Uniqueness failure for entropy solutions of hyperbolic systems of conservation laws

We consider a nonlinear system of conservation laws, which is strictly hyperbolic, genuinely nonlinear in the large, equipped with a convex entropy function and global Riemann invariants. Nevertheless, for such a system for dimension five, it is shown that uniqueness of the similarity solution of a Riemann problem satisfying the entropy condition can fail.     

Stratified solutions for systems of conservation laws

We study a class of weak solutions to hyperbolic systems of conservation (balance) laws in one space dimension, called stratified solutions. These solutions are bounded and ``regular' in the direction of a linearly degenerate characteristic field of the system, but not in other directions. In particular, they are not required to have finite total variation. We prove some results of local existence and uniqueness.

     

Uniqueness and error estimates in first order quasilinear conservation laws via the kinetic entropy defect measure

We give a new uniqueness proof for solutions to quasilinear scalar conservation laws. It is based on the kinetic formulation and does not make use of Kruzkov entropies and doubling of variables. It uses in a fundamental way the entropy defect measure appearing in the kinetic formulation. This measure also plays a central role for proving error estimates that we recast in our simplified approach.     

Lagrangian systems of conservation laws

Summary. We study the mathematical structure of 1D systems of conservation laws written in the Lagrange variable. Modifying the symmetrization proof of systems of conservation laws with three hypothesis, we prove that these models have a canonical formalism. These hypothesis are i) the entropy flux is zero, ii) Galilean invariance, iii) reversibility for smooth solutions. Then we study a family of numerical schemes for the solution of these systems. We prove that they are entropy consistent. We also prove from general considerations the symmetry of the spectrum of the Jacobian matrix. Received December 15, 1999 / Published online February 5, 2001     

Entropy functions for symmetric systems of conservation laws

Using a simple symmetrizability criterion, we show that symmetric systems of conservation laws are equipped with a one-parameter family of entropy functions.     

Stable viscosity matrices for systems of conservation laws

A natural class of appropriate viscosity matrices for strictly hyperbolic systems of conservation laws in one space dimension, u₁ + f(u)_x = 0, u?R^m, is studied. These matrices are admissible in the sense that small-amplitude shock wave solutions of the hyperbolic system are shown to be limits of smooth traveling wave solutions of the parabolic system u_t + f(u)_x = v(Du_x)_x as ifv → 0 if D is in this class. The class is determined by a linearized stability requirement: The Cauchy problem for the equation u₁ + f′(u₀) u_x = vDu_xx should be well posed in L² uniformly in v as v → 0. Previous examples of inadmissible viscosity matrices are accounted for through violation of the stability criterion.     

Decay rates for nonconvex systems of conservation laws

We study decay of solutions for hyperbolic systems of conservation laws which are not genuinely nonlinear. For a generic class of such systems, we determine sharp (algebraic) rates of decay in the total variation of the wave speed, for solutions with compact initial support. Our analysis involves generalized characteristic arguments and the random choice difference scheme of Glimm. © 1993 John Wiley & Sons, Inc.     

Local variational problems and conservation laws

We investigate globality properties of conserved currents associated with local variational problems admitting global Euler–Lagrange morphisms. We show that the obstruction to the existence of a global conserved current is the difference of two conceptually independent cohomology classes: one coming from using the symmetries of the Euler–Lagrange morphism and the other from the system of local Noether currents.     

Discrete shock profiles for systems of conservation laws

The existence of discrete shock profiles for difference schemes approximating a system of conservation laws is the major topic studied in this paper. The basic theorem established here applies to first-order accurate difference schemes; for weak shocks, this theorem provides necessary and sufficient conditions involving the truncation error of the linearized scheme which guarantee entropy satisfying or entropy violating discrete shock profiles. Several explicit difference schemes are used as examples illustrating the interplay between the entropy condition, monotonicity, and linearized stability. Entropy violating stationary shocks for second-order accurate Lax-Wendroff schemes approximating systems are also constructed. The only tools used in the proofs are local analysis and the center manifold theorem.     

高维非线性守恒律方程组

本文根据高维非线性守恒律方程组的研究历程将这一领域的研究大体分为四个阶段: 局部经典解、具扇状波结构弱解、具花状波结构弱解、整体解与混合型方程. 本文据此线索回顾与介绍多年来在该领域所获得的主要成果与进展, 并提出今后所面临的一些未解决的重要问题及困难.     

Two new moving boundary problems for scalar conservation laws

The application of the theory of scalar conservation laws to semiconductor device fabrication is described. This application is the source of a Stefan problem and another moving boundary problem for a class of such equations. The analogue of the Riemann problem for these problems is analyzed and solved. Conditions on the boundary values that characterize physically correct solutions are derived.     

Uniqueness of weak solutions of the Cauchy problem for general 2 × 2 conservation laws

In this paper we prove uniqueness theorems of the Cauchy problem for general 2 × 2 genuinely nonlinear conservation laws and of isentropic gas dynamics equations, not necessarily convex. We consider solutions which are piecewise continuous and have a finite number of centered rarefaction waves in each compact set. We require the solutions to satisfy an extended entropy condition (E) which reduces to Lax's shock inequalities (L) when the system is genuinely nonlinear.     

On first-order conservation laws for systems of hydrodynamic-type equations

First-order conservation laws quadratic in derivatives are considered for systems of hydrodynamic-type equations. Defining relationships for the densities of such conservation laws are derived in a form that is invariant with respect to pointwise changes of the variables. Examples of nondiagonalizable systems admitting quadratic conservation laws are given.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 1, pp. 109–128, July, 1996.     

The Riemann problem for general systems of conservation laws

An extended entropy condition (E) has previously been proposed, by which we have been able to prove uniqueness and existence theorems for the Riemann problem for general 2-conservation laws. In this paper we consider the Riemann problem for general n-conservation laws. We first show how the shock are related to the characteristic speeds. A uniqueness theorem is proved subject to condition (E), which is equivalent to Lax's shock inequalities when the system is “genuinely nonlinear.” These general observations are then applied to the equations of gas dynamics without the convexity condition P_vv(v, s) > 0. Using condition (E), we prove the uniqueness theorem for the Riemann problem of the gas dynamics equations. This answers a question of Bethe. Next, we establish the relation between the shock speed σ and the entropy S along any shock curve. That the entropy S increases across any shock, first proved by Weyl for the convex case, is established for the nonconvex case by a different method. Wendroff also considered the gas dynamics equations without convexity conditions and constructed a solution to the Riemann problem. Notice that his solution does satisfy our condition (E).     

A note on regularity and failure of regularity for systems of conservation laws via Lagrangian formulation

The paper reviews recent progresses on regularity results which have been studied since Ole?nik and Schaeffer. It also outlines a limit introducing heuristically an original counterexample obtained with L. Spinolo.