Well-Posedness for General 2 x 2 Systems of Conservation Laws

We consider the Cauchy problem for a strictly hyperbolic $2\\times 2$ system of conservation laws in one space dimension $u_t+[F(u)]_x=0, u(0,x)=\\bar u(x),$ which is neither linearly degenerate nor genuinely non-linear. We make the following assumption on the characteristic fields. If $r_i(u), \\i=1,2,$ denotes the $i$-th right eigenvector of $DF(u)$ and $\\lambda_i(u)$ the corresponding eigenvalue, then the set $\\{u:
abla \\lambda_i \\cdot r_i (u) = 0\\}$ is a smooth curve in the $u$-plane that is transversal to the vector field $r_i(u)$. Systems of conservation laws that fulfill such assumptions arise in studying elastodynamics or rigid heat conductors at low temperature.For such systems we prove the existence of a closed domain $\\mathcal{D} \\subset L^1,$ containing all functions with sufficiently small total variation, and of a uniformly Lipschitz continuous semigroup $S:\\mathcal{D} \\times [0,+\\infty)\\rightarrow \\mathcal{D}$ with the following properties. Each trajectory $t \\mapsto S_t \\bar u$ of $S$ is a weak solution of (1). Viceversa, if a piecewise Lipschitz, entropic solution $u= u(t,x)$ of (1) exists for $t \\in [0,T],$ then it coincides with the trajectory of $S$, i.e. $u(t,\\cdot) = S_t \\bar u. This result yields the uniqueness and continuous dependence of weak, entropy-admissible solutions of the Cauchy problem with small initial data, for systems satysfying the above assumption.