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 $\{u: \nabla \lambda_i \cdot r_i (u) = 0\}$ 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.