Intermittent solutions of the stationary 2D surface quasi-geostrophic equation

In this paper we construct non-trivial solutions to the stationary dissipative surface quasi-geostrophic equation on the two dimensional torus which lie strictly below the critical regularity threshold of $\dot{H}^{-1/2}(\mathbb{T}^2)$. Specifically, for any $\alpha < 1/2$ and any dissipation exponent $0 < \gamma \leq 2$ we construct non-trivial solutions such that $$u,\theta \in \dot{B}^{\alpha-1}_{\infty,\infty}(\mathbb{T}^2) \cap \dot{B}^{\alpha-1}_{2,2}(\mathbb{T}^2).$$ Due to the fact our solutions do not lie in $\dot{H}^{-1/2}(\mathbb{T}^2)$, this requires reinterpreting the notion of a solution. This leads us to formulate the notion of a weak paraproduct solution for the stationary SQG equation. The main new ingredient is the incorporation of intermittency into the construction of the solutions. This allows us to demonstrate non-trivial integrability results for certain fractional derivatives of our solutions. In particular, for highly intermittent solutions, we are able to conclude for every $1 \leq p < 4/3$ we can construct $u$ and $\theta$ lying in $L^p(\mathbb{T}^2)$.