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

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.