Sup-Norm Estimates in Glimm's Method.
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WISCONSIN UNIV-MADISON MATHEMATICS RESEARCH CENTER
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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.
- Numerical Mathematics