Measure-valued solutions to conservation laws. (English) Zbl 0616.35055

The topic of discussion is ”weak solutions” to the Cauchy problem for (i) \(F(u)=0\) where \(F(u(x,t))=(\partial u/\partial t)^{(x,t)}+(\partial /\partial x)(fu(x,t))\) and to the associated Cauchy problem for (ii) \(F(u)=\epsilon \partial^ 2u/\partial x^ 2\) and (iii) \(F(u)=\epsilon \partial^ 3u/\partial x^ 3\) all with the same Cauchy data: \(u(x,0)=u_ 0(x)\) where \(u_ 0: {\mathbb{R}}\to {\mathbb{R}}^ n\) and \(f: {\mathbb{R}}^ n\to {\mathbb{R}}^ n\) are given functions, \(\epsilon\) is a positive parameter and the unknown u has domain \({\mathbb{R}}\times [0,\infty)\) and takes values in \({\mathbb{R}}^ n\) in some weak sense. There are various concepts of weak solutions such as locally Lipschitz functions that satisfy the equation a.e., distributional solutions where the equation is required to be satisfied in the topology of the underlying function space and so on. In earlier papers the author considered another concept of weak solutions (viz. that of measurevalued solutions) generalizing the usual notion of distributional solutions. With this concept he has been able to prove the convergence of the so- called viscosity solutions of (ii) and (iii) to solutions of (i) when the parameter \(\epsilon\) tends to zero with an appropriate measure theoretic notion of convergence.
The present paper constitutes a more detailed analysis of the class of measure valued solutions which, it turns out, could be regarded as a convex subset of an appropriate Banach space. The concept of extreme solutions and their existence follow from this convexity via the Krein- Milman theorem.
Reviewer: P.Ramankutty


35L65 Hyperbolic conservation laws
35L45 Initial value problems for first-order hyperbolic systems
35D05 Existence of generalized solutions of PDE (MSC2000)
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