On Sobolev Regularizations of Hyperbolic Conservation Laws

1. INTRODUCI7ON We study certain Sobolev-type regularizations of the hyperbolic conservation laws that add terms simulating both dissipative and dispersive processes. These evolution equations have the form with the auxiliary specification The behaviour in L'(R) of the solutions of problem (S), as well as the value of (S) as an approximation to problem (C), is studied. Convergence results, with error estimates, are given as v and tend to zero. In a companion paper [19], finite-difference discretizations of (S) are studied as an approximation for (C). As a tool to study nonlinear evolution equations posed in L1(R), it is shown that any nonlinear mapping from L1(R) to itself t h a t preserves the integral, is a contraction, and commutes with translations satisfies a maximum principle. This lemma gives necessary and sufficient conditions that solutions of (S) satisfy a maximum principle, despite the dispersive nature of (S).