Showing papers in "Archive for Rational Mechanics and Analysis in 2006"

A Hierarchy of Plate Models Derived from Nonlinear Elasticity by Gamma-Convergence

Abstract: We derive a hierarchy of plate models from three-dimensional nonlinear elasticity by Γ-convergence. What distinguishes the different limit models is the scaling of the elastic energy per unit volume ∼hβ, where h is the thickness of the plate. This is in turn related to the strength of the applied force ∼hα. Membrane theory, derived earlier by Le Dret and Raoult, corresponds to α=β=0, nonlinear bending theory to α=β=2, von Karman theory to α=3, β=4 and linearized vK theory to α>3. Intermediate values of α lead to certain theories with constraints. A key ingredient in the proof is a generalization to higher derivatives of our rigidity result [29] which states that for maps v:(0,1)3→ℝ3, the L2 distance of ∇v from a single rotation is bounded by a multiple of the L2 distance from the set SO(3) of all rotations.

Contractions in the 2-Wasserstein Length Space and Thermalization of Granular Media

Abstract: An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow-through model representing the continuum limit of a gas of particles interacting through slightly inelastic collisions. This rate is obtained by reformulating the dynamical problem as the gradient flow of a convex energy on an infinite-dimensional manifold. An abstract theory is developed for gradient flows in length spaces, which shows how degenerate convexity (or even non-convexity) — if uniformly controlled — will quantify contractivity (limit expansivity) of the flow.

Low Mach Number Limit of the Full Navier-Stokes Equations

Abstract: The low Mach number limit for classical solutions of the full Navier-Stokes equations is here studied. The combined effects of large temperature variations and thermal conduction are taken into account. In particular, we consider general initial data. The equations lead to a singular problem, depending on a small scaling parameter, whose linearized system is not uniformly well-posed. Yet, it is proved that solutions exist and they are uniformly bounded for a time interval which is independent of the Mach number Ma ∈ (0,1], the Reynolds number Re ∈ [1,+∞] and the Peclet number Pe ∈ [1,+∞]. Based on uniform estimates in Sobolev spaces, and using a theorem of G. Metivier & S. Schochet [30], we next prove that the penalized terms converge strongly to zero. This allows us to rigorously justify, at least in the whole space case, the well-known computations given in the introduction of P.-L. Lions' book [26].

Quasistatic Evolution Problems for Linearly Elastic–Perfectly Plastic Materials

Abstract: The problem of quasistatic evolution in small strain associative elastoplasticity is studied in the framework of the variational theory for rate-independent processes. Existence of solutions is proved through the use of incremental variational problems in spaces of functions with bounded deformation. This approach provides a new approximation result for the solutions of the quasistatic evolution problem, which are shown to be absolutely continuous in time. Four equivalent formulations of the problem in rate form are derived. A strong formulation of the flow rule is obtained by introducing a precise definition of the stress on the singular set of the plastic strain.

Fractal first-order partial differential equations

Abstract: The present paper is concerned with semi-linear partial differential equations involving a particular pseudo-differential operator. It investigates both fractal conservation laws and non-local Hamilton–Jacobi equations. The idea is to combine an integral representation of the operator and Duhamel's formula to prove, on the one hand, the key a priori estimates for the scalar conservation law and the Hamilton–Jacobi equation and, on the other hand, the smoothing effect of the operator. As far as Hamilton–Jacobi equations are concerned, a non-local vanishing viscosity method is used to construct a (viscosity) solution when existence of regular solutions fails, and a rate of convergence is provided. Turning to conservation laws, global-in-time existence and uniqueness are established. We also show that our formula allows us to obtain entropy inequalities for the non-local conservation law, and thus to prove the convergence of the solution, as the non-local term vanishes, toward the entropy solution of the pure conservation law.

The Interaction between Quasilinear Elastodynamics and the Navier-Stokes Equations

Abstract: The interaction between a viscous fluid and an elastic solid is modeled by a system of parabolic and hyperbolic equations, coupled to one another along the moving material interface through the continuity of the velocity and traction vectors. We prove the existence and uniqueness (locally in time) of strong solutions in Sobolev spaces for quasilinear elastodynamics coupled to the incompressible Navier-Stokes equations. Unlike our approach in [5] for the case of linear elastodynamics, we cannot employ a fixed-point argument on the nonlinear system itself, and are instead forced to regularize it by a particular parabolic artificial viscosity term. We proceed to show that with this specific regularization, we obtain a time interval of existence which is independent of the artificial viscosity; together with a priori estimates, we identify the global solution (in both phases), as well as the interface motion, as a weak limit in strong norms of our sequence of regularized problems.

The Singular Set of Minima of Integral Functionals

Abstract: In this paper we provide upper bounds for the Hausdorff dimension of the singular set of minima of general variational integrals Open image in new window where F is suitably convex with respect to Dv and Holder continuous with respect to (x,v). In particular, we prove that the Hausdorff dimension of the singular set is always strictly less than n, where Open image in new window.

Stability of Contact Discontinuities for the 1-D Compressible Navier-Stokes Equations

Abstract: In this paper, we study the large-time asymptotic behavior of solutions of the one-dimensional compressible Navier-Stokes system toward a contact discontinuity, which is one of the basic wave patterns for the compressible Euler equations. It is proved that such a weak contact discontinuity is a metastable wave pattern, in the sense introduced in [24], for the 1-D compressible Navier-Stokes system for polytropic fluid by showing that a viscous contact wave, which approximates the contact discontinuity on any finite-time interval for small heat conduction and then runs away from it for large time, is nonlinearly stable with a uniform convergence rate provided that the initial excess mass is zero. This result is proved by an elaborate combination of elementary energy estimates with a weighted characteristic energy estimate, which makes full use of the underlying structure of the viscous contact wave.

Tractions, Balances, and Boundary Conditions for Nonsimple Materials with Application to Liquid Flow at Small-Length Scales

Abstract: Using a nonstandard version of the principle of virtual power, we develop general balance equations and boundary conditions for second-grade materials. Our results apply to both solids and fluids as they are independent of constitutive equations. As an application of our results, we discuss flows of incompressible fluids at small-length scales. In addition to giving a generalization of the Navier–Stokes equations involving higher-order spatial derivatives, our theory provides conditions on free and fixed boundaries. The free boundary conditions involve the curvature of the free surface; among the conditions for a fixed boundary are generalized adherence and slip conditions, each of which involves a material length scale. We reconsider the classical problem of plane Poiseuille flow for generalized adherence and slip conditions.

Regularity of the Inverse of a Planar Sobolev Homeomorphism

Abstract: Let be a domain. Suppose that f ∈ W1,1loc(Ω,R2) is a homeomorphism such that Df(x) vanishes almost everywhere in the zero set of J f . We show that f-1 ∈ W1,1loc(f(Ω),R2) and that Df−1(y) vanishes almost everywhere in the zero set of Sharp conditions to quarantee that f−1 ∈ W1, q (f(Ω),R2) for some 1

A Variational Model for Dislocations in the Line Tension Limit

Abstract: We study the interaction of a singularly-perturbed multiwell energy (with an anisotropic nonlocal regularizing term of H1/2 type) and a pinning condition. This functional arises in a phase field model for dislocations which was recently proposed by Koslowski, Cuitino and Ortiz, but it is also of broader mathematical interest. In the context of the dislocation model we identify the Γ-limit of the energy in all scaling regimes for the number Nɛ of obstacles. The most interesting regime is Nɛ≈|ln ɛ|/ɛ, where ɛ is a nondimensional length scale related to the size of the crystal lattice. In this case the limiting model is of line tension type. One important feature of our model is that the set of energy wells is periodic, and hence not compact. Thus a key ingredient in the proof is a compactness estimate (up to a single translation) for finite energy sequences, which generalizes earlier results from Alberti, Bouchitte and Seppecher for the two-well problem with a H1/2 regularization.

Gross-Pitaevskii Equation as the Mean Field Limit of Weakly Coupled Bosons

Abstract: We consider the dynamics of N boson systems interacting through a pair potential N−1Va(xi−xj) where Va(x)=a−3V(x/a). We denote the solution to the N-particle Schrodinger equation by ΨN, t. Recall that the Gross-Pitaevskii (GP) equation is a nonlinear Schrodinger equation and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if ut solves the GP equation, then the family of k-particle density matrices Open image in new window solves the GP hierarchy. Under the assumption that a=N−ɛ for 0<ɛ<3/5, we prove that as N→∞ the limit points of the k-particle density matrices of ΨN, t are solutions of the GP hierarchy with the coupling constant in the nonlinear term of the GP equation given by ∫V(x)dx. The uniqueness of the solutions of this hierarchy remains an open question.

Bifurcation From Stability to Instability for a Free Boundary Problem Arising in a Tumor Model

Abstract: We consider a time-dependent free boundary problem with radially symmetric initial data: σt − Δσ + σ = 0 if Open image in new window and σ(r,0)=σ0(r) in {r μ* then the stationary solution is unstable.

Kinetic Limit for Wave Propagation in a Random Medium

Abstract: We study crystal dynamics in the harmonic approximation. The atomic masses are weakly disordered, in the sense that their deviation from uniformity is of the order √ e. The dispersion relation is assumed to be a Morse function and to suppress crossed recollisions. We then prove that in the limit e → 0, the disorder-averaged Wigner function on the kinetic scale, time and space of order e −1 , is governed by a linear Boltzmann equation.

Crystalline Mean Curvature Flow of Convex Sets

Abstract: We prove a local existence and uniqueness result of crystalline mean curvature flow starting from a compact convex admissible set in Open image in new window. This theorem can handle the facet breaking/bending phenomena, and can be generalized to any anisotropic mean curvature flow. The method provides also a generalized geometric evolution starting from any compact convex set, existing up to the extinction time, satisfying a comparison principle, and defining a continuous semigroup in time. We prove that, when the initial set is convex, our evolution coincides with the flat φ-curvature flow in the sense of Almgren-Taylor-Wang. As a by-product, it turns out that the flat φ-curvature flow starting from a compact convex set is unique.

Long-Time Asymptotics of a Multiscale Model for Polymeric Fluid Flows

Abstract: In this paper, we investigate the long-time behavior of some micro-macro models for polymeric fluids (Hookean model and FENE model), in various settings (shear flow, general bounded domain with homogeneous Dirichlet boundary conditions on the velocity, general bounded domain with non-homogeneous Dirichlet boundary conditions on the velocity). We use both probabilistic approaches (coupling methods) and analytic approaches (entropy methods).

Nonlinear Stability of Rarefaction Waves for the Boltzmann Equation

Abstract: It is well known that the Boltzmann equation is related to the Euler and Navier-Stokes equations in the field of gas dynamics. The relation is either for small Knudsen number, or, for dissipative waves in the time-asymptotic sense. In this paper, we show that rarefaction waves for the Boltzmann equation are time-asymptotic stable and tend to the rarefaction waves for the Euler and Navier-Stokes equations. Our main tool is the combination of techniques for viscous conservation laws and the energy method based on micro-macro decomposition of the Boltzmann equation. The expansion nature of the rarefaction waves and the suitable microscopic version of the H-theorem are essential elements of our analysis.

Stability of semiconductor states with insulating and contact boundary conditions

Abstract: We prove the existence of global smooth solutions near a given steady state of the hydrodynamic model of the semiconductors in a bounded domain with physical boundary conditions. The steady state and the doping profile are permitted to be of large variation but the initial velocity must be small. Two cases are considered. In the first one the problem is three-dimensional, the boundary conditions are insulating and the steady state velocity vanishes. In the second one, the problem is one-dimensional, the boundary is of contact type and the steady state velocity does not vanish.

Macroscopic behavior of microscopic oscillations in harmonic lattices via wigner-husimi transforms

Abstract: We consider the dynamics of infinite harmonic lattices in the limit of the lattice distance ɛ tending to 0. We allow for general polyatomic crystals, but assume exact periodicity such that the system can be solved, in principle, by Fourier-transform and linear-algebra methods.

The Concept of a Minimal State in Viscoelasticity: New Free Energies and Applications to PDEs

Abstract: We show here the impact on the initial-boundary value problem, and on the evolution of viscoelastic systems of the use of a new definition of state based on the stress-response (see, e.g., [48, 16, 41]). Comparisons are made between this new approach and the traditional one, which is based on the identification of histories and states. We shall refer to a stress-response definition of state as the minimal state [29]. Materials with memory and with relaxation are discussed. The energetics of linear viscoelastic materials is revisited and new free energies, expressed in terms of the minimal state descriptor, are derived together with the related dissipations. Furthermore, both the minimum and the maximum free energy are recast in terms of the minimal state variable and the current strain. The initial-boundary value problem governing the motion of a linear viscoelastic body is re-stated in terms of the minimal state and the velocity field through the principle of virtual power. The advantages are (i) the elimination of the need to know the past-strain history at each point of the body, and (ii) the fact that initial and boundary data can now be prescribed on a broader space than resulting from the classical approach based on histories. These advantages are shown to lead to natural results about well-posedness and stability of the motion. Finally, we show how the evolution of a linear viscoelastic system can be described through a strongly continuous semigroup of (linear) contraction operators on an appropriate Hilbert space. The family of all solutions of the evolutionary system, obtained by varying the initial data in such a space, is shown to have exponentially decaying energy.

Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution

Abstract: We study a mathematical model describing dislocation dynamics in crystals. We consider a single dislocation line moving in its slip plane. The normal velocity is given by the Peach-Koehler force created by the dislocation line itself. The mathematical model is an eikonal equation with a velocity which is a non-local quantity depending on the whole shape of the dislocation line. We study the special case where the dislocation line is assumed to be a graph or a closed loop. In the framework of discontinuous viscosity solutions for Hamilton–Jacobi equations, we prove the existence and uniqueness of a solution for small time. We also give physical explanations and a formal derivation of the mathematical model. Finally, we present numerical results based on a level-sets formulation of the problem. These results illustrate in particular the fact that there is no general inclusion principle for this model.

A Variational View of Partial Brittle Damage Evolution

Abstract: Under time-dependent loading an elastic material undergoes the simplest form of damage in which it passes from its original state to a weaker elastic state. Elaborating on prior work [14], we establish the existence of a relaxed variational evolution where, at each time, the two states of the material combine to form a fine mixture, optimal from the standpoint of the applied load at that time, yet preserving the irreversibility of the damaging process.

Effective Cohesive Behavior of Layers of Interatomic Planes

Abstract: A simple model of cleavage in brittle crystals consists of a layer of material containing N atomic planes separating in accordance with an interplanar potential under the action of an opening displacement δ prescribed on the boundary of the layer. The problem addressed in this work concerns the characterization of the constrained minima of the energy EN of the layer as a function of δ as N becomes large. These minima determine the effective or macroscopic cohesive law of the crystal. The main results presented in this communication are: (i) the computation of the � limit E0 of EN as N →∞ ; (ii) the characterization of the minimum values of E0 as a function of the macroscopic opening displacement; (iii) a proof of uniform convergence of the minima of EN for the case of nearest-neighbor interactions; and (iv) a proof of uniform convergence of the derivatives of EN , or tractions, in the same case. The scaling on which the present � -convergence analysis is based has the effect of separating the bulk and surface contributions to the energy. It differs crucially from other scalings employed in the past in that it renders both contributions of the same order.

Finite Propagation Speed for Limited Flux Diffusion Equations

Abstract: We prove that the support of solutions of a limited flux diffusion equation known as a relativistic heat equation evolves at a constant speed, identified as the speed of light c. For that we construct entropy sub- and super-solutions which are fronts evolving at speed c and prove the corresponding comparison principle between entropy solutions and sub- and super-solutions, respectively. This enables us to prove the existence of discontinuity fronts moving at light's speed.

Existence and Stability of Supersonic Euler Flows Past Lipschitz Wedges

Abstract: It is well known that, when the vertex angle of a straight wedge is less than the critical angle, there exists a shock-front emanating from the wedge vertex so that the constant states on both sides of the shock-front are supersonic. Since the shock-front at the vertex is usually strong, especially when the vertex angle of the wedge is large, then a global flow is physically required to be governed by the isentropic or adiabatic Euler equations. In this paper, we systematically study two-dimensional steady supersonic Euler (i.e. nonpotential) flows past Lipschitz wedges and establish the existence and stability of supersonic Euler flows when the total variation of the tangent angle functions along the wedge boundaries is suitably small. We develop a modified Glimm difference scheme and identify a Glimm-type functional, by naturally incorporating the Lipschitz wedge boundary and the strong shock-front and by tracing the interaction not only between the boundary and weak waves, but also between the strong shock-front and weak waves, to obtain the required BV estimates. These estimates are then employed to establish the convergence of both approximate solutions to a global entropy solution and corresponding approximate strong shock-fronts emanating from the vertex to the strong shock-front of the entropy solution. The regularity of strong shock-fronts emanating from the wedge vertex and the asymptotic stability of entropy solutions in the flow direction are also established.

Cauchy Problem for the Vlasov–Poisson–Boltzmann System

Abstract: The dynamics of dilute electrons can be modelled by the fundamental Vlasov–Poisson–Boltzmann system which describes mutual interactions of the electrons through collisions in the self-consistent electric field. In this paper, it is shown that any smooth perturbation of a given global Maxwellian leads to a unique global-in-time classical solution when either the mean free path is small or the background charge density is large. Moreover, the solution converges to the global Maxwellian when time tends to infinity. The analysis combines the techniques used in the study of conservation laws with the decomposition of the Boltzmann equation introduced in [17, 19] by obtaining new entropy estimates for this physical model.

Asymptotic Variational Wave Equations

Abstract: We investigate the equation (ut + (f(u))x)x = f ′′ (u)(ux) 2 /2 where f(u) is a given smooth function. Typically f(u) = u 2 /2 or u 3 /3. This equation models unidirectional and weakly nonlinear waves for the variational wave equation utt − c(u)(c(u)ux)x = 0 which models some liquid crystals with a natural sinusoidal c. The equation itself is also the Euler-Lagrange equation of a variational problem. Two natural classes of solutions can be associated with this equation. A conservative solution will preserve its energy in time, while a dissipative weak solution loses energy at the time when singularities appear. Conservative solutions are globally defined, forward and backward in time, and preserve interesting geometric features, such as the Hamiltonian structure. On the other hand, dissipative solutions appear to be more natural from the physical point of view. We establish the well-posedness of the Cauchy problem within the class of conservative solutions, for initial data having finite energy and assuming that the flux function f has Lipschitz continuous second-order derivative. In the case where f is convex, the Cauchy problem is well-posed also within the class of dissipative solutions. However, when f is not convex, we show that the dissipative solutions do not depend continuously on the initial data.

Global Behavior of Compressible Navier-Stokes Equations with a Degenerate Viscosity Coefficient

Abstract: In this paper, we study a free boundary problem for compressible Navier-Stokes equations with density-dependent viscosity. Precisely, the viscosity coefficient μ is proportional to ρθ with Open image in new window, where ρ is the density, and γ > 1 is the physical constant of polytropic gas. Under certain assumptions imposed on the initial data, we obtain the global existence and uniqueness of the weak solution, give the uniform bounds (with respect to time) of the solution and show that it converges to a stationary one as time tends to infinity. Moreover, we estimate the stabilization rate in L∞ norm, (weighted) L2 norm and weighted H1 norm of the solution as time tends to infinity.

A Γ-Convergence Approach to Stability of Unilateral Minimality Properties in Fracture Mechanics and Applications

Abstract: We prove a stability result for a large class of unilateral minimality properties which arise naturally in the theory of crack propagation proposed by Francfort & Marigo in [14]. Then we give an application to the quasistatic evolution of cracks in composite materials. The main tool in the analysis is a � -convergence result for energies of the form

Semiclassical Limit of the Gross-Pitaevskii Equation in an Exterior Domain

Abstract: In this paper, we study the semiclassical limit of the Gross-Pitaevskii equation (a cubic nonlinear Schrodinger equation) with the Neumann boundary condition in an exterior domain. We prove that before the formation of singularities in the limit system, the quantum density and the quantum momentum converge to the unique solution of the compressible Euler equation with the slip boundary condition as the scaling parameter approaches 0.