Well-posedness of the Cauchy problem for nxn Systems of Conservation Laws

This paper is concerned with the initial value problem for a strictly hyperbolic $n\times n$ system of conservation laws in one space dimension: \begin{equation}u_t+\big[ F(u)\big]_x=0,\qquad\quad u(0,x)=\bar u(x).\label{aaa}\end{equation} 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.