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.
Categories: math.AP