Fabio Ancona 
Well-Posedness for General $2/times 2$ Systems of Conservation Laws [PDF ebook] 

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.