Scalar conservation laws seen as gradient flows: known results and new perspectives

Viscous Conservation Laws, Part I: Scalar Laws

Viscous conservation laws are the basic models for the dissipative phenomena. We aim at a systematic presentation of the basic ideas for the quantitative study of the nonlinear waves for viscous conservation laws. The present paper concentrates on the scalar laws; an upcoming Part II will deal with the systems. The basic ideas for scalar viscous conservation laws originated from two sources: th...

Discrete and continuous scalar conservation laws

The MIT Faculty has made this article openly available. Please share how this access benefits you. Your story matters. Abstract Motivated by issues arising for discrete second-order conservation laws and their continuum limits (applicable, for example, to one-dimensional nonlinear spring-mass systems), here we study the corresponding issues in the simpler setting of first-order conservation law...

Regularity through Approximation for Scalar Conservation Laws∗

In this paper it is shown that recent approximation results for scalar conservation laws in one space dimension imply that solutions of these equations with smooth, convex fluxes have more regularity than previously believed. Regularity is measured in spaces determined by quasinorms related to the solution’s approximation properties in L1(R) by discontinuous, piecewise linear functions. Using a...

L2 formulation of multidimensional scalar conservation laws

Oscillatory Waves in Discrete Scalar Conservation Laws

We study Hamiltonian difference schemes for scalar conservation laws with monotone flux function and establish the existence of a three-parameter family of periodic travelling waves (wavetrains). The proof is based on an integral equation for the dual wave profile and employs constrained maximization as well as the invariance properties of a gradient flow. We also discuss the approximation of w...