A Variational Approximation Scheme for¶Three-Dimensional Elastodynamics¶with Polyconvex Energy

We construct a variational approximation scheme for the equations of three-dimensional elastodynamics with polyconvex stored energy. The scheme is motivated by some recently discovered geometric identities (Qin [18]) for the null Lagrangians (the determinant and cofactor matrix), and by an associated embedding of the equations of elastodynamics into an enlarged system which is endowed with a convex entropy. The scheme decreases the energy, and its solvability is reduced to the solution of a constrained convex minimization problem. We prove that the approximating process admits regular weak solutions, which in the limit produce a measure-valued solution for polyconvex elastodynamics that satisfies the classical weak form of the geometric identities. This latter property is related to the weak continuity properties of minors of Jacobian matrices, here exploited in a time-dependent setting. Accepted November 18, 2000?Published online April 23, 2001     

Unique Global Solvability for¶Initial-Boundary Value Problems¶in One-Dimensional Nonlinear Thermoviscoelasticity

The balance laws of mass, momentum and energy are considered for a broad class of one-dimensional nonlinear thermoviscoelastic materials. For the initial-boundary value problem corresponding to pinned endpoints held at constant temperature, we establish existence and uniqueness of temporally global classical solutions for initial data of unrestricted size. Our approach also applies to all boundary conditions involving pinned or stress-free endpoints which are either held at constant temperature or insulated. An additional and novel feature of the theory is that solid-like and gaseous materials are treated in a unified way. Accepted: June 24, 1999     

Existence and Uniqueness of Solutions¶of an Asymptotic Equation¶Arising from a Variational Wave Equation¶with General Data

We establish here the global existence and uniqueness of admissible (both dissipative and conservative) weak solutions to a canonical asymptotic equation () for weakly nonlinear solutions of a class of nonlinear variational wave equations with any L ²(ℝ) initial datum. We use the method of Young measures and mollification techniques. Accepted April 25, 2000?Published online November 16, 2000     

A Variational Formulation of¶Rate-Independent Phase Transformations¶Using an Extremum Principle

We propose a rate-independent, mesoscopic model for the hysteretic evolution of phase transformations in shape-memory alloys. The model uses the deformation and phase-indicator function as basic unknowns and the potentials for the elastic energy and for the dissipation as constitutive laws. Using the associated functionals, admissible processes are defined to be the ones which are stable at all times and which satisfy the energy inequality.This concept leads to a natural time-incremental method which consists in a minimization problem. The mesoscopic model is obtained by a relaxation procedure. It leads to new functionals involving the cross-quasiconvexification of the elastic stored-energy density. For a special case involving two phases of linearized elastic materials we show that the incremental problem provides existence of admissible processes for the time-continuous problem, if we let the time-step go to 0. Dedicated to Erwin Stein on the occasion of his seventiethbirthday     

Isentropic Approximation¶of the Compressible Euler System¶in One Space Dimension

Using the stability results of Bressan & Colombo [BC] for strictly hyperbolic 2 × 2 systems in one space dimension, we prove that the solutions of isentropic and non-isentropic Euler equations in one space dimension with the respective initial data (ρ₀, u ₀) and (ρ₀, u ₀, &\theta;₀=ρ₀ ^{γ− 1}) remain close as soon as the total variation of (ρ₀, u ₀) is sufficiently small. Accepted April 25, 2000?Published online November 24, 2000     

Homogenization of Polycrystalline¶Aggregates

The effective elastic properties of a polycrystalline material depend on the single crystal elastic constants of the crystallites comprising the polycrystal and on the manner in which the crystallites are arranged. In this paper we apply the techniques of homogenization to put the problem of determining effective elastic constants in a precise mathematical framework that permits us to derive an expression for the effective elasticity tensor. We also study how the homogenized elasticity tensor changes as the probability characterizing the ensemble changes. Under the assumption that the field of orientations of the crystallographic axes of the crystallites is an independent random field, we show that our theory is compatible with the formulation used in texture analysis. In particular, we are able to prove that the physical assumption made by [10] in his study of weakly-textured polycrystals holds true. In addition, we establish some elementary bounds on the material constants that characterize the effective elasticity tensor of weakly-textured orthorhombic aggregates of cubic crystallites. Accepted: (June 15, 1999)     

Universal Attractors¶for the Navier-Stokes Equations¶of Compressible and Heat-Conductive Fluid¶in Bounded Annular Domains in ℝn

This paper is concerned with the dynamics for the Navier-Stokes equations for a polytropic viscous heat-conductive ideal gas in bounded annular domains Ω _n in ? ⁿ (n= 2, 3). One of the important features of this problem is that the metric spaces H ⁽¹⁾ and H ⁽²⁾ we work with are two incomplete metric spaces, as can be seen from the constraints θ >0 and u> 0, withθ and u being absolute temperature and specific volume respectively. For any constants δ₁, δ₂, δ₃, δ₄, δ₅ satisfying certain conditions, two sequences of closed subspaces H ^{( i )} _δ?H ^{( i )} (i= 1,2) are found, and the existence of two (maximal) universal attractors in H ⁽¹⁾ _δ and H ⁽²⁾ _δ is proved.     

Constitutive Relations for Monatomic Gases¶Based on a¶Generalized Rational Approximation¶to the Sum of the Chapman-Enskog Expansion

. We study the asymptotic behavior as time goes to infinity of solutions to the initial‐boundary‐value problem on the half space for a one‐dimensional model system for the isentropic flow of a compressible viscous gas, the so‐called p‐system with viscosity. As boundary conditions, we prescribe the constant state at infinity and require that the velocity be zero at the boundary . When the velocity at infinity is negative and satisfies a condition on the magnitude, we prove that if the initial data are suitably close to those for the corresponding outgoing viscous shock profile, which is suitably far from the boundary, then a unique solution exists globally in time and tends toward the properly shifted viscous shock profile as the time goes to infinity. The proof is given by an elementary energy method. (Accepted March 2, 1998)     

Initial Layers and Uniqueness of¶Weak Entropy Solutions to¶Hyperbolic Conservation Laws

We consider initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws through the scalar case. The entropy solutions we address assume their initial data only in the sense of weak-star in L ^∞ as t→0₊ and satisfy the entropy inequality in the sense of distributions for t>0. We prove that, if the flux function has weakly genuine nonlinearity, then the entropy solutions are always unique and the initial layers do not appear. We also discuss applications to the zero relaxation limit for hyperbolic systems of conservation laws with relaxation. Accepted: October 26, 1999     

Spinodal Decomposition for the¶Cahn-Hilliard Equation in Higher Dimensions:¶Nonlinear Dynamics

This paper addresses the phenomenon of spinodal decomposition for the Cahn-Hilliard equation

Network Approximation in the Limit of¶Small Interparticle Distance of the¶Effective Properties of a High-Contrast¶Random Dispersed Composite

We consider a high-contrast two-phase composite such as a ceramic/polymer composite or a fiberglass composite. Our objective is to determine the dependence of the effective conductivity (or the effective dielectric constant or the effective shear modulus) of the composite on the random locations of the inclusions (ceramic particles or fibers) when the concentration of the inclusions is high. We consider a two-dimensional model and show that the continuum problem can be approximated by a discrete random network (graph). We use variational techniques to provide rigorous mathematical justification for this approximation. In particular, we have shown asymptotic equivalence of the effective constant for the discrete and continuum models in the limit when the relative interparticle distance goes to zero. We introduce the geometrical interparticle distance parameter using Voronoi tessellation, and emphasize the relevance of this parameter due to the fact that for irregular (non-periodic) geometries it is not uniquely determined by the volume fraction of the inclusions. We use the discrete network to compute numerically. For this purpose we employ a computer program which generates a random distribution of disks on the plane. Using this distribution we obtain the corresponding discrete network. Furthermore, the computer program provides the distribution of fluxes in the network which is based on Keller's formula for two closely spaced disks. We compute the dependence of on the volume fraction of the inclusions V for monodispersed composites and obtaine results which are consistent with the percolation theory predictions. For polydispersed composites (random inclusions of two different sizes) the dependence is not simple and is determined by the relative volume fraction V _r of large and small particles. We found some special values of V _r for which the effective coefficient is significantly decreased. The computer program which is based on our network model is very efficient and it allows us to collect the statistical data for a large number of random configurations.     

On Global Infinite Energy Solutions¶to the Navier-Stokes Equations¶in Two Dimensions

This paper studies the bidimensional Navier–Stokes equations with large initial data in the homogeneous Besov space . As long as r,q < +∞, global existence and uniqueness of solutions are proved. We also prove that weak–strong uniqueness holds for the d-dimensional equations with data in L ²(? ^d ) for d/r+ 2/q≥ 1.     

Solitons in Several Space Dimensions:¶Derrick's Problem and¶Infinitely Many Solutions

In this paper we study a class of Lorentz invariant nonlinear field equations in several space dimensions. The main purpose is to obtain soliton-like solutions. These equations were essentially proposed by C. H. Derrick in a celebrated paper in 1964 as a model for elementary particles. However, an existence theory was not developed. The fields are characterized by a topological invariant, the charge. We prove the existence of a static solution which minimizes the energy among the configurations with nontrivial charge. Moreover, under some symmetry assumptions, we prove the existence of infinitely many solutions, which are constrained minima of the energy. More precisely, for every n∈:N there exists a solution of charge n. Accepted March 13, 2000?Published online September 12, 2000     

Heteroclinic Orbits and Bernoulli Shift¶for the Elliptic Collision Restricted¶Three-Body Problem

We consider two mass points of masses m ₁=m ₂= moving under Newton's law of gravitational attraction in a collision elliptic orbit while their centre of mass is at rest. A third mass point of mass m ₃≈ 0, moves on the straight line L, perpendicular to the line of motion of the first two mass points and passing through their centre of mass. Since m ₃≈ 0, the motion of the masses m ₁ and m ₂ is not affected by the third mass, and from the symmetry of the motion it is clear that m ₃ will remain on the line L. So the three masses form an isosceles triangle whose size changes with the time. The elliptic collision restricted isosceles three-body problem consists in describing the motion of m ₃. In this paper we show the existence of a Bernoulli shift as a subsystem of the Poincaré map defined near a loop formed by two heteroclinic solutions associated with two periodic orbits at infinity. Symbolic dynamics techniques are used to show the existence of a large class of different motions for the infinitesimal body. Accepted July 6, 2000?Published online February 14, 2001     

Global Existence in Critical Spaces¶for Flows of Compressible¶Viscous and Heat-Conductive Gases

We are concerned with global existence and uniqueness of strong solutions for a general model of viscous and heat-conductive gases. The initial data are supposed to be close to a stable equilibrium with constant density and temperature. Using uniform estimates for the linearized system with a convection term, we get global well-posedness in a functional setting invariant with respect to the scaling of the associated equations (in space dimension N≧3). We also show a smoothing effect on the velocity and the temperature, and a decay on the difference between the density and the constant reference state. These results extend a previous paper devoted to the barotropic case (see [5]).     

From the Boltzmann Equations¶to the Equations of¶Incompressible Fluid Mechanics, II

We consider here the problem of deriving rigorously from renormalized solutions of Boltzmann's equation, globally in time, for general initial conditions and without any additional assumption, solutions of Stokes' equations (together with the strong Boussinesq relation). We also obtain similar results for Euler equations where, however, we need to make an assumption on the high velocities of the solutions of Boltzmann's equation.     

Homogenization for¶Stochastic Hamilton-Jacobi Equations

Homogenization asks whether average behavior can be discerned from partial differential equations that are subject to high-frequency fluctuations when those fluctuations result from a dependence on two widely separated spatial scales. We prove homogenization for certain stochastic Hamilton-Jacobi partial differential equations; the idea is to use the subadditive ergodic theorem to establish the existence of an average in the infinite scale-separation limit. In some cases, we also establish a central limit theorem. Accepted: (April 23, 1999)     

Maps of Convex Sets and Invariant Regions¶for Finite-Difference Systems¶of Conservation Laws

General results about maps of convex sets in ? ⁿ are proved. We outline their extensions to an infinite-dimensional context. Such extensions have applications in nonlinear analysis such as in the study of the invariance of convex sets under nonlinear maps. Here, we explore applications only in the finite-dimensional context. More specifically, we apply the general results to the problem of finding sufficient conditions for a region of the state space to be globally or locally invariant under finite-difference schemes applied to systems of conservation laws in several space variables. In particular, we establish a final characterization of the invariant regions under the Lax-Friedrichs scheme and also give sufficient conditions for the local invariance. Further, we give sufficient conditions for the global and local invariance of regions under flux-splitting finite-difference schemes. An example of the multi-dimensional Euler equations for non-isentropic gas dynamics is discussed.     

Variational Problems¶on Multiply Connected Thin Strips II:¶Convergence of the Ginzburg-Landau

Let M be a planar embedded graph whose arcs meet transversally at the vertices; Let ?(M) be a strip-shaped domain around M, of width M except in a neighborhood of the singular points. Assume that the boundary of ?(M) is smooth. We consider the Ginzburg-Landau energy functional for superconductivity on ?(M). We prove that its minimizers converge in a suitable sense to the minimizers of a simpler functional on M. The supercurrents in ?(M) are shown to converge to one-dimensional currents in M.     

Existence of a Global Smooth Solution¶for a Degenerate Goursat Problem¶of Gas Dynamics

The Goursat problem of a mixed type equation , P≥ 0, is considered. At the ends of its supports we have P=0, which means it is degenerate hyperbolic. We prove the global existence of a smooth solution to the degenerate Goursat problem up to a boundary where P=0. This problem comes from the expansion of a wedge of gas with constant velocity into vacuum, in two-dimensional pressure-gradient equations in gas dynamics, where P is the pressure and P=0 means vacuum. Accepted June 16, 2000?Published online December 6, 2000