Delayed Dissipation for Two-Dimensional Vortex Sheets

Abstract

We quantify viscous energy loss for two-dimensional Delort vortex sheets. Let $u^\nu$ be Leray-Hopf solutions on $\mathbb{T}^2$ with uniformly bounded kinetic energy and total vorticity variation, and write $\omega_0^\nu=\mu_0^\nu+f_0^\nu$, where $\mu_0^\nu\geq0$ and $f_0^\nu$ is bounded in $L^p$, $p>1$. For every fixed $0<\delta 0$. Previous estimates covered only $T_\nu=o(\exp(|\log\nu|^\kappa))$, $\kappa<1/2$, so this gives a polynomial lower bound on the energetic lifetime of the inviscid vortex-sheet model. If instead $f_0^\nu$ is bounded in $L(\log L)^\alpha$, the rate is $O(|\log\nu|^{-q_\alpha})$, $q_\alpha=\min\{2\alpha,1\}$, and the loss vanishes when $\log T_\nu=o(|\log\nu|^{q_\alpha})$. On $\mathbb{R}^2$, exact radial solutions attain these exponents for $0<\alpha\leq1/2$. At the endpoint, a bounded-energy $L^p$ family attains the rate $1/|\log\nu|$, while every fixed radial datum dissipates $o(1/|\log\nu|)$ and can lose a fixed amount of energy only on the diffusive scale $1/\nu$.