inviscid damping

Consider the classical incompressible Euler equations in two dimension; namely written in the vorticity formulation, the transport equation for the unknown scalar vorticity ${\omega = \omega(t,x,y)}$ ,

$\displaystyle \partial_t \omega + (u \cdot \nabla) \omega =0$

posed on a spatial domain ${\Omega \subset \mathbb{R}^2}$ , where the velocity field ${u\in \mathbb{R}^2}$ is obtained through the Biot-Savart law ${u = \nabla^\perp \Delta^{-1} \omega}$ (with convention $(a_1,a_2)^\perp = (a_2,-a_1)$ ). By construction, the velocity field ${u}$ is incompressible: ${\nabla \cdot u=0}$ . When dealing with domains with a boundary, ${\Delta^{-1} }$ is defined together with a Dirichlet boundary condition that corresponds to the no-penetration condition of fluids ${u\cdot n =0}$ on the boundary.

The global well-posedness theory of 2D Euler is classical: (1) smooth initial data give rise to solutions that remain smooth for all times (e.g. the Beale-Kato-Majda criterium holds, as vorticity is uniformly bounded for all times; in fact, being transported along the volume-preserving flow, all ${L^p}$ norms of vorticity are conserved), and (2) weak solutions with bounded vorticity are unique (see Yudovich ’63).

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In his famous 1981 paper, Mourre gave a sufficient condition for a self-adjoint operator ${H}$ to assure the absence of its singular continuous spectrum. More precisely, consider a self-adjoint operator ${H}$ on a Hilbert space ${\mathcal{H}}$ (e.g., ${L^2}$ with the usual norm), and assume that there is a self-adjoint operator ${A}$ , called a conjugate operator of ${H}$ on an interval ${I\subset \mathbb{R}}$ , so that

$\displaystyle P_I i[H,A] P_I \ge \theta_I P_I + P_I K P_I \ \ \ \ \ (1)$

for some positive constant ${\theta_I}$ and some compact operator ${K}$ on ${\mathcal{H}}$ , where ${P_I}$ denotes the spectral projection of ${H}$ onto ${I}$ , the commutator ${[H,A] = HA - AH}$ , and the inequality is understood in the sense of self-adjoint operators.