Well-posedness of the Cauchy problem for nxn systems of conservation laws

Alberto Bressan, Graziano Crasta, and Benedetto Piccoli

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:
$$u_t+\big[ F(u)\big]_x=0,\qquad\quad u(0,x)=\bar u(x).\eqno(*)$$
Each characteristic field is assumed to be either linearly degenerate or genuinely nonlinear. We prove that there exists a domain $\D\subset\L^1$, containing all functions with sufficiently small total variation, and a uniformly Lipschitz continuous semigroup $S:\D\times [0,\infty[\,\mapsto\D$ with the following properties. Every trajectory $t\mapsto u(t,\cdot)=S_t\bar u$ of the semigroup is a weak, entropy-admissible solution of (*). Viceversa, if a piecewise Lipschitz, entropic solution $u=u(t,x)$ of ($*$) exists for $t\in [0,T]$, then it coincides with the semigroup trajectory, i.e. $u(t,\cdot)=S_t\bar u$. For a given domain $\D$, 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.

Paper:
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Title:
Well-posedness of the Cauchy problem for nxn systems of conservation laws
Author(s):
Alberto Bressan, <bressan@sissa.it>
Graziano Crasta,
Benedetto Piccoli, <piccoli@sissa.it>
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