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\(L^1\)-theory of scalar conservation law with continuous flux function. (English) Zbl 0944.35048

The authors study uniqueness of a generalized entropy solution (g.e.s.) to the Cauchy problem for \(N\)-dimensional scalar conservation law \(u_t+\text{div}_xF(u)=g\), \(u(.,0)=h\), where the flux function \(F\) is only continuous. For data \((h,g)\) vanishing at infinity, it is shown that there exists a maximal and a minimal g.e.s. to the Cauchy problem and to the associated stationary problem \(u+\text{div}_xF(u) =h\). More precisely, \(h\) is supposed to be of the form \(h=c+h_0\), \(c\in\mathbb{R}\), \(h_0\in L^\infty_0(\mathbb{R}^N)\). Then the minimal and the maximal solutions coincide except for a countable set of values of \(c\) depending on \(F\), \(h_0\) and \(g\). In the case of \(L^1\) data it is proved that there is uniqueness for all data of g.e.s. to the Cauchy problem iff there is uniqueness for all data of g.e.s. to the related stationary problem. The proof uses nonlinear semigroup theory. The results are applied on the dimension \(N>1\) and extends the uniqueness results for flux having some monotonicity properties.
Reviewer: A.Doktor (Praha)

MSC:

35L65 Hyperbolic conservation laws
35L45 Initial value problems for first-order hyperbolic systems
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