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

TL;DR

This paper investigates the existence and regularity of forward self-similar solutions to the 2D hypodissipative Navier-Stokes equations with fractional diffusion, establishing conditions for existence, smoothness, and decay properties.

Contribution

It demonstrates the existence of weak solutions for large initial data and proves smoothness and decay estimates for solutions when the fractional power exceeds 2/3.

Findings

Existence of weak solutions for large initial data.

Solutions are smooth and satisfy decay estimates when  > 2/3.

Solutions differ from fractional heat equation profiles by an element in H^lpha(R^2).

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 $H^\alpha(\mathbb{R}^2)$. 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.