## 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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