STABILITY OF RAREFACTION WAVES IN VISCOUS MEDIA

We study the time-asymptotic behavior of weak rarefaction waves of sys tems of conservation laws describing one-dimensional viscous media, wi th strictly hyperbolic flux functions. Our main result is to show that solutions of perturbed rarefaction data converge to an approximate, ' 'Burgers'' rarefaction wave, for initial perturbations w(o) with small mass and localized as w(o)(x)= O(\x\(-1)). The proof proceeds by iter ation of a pointwise ansatz for the error, using integral representati ons of its various components, based on Green's functions. We estimate the Green's functions by careful use of the Hopf-Cole transformation, combined with a refined parametrix method. As a consequence of our me thod, we also obtain rates of decay and detailed pointwise estimates f or the error. This pointwise method has been used successfully in stud ying stability of shock and constant-state solutions. New features in the rarefaction case are time-varying coefficients in the linearized e quations and error waves of unbounded mass O(log(t)). These ''diffusio n waves'' have amplitude O(t(-1/2) log t) in linear degenerate transve rsal fields and O(t(-1/2)) in genuinely nonlinear transversal fields, a distinction which is critical in the stability proof.