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Well-posedness of the Cauchy problem for \(n\times n\) systems of conservation laws. (English) Zbl 0958.35001

Mem. Am. Math. Soc. 694, 134 p. (2000).
Authors’ abstract: This paper is concerned with the initial value problem for a strictly hyperbolic \(n\times n\) system of conservation laws in one space dimension: \[ u_t+\bigl[F(x)\bigr]_x=0,\quad u(0,x)=\overline u(x). \tag{*} \] Each characteristic field is assumed to be either linearly degenerate or genuinely nonlinear. We prove that there exist a domain \({\mathcal D}\subset L^1\), containing all functions with sufficiently small total variation, and a uniformly Lipschitz continuous semigroup \(S:{\mathcal D}\times[0,\infty [\mapsto {\mathcal D}\) with the following properties. Every trajectory \(t\mapsto u(t,\cdot) =S_t \overline u\) of the semigroup is a weak, entropy-admissible solution of (*). Viceversa, if a picewise Lipschitz, entropic solution \(u=u(t,x)\) of (*) exists for \(t\in[0,T]\), then it coincides with the semigroup trajectory, i.e. \(u(t,\cdot)=S_t\overline u\). For a given domain \({\mathcal D}\), the semigroup \(S\) with the above properties is unique.
These results yield the uniqueness, continuous dependence and global stability of weak, entropy-admissible solutions of the Cauchy problem (*), for general \(n\times n\) systems of conservation laws, with small initial data.
Reviewer: A.Doktor (Praha)

MSC:

35-02 Research exposition (monographs, survey articles) pertaining to partial differential equations
35L65 Hyperbolic conservation laws
35L67 Shocks and singularities for hyperbolic equations
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