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). Vice versa, 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 (1) with small initial
data, for systems satisfying the above assumption.

Readership

Graduate students and research mathematicians interested in
partial differential equations.