Existence and Stability of Multidimensional Shock Fronts in the Vanishing Viscosity Limit
Résumé
In this paper we present a new approach to the study of linear and nonlinear stability of inviscid multidimensional shock waves under small viscosity perturbation, yielding optimal estimates and eventually an extension to the viscous case of the celebrated theorem of Majda on existence and stability of multidimensional shock waves. More precisely, given a curved Lax shock solution u0 to a hyperbolic system of conservation laws, we construct nearby viscous shock solutions uε to a parabolic viscous perturbation of the hyperbolic system which converge to u0 as viscosity ε → 0 and satisfy an appropriate (conormal) version of Majda's stability estimate. The main new feature of the paper is the derivation of maximal and optimal estimates for the linearization of the parabolic problem about a highly singular approximate solution. These estimates are more robust than the singular estimates obtained in our previous work, and permit us to remove an earlier assumption limiting how much the inviscid shock we start with can deviate from flatness. The key to the new approach is to work with the full linearization of the parabolic problem, that is, the linearization with respect to both uε and the unknown viscous front, and to allow variation of the front at all stages – not only in the construction of the approximate solution as done in previous work, but also in the final error equation. After reformulating the problem as a transmission problem, we show that the linearized problem can be desingularized and optimal estimates obtained by adding an appropriate extra boundary condition involving the front. The extra condition determines a local evolution rule for the viscous front.