Proceedings of Symposia in Applied Mathematics

Multidimensional shock waves and surface waves

Sylvie

Benzoni-Gavage∗

and Jean-Fran¸ cois Coulombel

Abstract. The theory of weakly stable multidimensional shocks is reviewed

from two diﬀerent points of view: a completely nonlinear one, which yields

the astonishing result – to the price of a huge machinery – that weak stability

implies nonlinear well-posedness, and a weakly nonlinear one, which leads

formally to an amplitude equation for perturbations of linear surface waves, in

the form of a nonlocal Burgers equation for which a well-posedness condition

is investigated.

Introduction

This paper is concerned with special, piecewise smooth solutions of multidimen-

sional hyperbolic systems of conservation laws, namely solutions that are smooth

on either side of an interface of discontinuity, which we shall call shock-wave solu-

tions. The mathematical theory of shock-wave solutions dates back to the seminal

work of Majda in the 1980’s [Maj83b, Maj83a, Maj84] and was developed by

M´ etivier and co-workers (see [M´ et01] and references therein). In the present work

we are interested in shock-wave solutions that are perturbations of weakly stable

reference planar shock-wave solutions. Such shocks arise for instance in ‘real’ gas

dynamics (but not in perfect gases, see [BGS07, ch. 15]), and also, if we allow

undercompressive shocks, in ideal liquid-vapor flows [BG98]. In the first part

we review the theory and in particular explain what ‘weakly stable’ means in the

context of constant-coeﬃcient linearized problems, in terms of so-called neutral

modes. The second part gives an overview of recent results obtained by Coulombel

and Secchi [Cou04, CS04, CS08] regarding linear and nonlinear stability in the

presence of neutral modes. A striking fact is that weak spectral stability is even-

tually suﬃcient to obtain the well-posedness of the fully nonlinear, free boundary

value problem. This is a highly nontrivial extension of Madja’s main result in

[Maj83a], which required the stronger assumption of uniform stability (a condition

analogous to the uniform Kreiss-Lopatinski˘ı condition for classical initial boundary

value problems). Its proof uses the Nash-Moser method and many other tools, in

particular generalized (degenerate) Kreiss symmetrizers and symbolic calculus. It

will be only lightly touched here, the reader being referred to the series of papers

1991 Mathematics Subject Classification. Primary 35L65; Secondary 35L50.

1

Proceedings of Symposia in Applied Mathematics

Volume 67.1, 2009

3

http://dx.doi.org/10.1090/psapm/067.1/2605210