Forward Self-Similar Solutions to the 2D Hypodissipative Navier-Stokes Equations

Abstract

We study the forward self-similar solutions to the $2$D hypodissipative Navier-Stokes equation with fractional diffusion $(-\Delta)^\alpha$ for $\frac{1}{2}<\alpha<1$. We first show that for arbitrarily large $(1-2\alpha)$-homogeneous initial data which are locally Lipschitz, there exists at least one weak solution whose profile differs from the self-similar profile of the fractional heat equation by an element of $H^\alpha(\reall^2)$. Moreover, when $\alpha\in(\frac{2}{3},1)$ we show that any such weak solution is actually smooth, hence a strong solution, and satisfies certain far field decay estimates. Finally, we provide numerical evidence for the nonuniqueness of the related $2$D Navier-Stokes equation with time-dependent viscosity.