Memoirs of the American Mathematical Society 2004; 170 pp; softcover Volume: 169 ISBN10: 0821834355 ISBN13: 9780821834350 List Price: US$71 Individual Members: US$42.60 Institutional Members: US$56.80 Order Code: MEMO/169/801
 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 nonlinear. 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, entropyadmissible 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. Table of Contents  Introduction
 Preliminaries
 Outline of the proof
 The algorithm
 Basic interaction estimates
 Bounds on the total variation and on the interaction potential
 Estimates on the number of discontinuities
 Estimates on shift differentials
 Completion of the proof
 Conclusion
 Bibliography
