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

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 Lemarié-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.

19 pages