Sup-Norm Estimates in Glimm's Method.
Abstract
A system of two conservation laws in one dimension is a set of first order nonlinear partial differential equations of a certain form (1) where (u,v) is a vector function of (x,t), x epsilon R, t > or = 0. The Cauchy problem asks for a solution of (1) given the initial values of u and v at time t = 0. Equations of type (1) arise, for example, in gas dynamics where they express the conservation of quantities like mass, momentum and energy, when diffusion is neglected. Typically, smooth solutions of (1) cannot be found. This is due to the formation of shock waves. Shock waves are the mechanism by which entropy is dissipated in solutions of (1). This paper gives a proof that solutions exist even after shock waves form, so long as the amplitude of the waves are not too great initially. Keywords: Riemann invariants; Conservation laws; Stability.
Document Details
- Document Type
- Technical Report
- Publication Date
- Aug 01, 1985
- Accession Number
- ADA160981
Entities
People
- Blake Temple
Organizations
- University of Wisconsin–Madison