Transverse instability of solitary waves in Korteweg fluids

Sylvie Benzoni-Gavage (1)

1 ICJ - Institut Camille Jordan

The Euler-Korteweg model is made of the standard Euler equations for compressible fluids supplemented with the Korteweg tensor, which is intended to take into account capillary effects. For nonmonotone `pressure' laws, the Euler-Korteweg model is known to admit solitary waves, even though their physical significance remains unclear. In fact, several kinds of solitary waves, with various endstates, can be identified. In one space dimension, all these solitary waves may be viewed as critical points under constraint of the total energy, the constraint being linked to translational invariance. In an earlier work with Danchin, Descombes and Jamet [Interf. Free Bound. 2005], a sufficient condition was obtained for their orbital stability, by the method of Grillakis, Shatah and Strauss [Journal of Functional Analysis, 1987], relying on the Hamiltonian structure and on the translational invariance. Numerical evidence was given that this condition is satisfied by some dynamic solitary waves, whereas it fails for solitary waves closer to thermodynamic equilibrium. That condition is of the form $m''(\sigma)>0$, with $\sigma$ the speed and $m$ the constrained energy of the wave. It turns out that, as was already known in other contexts, $m''(\sigma)$ is linked to the low frequency behavior of the Evans function associated with the linearized equations. This link was investigated by Zumbrun [Z. Anal. Anwend. 2008] (and independently by Bridges and Derks) for simplified equations (with constant capillarity) in Lagrangian coordinates. Zumbrun proved in that context that $m''(\sigma)\geq 0$ is necessary for linearized stability. This result is revisited here with general capillarities in Eulerian coordinates, and the main purpose is to investigate the {\em multidimensional} stability of planar solitary waves. In this respect, variational tools are not much appropriate. Nevertheless, the Evans function technique does extend to arbitrary space dimensions, and its low-frequency behavior can be computed explicitly. It turns out from this behavior and an argument pointed out by Zumbrun and Serre [Indiana Univ. Math. J 1999] that planar solitary wave solutions of the Euler-Korteweg model are linearly unstable with respect to transverse perturbations of large wave length.

Domaines: Equations aux dérivées partielles [math.AP]

Sylvie Benzoni-Gavage. Transverse instability of solitary waves in Korteweg fluids. Journal of Mathematical Analysis and Applications, 2010, 361 (2), pp.338-357. ⟨hal-00342400⟩