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
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.
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Proceedings of Symposia in Applied Mathematics
Volume 67.1, 2009
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http://dx.doi.org/10.1090/psapm/067.1/2605210
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