Athanasios E. Tzavaras

Contractive relaxation systems and the scalar multidimensional conservation law

It is a classical result, Kruzhkov, that the Cauchy problem for the scalar multidimensional conservation law, with u{sub 0} {element_of} L{sup 1}(R{sup n}) {intersection} L{sup {infinity}}(R{sup n}) has a unique global weak solution u(x, t) satisfying the Kruzhkov entropy conditions Weak solutions of are constructed via finite difference approximations, Conway and Smoller, or as zero-viscosity limits of parabolic regularizations, Volpert and Kruzhkov, and the solution operator defines a contraction semigroup in L{sup 1}(R{sup n}), Crandall. 28 refs.

Representation of weak limits and definition of nonconservative products

The goal of this article is to show that the notion of generalized graphs is able to represent the limit points of the sequence { g(un), dun} in the weak-

Wave interactions and variation estimates for self-similar zero-viscosity limits in systems of conservation laws

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KINETIC DECOMPOSITION OF APPROXIMATE SOLUTIONS TO CONSERVATION LAWS: APPLICATION TO RELAXATION AND DIFFUSION-DISPERSION APPROXIMATIONS*

topology of measures when {un} is a sequence of continuous functions of uniformly bounded variation. The representation theorem induces a natural definition for the nonconservative product g(u), du in a BV context. Several existing definitions of nonconservative products are then compared, and the theory is applied to provide a notion of solutions and an existence theory to the Riemann problem for quasi-linear, strictly hyperbolic systems.

Weak solutions for a nonlinear system in viscoelasticity

We consider the problem of self-similar zero-viscosity limits for systems ofN conservation laws. First, we give general conditions so that the resulting boundary-value problem admits solutions. The obtained existence theory covers a large class of systems, in particular the class of symmetric hyperbolic systems. Second, we show that if the system is strictly hyperbolic and the Riemann data are sufficiently close, then the resulting family of solutions is of uniformly bounded variation and oscillation. Third, we construct solutions of the Riemann problem via self-similar zero-viscosity limits and study the structure of the emerging solution and the relation of self-similar zero-viscosity limits and shock profiles. The emerging solution consists ofN wave fans separated by constant states. Each wave fan is associated with one of the characteristic fields and consists of a rarefaction, a shock, or an alternating sequence of shocks and rarefactions so that each shock adjacent to a rarefaction on one side is a contact discontinuity on that side. At shocks, the solutions of the self-similar zero-viscosity problem have the internal structure of a traveling wave.

DIFFUSIVE N-WAVES AND METASTABILITY IN THE BURGERS EQUATION ∗

*Research partially supported by the National Science Foundation.

Weak–Strong Uniqueness of Dissipative Measure-Valued Solutions for Polyconvex Elastodynamics

We consider a one-dimensional model problem for the motion of a viscoelastic material with fading memory governed by a quasilinear hyperbolic system of integrodifferential equations of Volterra type. For given Cauchy data in , we use the method of vanishing viscosity and techniques of compensated compactness to obtain the existence of a weak solution (in the class of bounded measurable functions) in a special case

Stability of discontinuous steady states in shearing motions of a non-Newtonian fluid

We study the effect of viscosity on the large time behavior of the viscous Burgers equation by using a transformed version of Burgers (in self-similar variables) that captures efficiently the mechanism of transition to the asymptotic states and allows us to estimate the time of evolution from an N-wave to the final stage of a diffusion wave. Then we construct certain special solutions of diffusive N-waves with unequal masses. Finally, using a set of similarity variables and a variant of the Cole-Hopf transformation, we obtain an integrated Fokker-Planck equation. The latter is solvable and provides an explicit solution of the viscous Burgers equation in a series of Hermite polynomials. This format captures the long-time-small-viscosity interplay, as the diffusion wave and the diffusive N-waves correspond, respectively, to the first two terms in the Hermite polynomial expansion.

Stability and Convergence of a Class of Finite Element Schemes for Hyperbolic Systems of Conservation Laws

For the equations of elastodynamics with polyconvex stored energy, and some related simpler systems, we define a notion of a dissipative measure-valued solution and show that such a solution agrees with a classical solution with the same initial data, when such a classical solution exists. As an application of the method we give a short proof of strong convergence in the continuum limit of a lattice approximation of one dimensional elastodynamics in the presence of a classical solution. Also, for a system of conservation laws endowed with a positive and convex entropy, we show that dissipative measure-valued solutions attain their initial data in a strong sense after time averaging.

Convergence of relaxation schemes to the equations of elastodynamics

Abstract : This document studies the nonlinear stability of discontinuous steady states of a model initial-boundary value problem in one space dimension for incompressible, isothermal shear flow of a non-Newtonian fluid driven by a constant pressure gradient. The non-Newtonian contribution to the shear stress is assumed to satisfy a simple differential constitutive law. The key feature is a non-monotone relation between the total steady shear stress and shear strain- rate that results in steady states having, in general, discontinuities in the strain rate. It is shown that every solution tends to a steady state as t approaches limit of infinity, and we identify steady states that are stable.