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

Abstract

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 $0<\gamma \le 2$ we construct non-trivial solutions such that $$u,\theta \in {\dot{B}}_{\infty ,\infty }^{\alpha -1}({\mathbb{T}}^2)\cap {\dot{B}}_{2,2}^{\alpha -1}({\mathbb{T}}^2).$$ 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 $1\le p<4/3$ we can construct $u$ and $\theta$ lying in $L^p({\mathbb{T}}^2)$.

42 pages; final version