Well-posedness for General 2x2 Systems of Conservation Laws

ABSTRACT. We consider the Cauchy problem for a strictly hyperbolic 2x2 system of conservation laws in one space dimension u_t+[F(u)]_x=0, u(0,x)=u_0(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 : D lambda_i . 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 D in L^1, containing all functions with sufficiently small total variation, and of a uniformly Lipschitz continuous semigroup S: Dx [0,+infty) --->D with the following properties. Each trajectory t --> S_t (u_0) of S is a weak solution of (1). Viceversa, if a piecewise Lipschitz, entropic solution u= u(t,x) of (1) exists for t