Global regularity and decay behavior for Leray equations with critical-dissipation and Its Application to Self-similar Solutions

TL;DR

This paper establishes the global regularity and decay rates for solutions to the Leray equations with critical dissipation, and constructs a self-similar solution to 3D Navier-Stokes equations, advancing understanding of their long-term behavior.

Contribution

It introduces a novel approach combining maximal smoothing, $L^{p}$-regularity, and heat semigroup actions to prove global regularity and decay, and constructs a new self-similar solution.

Findings

Proves global regularity of solutions with critical dissipation.

Establishes optimal decay rates for solutions.

Constructs a new self-similar solution to 3D Navier-Stokes.

Abstract

In this paper, we show the global regularity and the optimal decay of weak solutions to the generalized Leray problem with critical dissipation. Our method is based on the maximal smoothing effect, $L^{p}$-type elliptic regularity of linearization, and the action of the heat semigroup generated by the fractional powers of Laplace operator on distributions with Fourier transforms supported in an annulus. As a by-product, we shall construct a self-similar solution to the three-dimensional incompressible Navier-Stokes equations, and more importantly, prove the global regularity and the optimal decay without additional requirement of existing literatures.