On the forward self-similar solutions to the two-dimensional Navier-Stokes equations

TL;DR

This paper proves the existence of global forward self-similar solutions to the 2D incompressible Navier-Stokes equations with initial data that is homogeneous of degree -1, overcoming challenges posed by infinite local energy.

Contribution

It establishes the existence of self-similar solutions without smallness assumptions, using a novel decomposition and cancellation approach to handle critical initial data.

Findings

Existence of global self-similar solutions for 2D Navier-Stokes with homogeneous initial data.

Development of a decomposition method into linear and perturbation parts for analysis.

Derivation of optimal decay estimates and compactness results for the solutions.

Abstract

We establish the global existence of forward self-similar solutions to the two-dimensional incompressible Navier-Stokes equations for any divergence-free initial velocity that is homogeneous of degree $-1$ and locally H\"older continuous. This result requires no smallness assumption on the initial data. In sharp contrast to the three-dimensional case, where $(-1)$-homogeneous vector fields are locally square-integrable, the major difficulty for the 2D problem is the criticality in the sense that the initial kinetic energy is locally infinite at the origin, and the initial vorticity fails to be locally integrable, so that the classical local energy estimates are not available. Our key ideas are to decompose the solution into a linear part solving the heat equation and a finite-energy perturbation part, and to exploit a kind of inherent cancellation relation between the linear part and the perturbation part. These, together with suitable choices of multipliers, enable us to control the interaction terms and to establish the $H^1$-estimates for the perturbation part. Furthermore, we can get an optimal pointwise estimate via investigating the corresponding Leray equations in weighted Sobolev spaces.This gives the faster decay of the perturbation part at infinity and compactness, which play important roles in proving the existence of global-in-time self-similar solutions.