Structure of Korteweg models and stability of diffuse interfaces

Abstract

The models considered are supposed to govern the motion of compressible fluids such as liquid-vapor mixtures endowed with a variable internal capillarity. Several formulations and simplifications are discussed, from the full multi-dimensional equations for non-isothermal motions in Eulerian coordinates to the one-dimensional equations for isothermal motions in Lagrangian coordinates. Hamiltonian structures are pointed out in each case, and in the one-dimensional isothermal case, they are used to study the stability of two kinds of non-linear waves: the solitary, or homoclinic waves, and the heteroclinic waves, which correspond to propagating phase boundaries of non-zero thickness, also called diffuse interfaces. It is known from an earlier work by Benzoni–Gavage [Physica D, 2001] that the latter are (weakly) spectrally stable. Here, diffuse interfaces are shown to be orbitally stable. The proof relies on their interpretation as critical points of the Hamiltonian under constraints, whose justification requires some care because of the different endstates at infinity. Another difficulty comes from higher order derivatives that are not controlled by the Hamiltonian. In the case of a variable capillarity, our stability result unfortunately does not imply global existence. As regards the solitary waves, which come into families parametrized by the wave speed, they are not stable from the variational point of view. However, using a method due to Grillakis, Shatah and Strauss [Journal of Functional Analysis, 1987], it is possible to show that some solitary waves, depending on their speed, are orbitally stable. Namely, the convexity of a function of the wave speed called moment of instability determines the stability of solitary waves. This approach, already used by Bona and Sachs [Communications in Mathematical Physics, 1988] for the Boussinesq equation, is here adapted to solitary waves in Korteweg models, which are first classified according to their endstate and internal structure. The corresponding moments of instability are computed by quadrature. They exhibit both convexity and concavity regions.