Delayed Dissipation for Two-Dimensional Vortex Sheets

Abstract

We quantify viscous energy loss for two-dimensional Delort vortex sheets. Let $u^ν$ be Leray-Hopf solutions on $\mathbb{T}^2$ with uniformly bounded kinetic energy and total vorticity variation, and write $ω_0^ν=μ_0^ν+f_0^ν$, where $μ_0^ν\geq0$ and $f_0^ν$ is bounded in $L^p$, $p>1$. For every fixed $0<δ 0$. Previous estimates covered only $T_ν=o(\exp(|\logν|^κ))$, $κ<1/2$, so this gives a polynomial lower bound on the energetic lifetime of the inviscid vortex-sheet model. If instead $f_0^ν$ is bounded in $L(\log L)^α$, the rate is $O(|\logν|^{-q_α})$, $q_α=\min\{2α,1\}$, and the loss vanishes when $\log T_ν=o(|\logν|^{q_α})$. On $\mathbb{R}^2$, exact radial solutions attain these exponents for $0<α\leq1/2$. At the endpoint, a bounded-energy $L^p$ family attains the rate $1/|\logν|$, while every fixed radial datum dissipates $o(1/|\logν|)$ and can lose a fixed amount of energy only on the diffusive scale $1/ν$.