The corresponding bound-ary value problem was established.As particular problem the couple variational formulation of the Navie-Stockes problem in La-grangian coordinates was obtained, allowing to formulate the consistent system of initial andboundary conditions. The algorithm of calculating of the effective damping properties for thecomposite with viscoelasticity components was elaborated using the proposed variational modelof nonholonomic 4D-media.
∗ Joint work with A. Leontiev and N. Tuchkova.
A stability result in a memory-type
Timoshenko system
Salim A. Messaoudi
King Fahd University of Petroleum and MineralsDepartment of Mathematics and Statistics
Dhahran 31261, Saudi ArabiaE-mail: [email protected]
In this work, we consider the following Timoshenko system
ρ1ϕtt − K(ϕx + ψ)x = 0, (0, 1) × IR+
ρ2ψtt − bψxx + K (ϕx + ψ) +∫ t0 g(t − τ)ψxx(τ)dτ = 0, (0, 1) × IR+
ϕ(0, t) = ϕ(1, t) = ψ(0, t) = ψ(1, t) = 0, t ≥ 0ϕ(x, 0) = ϕ0(x), ϕt(x, 0) = ϕ1(x), x ∈ (0, 1)ψ(x, 0) = ψ0(x), ψt(x, 0) = ψ1(x), x ∈ (0, 1)
and establish a general stability result, from which the usual exponential and polynomial stabilityare only special cases. Our result is proved under the usual equal wave speeds (ρ1/ρ2 = K/b)and relatively weaker conditions on the relaxation function g.
∗ Joint work with Muhammad I. Mustafa.
20
Decay properties of an anisotropic coupledmodel in electromagnetism/elasticity
Cleverson R. da Luz
Universidade Federal do Rio de Janeiro - UFRJInstituto de Matematica
Cidade Universitaria, Ilha do Fundao, P. O. Box 6853021945-970 Rio de Janeiro - RJ, Brasil
E-mail: [email protected]
In this work we study decay properties of the solutions of the following initial boundary-valueproblem in an exterior domain
utt −3∑
i,j =1
∂
∂xi
(A ij
∂u
∂xj
)+ ut + κ curl E = f(ut,u) inΩ × (0,∞), (1)
ǫEt − curl H + σE − κ curl ut = 0 in Ω × (0,∞), (2)
µHt + curl E = 0 in Ω × (0,∞), (3)
div (µH) = 0 in Ω × (0,∞), (4)
u(x, 0) = u0(x), ut(x, 0) = u1(x) in Ω (5)
E(x, 0) = E0(x), H(x, 0) = H0(x) in Ω, (6)
u = 0, E × η = 0 on ∂Ω × (0,∞) (7)
where Ω is the exterior of a compact, connected body in R3 with Lipschitz boundary, u =
u(x, t) is the vector displacement, E = E(x, t) denote the electric field and H = H(x, t) denotethe magnetic field, (x) and µ(x) denote the electric permeability and magnetic permittivityrespectively. They are 3 × 3 symmetric matrices, uniformly positive definite whose entries arereal-valued functions and belong to L∞(Ω). The vector-value function f(ut,u) is nonlinear.The parameter σ > 0 is called the conductivity constant and κ is a coupling constant. Let Aij
3 × 3 matrices, will assume that there exists a constant C0 > 0 such that
3∑
i, j=1
Aij vj . vi ≥ C0
3∑
i, j=1
|vi|2 ∀ vi = (v1i , v
2i , v
3i ) ∈ R
3.
This work is devoted to find uniform rates of decay of the total energy associated with problem(1)-(7). In order to study the semilinear problem for the above system we consider first theassociated linear problem in detail. The main difficulty here is the anisotropic character ofMaxwell’s equation which unable us to treat (1)-(7) as two vector wave equations and useavaliable know results.
∗ Joint work with G. Perla Menzala.
21
Numerical approximation of the boundarycontrol of the wave equation
Sorin Micu
Universitatea din CraiovaFacultatea de Matematica si Informatica
200585 Craiova, RomaniaE-mail: [email protected]
In the last years there was an increasing interest for the numerical approximations of thecontrols. For instance, Hilbert Uniqueness Method was used in [2, 3] to deduce numericalalgorithms with finite differences in the context of the two dimensional wave equation. In thesereferences a bad behavior of the approximate controls was observed as a consequence of thespurious high-frequency oscillations generated by the space semi-discrete dynamics. Preciselythe controls of the highest modes may have large norms which are not uniformly bounded asthe discretization parameter h goes to zero. All these phenomena occur in the semi-discretemodels obtained by finite differences or the classical finite element method (see [5]). Therefore,in any of these models, the controllability property is not uniform in h and the sequences ofapproximations may diverge even for regular initial data.
Since the main problem is the existence of the spurious high frequencies generated by thediscretization process, the idea of eliminating or reducing them in one way or another arisesnaturally. All the numerical experiments using Tychonoff regularization technique [2], bi-gridalgorithm [3, 6] or mixed finite element approximation [1, 4] are based on this idea. This paperconsiders a new method to achieve the uniform controllability. The idea is to introduce in thediscrete equation a numerical viscous term which vanishes when the mesh size h tends to zero.The dissipation has the role to damp out the bad spurious high frequencies responsible for thelarge norm controls, eventually ensuring the uniform controllability of the system. The methodhas the advantage of being simple, general and quite natural.
References
[1] Castro C. and Micu S., Boundary controllability of a linear semi-discrete 1-D wave equation derivedfrom a mixed finite element method, Numer. Math., 102 (2006), pp. 413-462.
[2] R. Glowinski, C.H. Li and J.-L. Lions, A numerical approach to the exact boundary controllabilityof the wave equation (I). Dirichlet controls: Description of the numerical methods, Jap. J. Appl.Math., vol. 7, 1990, pp. 1-76.
[3] R. Glowinski and J.-L. Lions, Exact and approximate controllability for distributed parameter sys-tems, Acta Numerica 1996, pp. 159-333.
[4] Glowinski R., Kinton W. and Wheeler M. F., A mixed finite element formulation for the boundarycontrollability of the wave equation, Int. J. Numer. Methods Eng., 27 (1989), pp. 623-636.
[5] J.A. Infante and E. Zuazua, Boundary observability for the space semi-discretization of the 1-D waveequation, M2AN, vol. 33 (2), 1999, pp. 407-438.
[6] Negreanu M. and Zuazua E., Uniform boundary controllability of a discrete 1-D wave equation,System and Control Letters, 48 (2003), pp. 261-280.
Partially Supported by Grant MTM2005-00714 of MCYT (Spain) and Grant CEEX-05-D11-36/2005 (Romania).
22
Persistence properties and unique continuationresults for canonical dispersive equations
Gustavo Ponce
University of California, Santa Barbara, UCSBDepartment of Mathematics
93106 CA, USAE-mail: [email protected]
The first part of this talk is concerned with persistence properties of solutions of classicaldispersive equations, i.e. NLS, generalized KdV, B-O, in weighted Sobolev spaces. We shall usea refined version of the Leibnitz’ rule for fractional derivatives to obtain an extension of previousworks.
In the second part, we present results concerning some unique continuation properties of thesedispersive models. The aim is to obtain sufficient conditions on the behavior of the differenceu1 − u2 of two solutions u1, u2 of which guarantee that u1 ≡ u2.
In joint works with L. Escauriaza, C. E. Kenig, and L. Vega sufficient conditions, on theasymptotic behavior of the difference of two solutions u1, u2 at two different times t1 < t2, havebeen deduced which guarantee that u1 ≡ u2.
Asymptotic behavior in nonlinear hyperbolic
thermoelasticity
Reinhard Racke
University of KonstanzDepartment of Mathematics and Statistics
78457 Konstanz, GermanyE-mail: [email protected]
We consider hyperbolic models in thermoelasticity and we discuss the asymptotic behavioras time tends to infinity of nonlinear problems in one space dimension. The we describe thepropagation of singularities for linear and semilinear models in three space dimensions. Finally,we look at resolvent expansions in exterior domains.
23
Existence and multiplicity resultsfor elliptic equation of p-Kirchhoff type
with discontinuous nonlinearities
Rubia G. Nascimento
Universidade Federal do Para - UFPACampus de Abaetetuba
68440-000 Abaetetuba - PA, BrasilE-mail: [email protected]
We use variational techniques to study the existence and multiplicity of nonnegatine solutionsof equations of the form
−[M
(||u||p1,p
)]p−1
= λH(u − a)uq + h(x)us in Ω,
where Ω is a bounded domain, λ, a > 0 are parameters, h ∈ L∞(Ω), h > 0 in Ω, 1 < q + 1 <p < s + 1, M is a continuous function satisfying some conditions to appear in the work, H isthe Heaviside function and ∆p is the p-Laplacian operator. Moreover, ||u||p1,p is usual norm in
Sobolev space W 1,p0 (Ω) given by
||u||1,p =
(∫
Ω|∇u|p
) 1
p
.
2000 Mathematics Subject Classification : 35A15, 35J60, 35H30
Keywords: nonlocal problem, p-Kirchhoff equation, discontinuous nonlinearities, MountainPass Theorem, Ekeland variational principle.
Explicit multidimensional Ingham-Beurlingtype estimates
Vilmos Komornik
Universite de Strasbourg I (Louis Pasteur)Departement de Mathematiques
67084 [email protected]
Years ago, we combined some theorems of Kahane and Haraux in order to solve an exactcontrollability problem for plate-like Petrovski systems. We show that this approach also allowsus to obtain rather precise explicit constants in the estimates. They improve and generalizesome recent results established by much deeper complex analytic tools.
24
On the vibrations of an extensiblethermoelastic beam
Maria Grazia Naso
Universita di BresciaDipartimento di Matematica, Facolta di Ingegneria
Via Valotti 9, 25133 Brescia, ItaliaEmail: [email protected]
We analyze the dissipative system
∂ttu + ∂xxxxu + ∂xxθ −
[β + ‖∂xu‖2
L2(0,1)
]∂xxu = f,
∂tθ − ∂xxθ − ∂xxtu = g,
describing the thermomechanical evolution of an extensible thermoelastic beam. The dissipationmechanism is contained only in the second equation, ruling the evolution of θ. Under naturalboundary conditions, we prove the existence of bounded absorbing sets. When the externalsources f and g are time-independent, the related semigroup of solutions is shown to possessthe global attractor of optimal regularity for all parameters β ∈ R.
Controllability results for 1-d coupleddegenerate parabolic equations
Luz de Teresa
Universidad Nacional Autonoma de MexicoInstituto de MatematicasC.U., 04510, D.F. Mexico
E-mail: [email protected]
In this talk we are concerned with the controllability properties of coupled degenerate 1-d parabolic equations. We are going to consider two different kind of systems: the first one,consists of two forward equations and the second one, consists of one equation forward and thesecond equation backward. We will see that in the treated cases the controllability results areclose to the results obtained in the non degenerate case. The key point in the results is theobtention of an observability inequality for the adjoint systems. This is done by a combinationof an appropriate Carleman inequality and Hardy type inequalities needed in the degeneratecase. This is a joint work with Piermarco Cannarsa.
∗ Joint work with Piermarco Cannarsa.
25
Topological derivative in multi-scale linearelasticity models
Andre Novotny
Laboratorio Nacional de Computacao Cientıfica - LNCCAvenida Getulio Vargas, 333 - Quitandinha
25651-075 Petropolis - RJ, BrasilEmail: [email protected]
Composite materials have become one of the most important classes of engineering materi-als. Then, the macroscopic mechanical behavior of this materials has a paramount importancein the design of mechanical components for a vast number of applications in civil, aerospace,biomedical and nuclear industries. In a broad sense, one can argue that much of material sci-ence is about topological and shape modifications on materials’ microstructures. For example,changing the shape of the graphite inclusions in a cast iron matrix dramatically changes all themacroscopic properties of the material. In this context, the ability to accurately predict themacroscopic mechanical behavior from the corresponding microstructural properties becomesessential in the analysis and potential purpose-design and optimisation of the underlying het-erogeneous medium. The goal of this design and optimization is to control the macroscopicmaterial properties, either to improve them, or to build materials with prescribed properties.Of crucial importance to the potential optimisation of the medium in this case, is the sensi-tivity of the effective macroscopic parameters to changes in the microstructure. In this sense,we can cite many authors that have been building new microscopic topologies based on heuris-tics arguments, for example, microstructures with macroscopic negative Poisson’s ratio (see,for instance, Almgreen, 1985 and Lakes, 1987). This paper proposes a general analytical ex-pression for the sensitivity of the two-dimensional macroscopic elasticity tensor to topologicalchanges of the microstructure of the underlying material. The macroscopic elasticity responseis estimated by means of a homogenisation-based multi-scale constitutive theory for elasticityproblems where, following closely the ideas of Germain et al. , 1983, the macroscopic strain andstress tensors at each point of the macroscopic continuum are defined as the volume averages oftheir microscopic counterparts over a Representative Volume Element (RVE) of material asso-ciated with that point. It is analogous to the multi-scale strategy presented, among others, byMiehe et al. , 1999 and Michel et al. , 1999 – and whose variational structure is described indetail in de Souza Neto & Feijoo, 2006. In this context and based on the axiomatic construc-tion of multi-scale constitutive models, the proposed sensitivity leads to a symmetric fourthorder tensor field over the RVE that measures how the macroscopic elasticity parameters es-timated within the multi-scale framework changes when a small circular hole is introduced atthe micro-scale. Its analytical formula is derived by making use of the concepts of topologicalasymptotic expansion and topological derivative (Sokolowski and Zochowski, 1999; Cea et al. ,2000) within the adopted multi-scale theory. These relatively new mathematical concepts allowthe closed form calculation of the sensitivity, whose value depends on the solution of a set ofequations over the original domain, of a given shape functional with respect to an infinitesimaldomain perturbation, like the insertion of holes, inclusions or source term (see, for instance,Nazarov & Sokolowski, 2003). Among the methods for calculation of the topological derivativecurrenlty available in the literature, here we shall adopt the one proposed by Novotny et al. ,2003. In the present context, the variational setting in which the underlying multi-scale theoryis cast, is found to be particularly well-suited for the application of the topological derivative
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formalism. The final format of the proposed analytical formula is strikingly simple and canbe potentially used in applications such as the synthesis and optimal design of microstructuresto meet a specified macroscopic behavior. In this work, initially we describes the multi-scaleconstitutive framework adopted in the estimation of the macroscopic elasticity tensor. A clearvariational foundation of the theory is established which is essential for the main developmentsto be presented later. Next, we presents the main result of the paper – the closed formula forthe sensitivity of the macroscopic elasticity tensor to topological microstructural perturbations.Here, a brief description of the topological derivative concept is initially given. This is followedby its application to the problem in question which leads to the identification of the requiredsensitivity tensor. A simple finite element-based numerical example is also provided for thenumerical verification of the analytically derived topological derivative formula. The paper endswith some concluding remarks.
References
[1] R.F. Almgreen. An Isotropic Three-Dimensional Structure with Poisson Ratio -1. Journalof Elasticity, 15:427-430, 1985.
[2] J. Cea, S. Garreau, Ph. Guillaume & M. Masmoudi, The shape and Topological Optimiza-tions Connection. Computer Methods in Applied Mechanics and Engineering, 188(4):713–726, 2000.
[3] E.A. de Souza Neto & R.A. Feijoo. Variational foundations of multi-scale constitutive mod-els of solid: Small and large strain kinematical formulation. LNCC Research & DevelopmentReport, No. 16/2006, National Laboratory for Scientific Computing, Brasil, 2006, also sub-mitted for publication to International Journal of Plasticity, 2007.
[4] P. Germain, Q.S. Nguyen & P. Suquet. Continuum thermodynamics. Journal of AppliedMechanics, Transactions of the ASME, 50:1010–1020, 1983.
[5] R. Lakes. Foam Structures with Negative Poisson’s Ratio. Science, AAAS, 235:1038–1040,1987.
[6] J.C. Michel, H. Moulinec & P. Suquet. Effective properties of composite materials with pe-riodic microstructure: a computational approach. Computer Methods in Applied Mechanicsand Engineering, 172:109–143, 1999.
[7] C. Miehe, J. Schotte, & J. Schroder. Computational Micro-Macro Transitions and OverallModuli in the Analysis of Polycrystals at Large Strains. Computational Materials Science,16:372–382, 1999.
[8] S. A. Nazarov & J. Sokolowski. Asymptotic analysis of shape functionals. Journal de Math-ematiques Pures et Appliquees, 82(2):125–196, 2003.
[9] A.A. Novotny, R.A. Feijoo, C. Padra & E. Taroco. Topological Sensitivity Analysis. Com-puter Methods in Applied Mechanics and Engineering, 192:803–829, 2003
[10] J. Sokolowski & A. Zochowski. On the Topological Derivatives in Shape Optimization.SIAM Journal on Control and Optimization, 37(4):1251-1272, 1999.
∗ Joint work with S.M. Giusti, E.A. de Souza Neto and R.A. Feijoo.
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Short wave stability for inviscid shear flow
Michael Renardy
Virginia TechMathematics Department
24061-0123 Blacksburg, VA, USAEmail: [email protected]
Linear stability studies of inviscid shear flows go back more than a century. In contrast tothe Kelvin-Helmholtz instability, which has growth rate which increase with wave number, allknown instabilities of smooth velocity fields are long wave instabilities with a short wave cutoff.In this lecture, we shall prove a general result to this effect.
Exact controllability of the nonlinearSchrodinger equation
Lionel Rosier
Institut Elie CartanUniversite Henri Poincare Nancy 1
B.P. 239, 54506 Vandoeuvre-les-Nancy Cedex, FranceE-mail: [email protected]
We consider the cubic Schrodinger equation on a bounded domain in RN and derive several
exact controllability results by using Strichartz estimates and Bourgain analysis.
∗ Joint work with Bing-Yu Zhang.
Gain of regularity for the KP-I equation
Octavio Vera Villagran
Departamento de MatematicaFacultad de Ciencias
Universidad del Bıo-Bıo - ConcepcionE-mail: [email protected]
In this work we study the smoothness properties of solutions to the KP-I equation. We showthat the equation’s dispersive nature leads to a gain in regularity for the solution.
∗ Joint work with J. Levandosky and M. Sepulveda.
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Stability of linear porous-thermo-elasticitysystem
Paulo X. Pamplona
Universidade Federal do Rio de Janeiro - UFRJInstituto de Matematica
Cidade Universitaria, Ilha do Fundao, P. O. Box 6853021945-970 Rio de Janeiro - RJ, Brasil
E-mail: [email protected]
We study the asymptotic behavior and the analyticity of the solutions of the one-dimensionalporous-elasticity problem with thermal effect. We assume the following constitutive equations
ρutt = µuxx + bϕx − βθx + γuxxt
Jϕtt = δϕxx − bux − ξϕ + mθ + ηϕxxt + k1θxx
cθt = kθxx − βuxt − mϕt + k2ϕxxt
with the boundary conditions
u(0, t) = u(π, t) = ϕx(0, t) = ϕx(π, t) = θx(0, t) = θx(π, t) = 0, t ∈ (0,∞)
and the initial conditions
u(x, 0) = u0(x), ut(x, 0) = u1(x), ϕ(x, 0) = ϕ0(x), ϕt(x, 0) = ϕ1(x),θ(x, 0) = θ0(x),
x ∈ (0, π). Here the variables u, ϕ and θ are the displacement of the solid elastic material, thevolume fraction and the temperature respectively. As coupling is considered, b must be differentfrom 0, but its sign does not matter in the analysis. As thermal effects is considered, we assumethat the thermal capacity c and the thermal conductivity k are strictly positive. The sign ofthe coupling term β does not matter in the analysis neither. And as viscoelastic dissipation isassumed in the system, γ > 0. In the first part of the work we assume that the porous dissipationis absent (η = k1 = k2 = 0). To guarantee that the system dissipates energy we need to assumethat the constitutive coefficients η, k1 and k2 satisfy the condition
(k1 + k2)2 < 4kη, k1, k2 ≥ 0.
We prove the lack of exponential stability and the polynomial stability in case k1 = k2 = η = 0.Moreover using a result true Pruss, we are able to improve the polynomial rate of decay bytaking more regular initial data. Finally, in case general, we show that the system is analytic,which in particular implies the exponential decay and the spectrum determined growth property(SDG-property).
∗ Joint work with R. Quintanilla and Jaime E. Munoz Rivera.
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On the stability of Mindlin-Timoshenko plates
Hugo D. Fernandez Sare
Laboratorio Nacional de Computacao Cientıfica - LNCCAvenida Getulio Vargas, 333 - Quitandinha
25651-075 Petropolis - RJ, BrasilE-mail: [email protected]
We consider a Mindlin-Timoshenko model with frictional dissipations acting on the equationsfor the rotation angles. We prove that this system is not exponentially stable independent ofany relations between the constants of the system, which is different from the analogous’ one-dimensional case. Moreover, we show that the solution decays polynomially to zero, with ratesthat can be improved depending on the regularity of the initial data.
2000 Mathematics Subject Classification : 35 B 40, 74 H 40.
Keywords : Timoshenko plates, non-exponential stability, polynomial stability.
Uniform stabilization of a coupled parabolic-
hyperbolic fluid-structure interaction modelwith dissipation at the interface
Roberto Triggiani
University of VirginiaDepartment of Applied Mathematics
Charlottesville, VA 22901, [email protected]
We consider an established parabolic-hyperbolic fluid-structure interaction model (Stokes-Lame) consisting of two partial differential equations coupled at the interface. Such problem isnot stable, and its degree of instability depends on the geometry of the structure immersed in thefluid (The worst case is when the structure is a ball). Accordingly, we then introduce a dampingterms at the interface between the two media and establish the desired uniform stabilizationdecay in the energy space, with no geometric conditions assumed (except for some smoothness).In contrast, a velocity damping term acting on the wave-component over its entire domain isnot enough even to produce strong stabilization. Joint work with George Avalos. The caseof nonlinear damping at the interface will also be discussed.
This work supports by the Russian Foundation of the Basic Researches (Projects 06-01-00051,07-01-13525 and 07-07-00082).
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Uniform decay for the wave equationof Kirchhoff type with viscoelastic
boundary condition and source term
Eugenio Cabanillas Lapa
Instituto de InvestigacionFCM-UNMSM
Lima, PeruE-mail: [email protected]
Our aim in this paper is to prove the existence of global solutions of the wave equationof Kirchhoff type with a viscoelastic boundary condition and source term,for small initialdata.Furthermore,we prove that the energy of system decays exponentially.
References
[1] Aassila M., Cavalcanti M.M., Soriano J.A. Asymptotic stability and energy decay rates forsolutions of the wave equation with memory a star-shaped domain.,SIAM J. Control Optim.38(5) (2000) 1581-1602.
[2] Bae J.J. On uniform decay of coupled wave equation of Kirchhoff type subject to memorycondition the boundary., Nonlinear Anal.TMA 61(2005)351-372.
[3] Bae J.J., Jeong J.M. On uniform decay of solutions for wave equatin of Kkirchhoff typewith nonlinear boundary damping and memory source term.,Appl.Math.Comput. 138(2003)463-478.
[4] Cavalcanti M.M., Domingos Cavalcanti V.N.,Ma T.F., Soriano J.A.Global existence andasymptotic stability for viscoelastic problems. ,Diff.Int.Eqs. 15(2002) 731-748.
[5] Cavalcanti M.M., Domingos Cavalcanti V.N.,Santos M.L. Existence and uniform decayrates of solutions to a degenerate system with memory condition at the boundary.,Appl.Math.Comput. 150(2004)439-465.
[6] Cavalcanti M.M. Existence and uniform decay for the Euler-Bernoulli viscoelastic equation-with nonlocal boundary dissipation., Disc.Cont.Dyn.Syst. 8(2002)675-695.
[7] Cavalcanti M.M., Domingos Cavalcanti V.N.,Prates Filho J.S., Soriano J.A.Existence anduniform decay rates for viscoelastic problems with nonlinear boundary damping., Diff. Integ.Eqs.14(2001) 85-116.
[8] Muoz Rivera J. Andrade D. Exponential decay of non-linear wave equation with viscoelasticboundary condition.,Math. Meth. Appl. Sci. 23(2000)41-61.
[9] Park J.Y.,Bae J.J. On coupled wave equation of Kirchhoff type with nonlinear boundarydamping and memory term., Appl. math. Comput. 129(2002) 87-105
[10] Santos M.L. Decay rates for solutions of a system of wave equation with memory. Elect. J.Diff. Eqs. 38(2002)1-17.
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[11] Santos M.L. Decay rates for solutions of semilinear wave equation with memory conditionat the boundary. EJQTDE. 7(2002)1-17.
∗ Joint work with J. Bernui and Z. Huaringa.
Partially supported by FCM -UNMSM
Mathematics Subject Classifications: 35B35,35B40,35L70.
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