On the instability of travelling wave solutions for the transport-Stokes equation

Abstract

In this paper, we investigate the instability of the spherical travelling wave solutions for the Transport-Stokes system in $\mathbb{R}^3$. First, a classical scaling argument ensures instability among all probability measures for the Wasserstein metric and the $L^1$ norm. Secondly, we address the instability among patch solutions with a perturbed surface. To this end, we study the linearized system of a contour dynamics equation derived in [18] in the case where the support of the patch is axisymmetric and described by spherical parametrization. We investigate numerically the existence of positive eigenvalues, which ensures the instability of the linearized system. Eventually we recover numerically the instability of the travelling wave by solving the Transport-Stokes equation using a finite element method on FreeFem.