Intermittent singular solutions of the stationary 2D Navier-Stokes equations in sharp Sobolev spaces

TL;DR

This paper constructs non-trivial stationary solutions to the 2D Navier-Stokes equations with singularities in sharp Sobolev spaces, extending previous results by incorporating intermittency into the solution construction.

Contribution

It introduces a novel method of constructing stationary solutions with intermittent singularities in sharp Sobolev spaces, broadening the understanding of Navier-Stokes solutions.

Findings

Solutions lie in intersection of L^{2-ε} and -psilon Sobolev spaces

Solutions are not square integrable, requiring redefinition of solutions

Incorporates intermittency into the construction of singular solutions

Abstract

In this paper we construct non-trivial solutions to the stationary Navier-Stokes equations on the two dimensional torus which lie in $\bigcap_{\epsilon \in (0,1)} L^{2-\epsilon}(\mathbb{T}^2) \cap \dot H^{-\epsilon}(\mathbb{T}^2)$. Due to the fact that our solutions are not square integrable, we must redefine the notion of solution. Our result gives a sharp extension of recent work of Lemari\'e-Rieusset, who proved a similar result in the space $\dot{H}^{-1} \cap {BMO}^{-1}$. The main new ingredient is the incorporation of intermittency into the construction of the solutions.