On travelling-wave solutions to systems of conservation laws with singular viscosity

Abstract. The method of vanishing artificial viscosity is used to obtain smooth, largedata travelling-wave solutions to a class of conservation laws with semidefinite viscosity. The one-dimensional Navier-Stokes equations serve as an illustrating example.     

Entropy-based nonlinear viscosity for Fourier approximations of conservation laws

An Entropy-based nonlinear viscosity for approximating conservation laws using Fourier expansions is proposed. The viscosity is proportional to the entropy residual of the equation (or system) and thus preserves the spectral accuracy of the method. To cite this article: J.-L. Guermond, R. Pasquetti, C. R. Acad. Sci. Paris, Ser. I 346 (2008).     

An adaptive wavelet viscosity method for hyperbolic conservation laws

We extend the multiscale finite element viscosity method for hyperbolic conservation laws developed in terms of hierarchical finite element bases to a (pre‐orthogonal spline‐)wavelet basis. Depending on an appropriate error criterion, the multiscale framework allows for a controlled adaptive resolution of discontinuities of the solution. The nonlinearity in the weak form is treated by solving a least‐squares data fitting problem. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2008     

Vanishing viscosity solutions for conservation laws with regulated flux

In this paper we introduce a concept of “regulated function” $v(t,x)$ of two variables, which reduces to the classical definition when v is independent of t. We then consider a scalar conservation law of the form $u_t+F{(v(t,x),u)}_x=0$, where F is smooth and v is a regulated function, possibly discontinuous w.r.t. both t and x. By adding a small viscosity, one obtains a well posed parabolic equation. As the viscous term goes to zero, the uniqueness of the vanishing viscosity limit is proved, relying on comparison estimates for solutions to the corresponding Hamilton–Jacobi equation.As an application, we obtain the existence and uniqueness of solutions for a class of $2\times 2$ triangular systems of conservation laws with hyperbolic degeneracy.     

Vanishing viscosity approximation to hyperbolic conservation laws

We study high order convergence of vanishing viscosity approximation to scalar hyperbolic conservation laws in one space dimension. We prove that, under suitable assumptions, in the region where the solution is smooth, the viscous solution admits an expansion in powers of the viscosity parameter ε. This allows an extrapolation procedure that yields high order approximation to the non-viscous limit as ε→0. Furthermore, an integral across a shock also admits a power expansion of ε, which allows us to construct high order approximation to the location of the shock. Numerical experiments are presented to justify our theoretical findings.     

非线性守恒律高阶谱粘性法的收敛性

讨论守恒型方程周期边界问题的高阶谱粘性方法逼近解的收敛性.在逼近解一致有界的假设下,通过建立其高阶导数的上界估计,证明了高阶谱粘性方法逼近解具有同二阶谱粘性方法逼近解相类似的高频衰减性质.以此为基础,用补偿列紧法证明了高阶谱粘性方法逼近解收敛于守恒型方程的物理解.     

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.

     

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.     

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.     

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.     

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).     

Vanishing viscosity with short wave-long wave interactions for multi-D scalar conservation laws

We consider a system coupling a multidimensional semilinear Schrödinger equation and a multidimensional nonlinear scalar conservation law with viscosity, which is motivated by a model of short wave-long wave interaction introduced by Benney (1977). We prove the global existence and uniqueness of the solution of the Cauchy problem for this system. We also prove the convergence of the whole sequence of solutions when the viscosity ε and the interaction parameter α approach zero so that α=o(ε1/2). We also indicate how to extend these results to more general systems which couple multidimensional semilinear systems of Schrödinger equations with multidimensional nonlinear systems of scalar conservation laws mildly coupled.     

Hyperbolic systems of balance laws via vanishing viscosity

Global weak solutions of a strictly hyperbolic system of balance laws in one-space dimension are constructed by the vanishing viscosity method of Bianchini and Bressan. For global existence, a suitable dissipativeness assumption has to be made on the production term g. Under this hypothesis, the viscous approximations u^?, that are globally defined solutions to , satisfy uniform BV bounds exponentially decaying in time. Furthermore, they are stable in L¹ with respect to the initial data. Finally, as ?→0, u^? converges in to the admissible weak solution u of the system of balance laws u_t+(f(u))_x+g(u)=0 when A=Df.     

On infinite-dimensional Keyfitz-Kranzer systems of conservation laws

For infinite-dimensional generalizations of the Keyfitz-Kranzer system of conservation laws in which the unknown vector ranges in an arbitrary Banach space, we single out the class of strong generalized entropy solutions of the Cauchy problem. Existence and uniqueness theorems are proved in this class.     

Relations between symmetries and conservation laws for difference systems

This paper presents a relation between divergence variational symmetries for difference variational problems on lattices and conservation laws for the associated Euler–Lagrange system provided by Noether's theorem. This inspires us to define conservation laws related to symmetries for arbitrary difference equations with or without Lagrangian formulations. These conservation laws are constrained by partial differential equations obtained from the symmetries generators. It is shown that the orders of these partial differential equations have been reduced relative to those used in a general approach. Illustrative examples are presented.