Mixing for the Euler equations with Ekman damping

Problem's Description

Let us consider 2D Euler equations in a bounded domain or on a torus, with a linear damping and a random external force: \begin{equation*} \partial_t u + \langle u,\nabla\rangle u+\gamma u+\nabla p=\eta(t,x), \quad \mathrm{div} u=0. \end{equation*} Here $u=(u_1,u_2)$ and $p$ are unknown velocity field and pressure, $\gamma>0$ is a parameter, and $\eta$ is a random force such that the equation in question generates a Markov process. For instance, $\eta$ can be a process white in time and regular in space (see Section 2.2 in [1]) or a piecewise independent random process (see Section 1 in [2]). Recall that a measure on the phase space is said to be stationary if it does not change under the evolution (see Section 1.3 in [1]).

Problem: Prove that the Euler equations have a unique stationary measure, and the law of any solution converges to it as $t\to+\infty$.

1. Book Mathematics of Two-Dimensional Turbulence

   • S. Kuksin,
   • A. Shirikyan

   year of publication: 2012

2. Article Control and mixing for 2D Navier-Stokes equations with space-time localised noise

   • A. Shirikyan

   Ann. Sci. Éc. Norm. Supér. (4) 48, 253-280, 2015