Existence and stability of multidimensional shock fronts in the vanishing viscosity limit. (English) Zbl 1072.35122

This interesting paper is devoted to the qualitative theory of shock waves resulting in \(N\times N\)-hyperbolic systems of partial differential equations describing conservation laws, i.e. \[ \partial_t u+\sum\limits_{j=1}^{d}\partial_jf_j(u)=0. \] Next, consider a parabolic viscous perturbation of that system \[ \partial_t u+\sum\limits_{j=1}^{d}\partial_jf_j(u)-\varepsilon \sum\limits_{j,k=1}^{d}\partial_j(B_{j,k}(u)\partial_ku)=0, \] where \(\varepsilon >0\) is a small number describing the rate of viscosity. The authors present a new approach to the study of linear and nonlinear stability of inviscid multidimensional shock waves under small viscosity perturbation. They obtain optimal estimates and an extension to the viscous case of the celebrated theorem of Majda on existence and stability of multidimensional shock waves.
The main new idea is the derivation of maximal and optimal estimates for the linearization of the parabolic problem about a highly singular approximate solution. The key to the new approach is to use the full linearization of the parabolic problem. A transmission problem is obtained. It is shown that the linearized problem can be desingularized. Then optimal estimates are obtained by adding an appropriate extra boundary condition involving the front. This condition determines a local evolution rule for the viscous front. For the convenience of the reader, the authors collect the results about para-differential calculus which are used to prove the linear stability estimates.


35L67 Shocks and singularities for hyperbolic equations
35L65 Hyperbolic conservation laws
35L45 Initial value problems for first-order hyperbolic systems
35L60 First-order nonlinear hyperbolic equations
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