Surface quasi-geostrophic equation perturbed by derivatives of space-time white noise

Abstract

We consider a family of singular surface quasi-geostrophic equations

$$\begin{aligned} \partial _{t}\theta +u\cdot \nabla \theta =-\nu (-\Delta )^{\gamma /2}\theta +(-\Delta )^{\alpha /2}\xi ,\qquad u=\nabla ^{\perp }(-\Delta )^{-1/2}\theta , \end{aligned}$$

on $[0,\infty )\times {\mathbb {T}}^{2}$, where $\nu \geqslant 0$, $\gamma \in [0,3/2)$, $\alpha \in [0,1/4)$ and $\xi$ is a space-time white noise. For the first time, we establish the existence of infinitely many non-Gaussian

• probabilistically strong solutions for every initial condition in $C^{\eta }$, $\eta >1/2$;

• ergodic stationary solutions.

The result presents a single approach applicable in the subcritical, critical as well as supercritical regime in the sense of Hairer (Invent Math 198(2):269–504, 2014). It also applies in the particular setting $\alpha =\gamma /2$ which formally possesses a Gaussian invariant measure. In our proof, we first introduce a modified Da Prato–Debussche trick which, on the one hand, permits to convert irregularity in time into irregularity in space and, on the other hand, increases the regularity of the linear solution. Second, we develop a convex integration iteration for the corresponding nonlinear equation which yields non-unique non-Gaussian solutions satisfying powerful global-in-time estimates and generating stationary as well as ergodic stationary solutions.

Appendix A. Some auxiliary results

In this part, we recall some auxiliary lemmas from [12]. The first result is rather classical and permits to rewrite the nonlinearity and formulate the notion of weak solution as in Definition 1.1.

Proposition A.1

[12, Proposition 5.1] Let ${\mathcal {R}}={\mathcal {R}}_{j}$, $j=1,2$. Assume that $\phi \in H^{3}$ and $\theta \in {\dot{H}}^{-1/2}$. Then we have

$$\begin{aligned} \Vert [{\mathcal {R}},\phi ]\theta \Vert _{{\dot{H}}^{1/2}}\lesssim \Vert \phi \Vert _{{\dot{H}}^{3}}\Vert \theta \Vert _{{\dot{H}}^{-1/2}}. \end{aligned}$$

The second result is a kind of Leibniz rule useful in the estimate of the new stress in Sect. 6.5.

Lemma A.2

[12, Lemma 2.1] Let $|l|=1,\lambda l\in {\mathbb {Z}}^2$, and $g(x)=a(x)\cos (\lambda l\cdot x).$ Then

$$\begin{aligned} \Lambda g=\lambda g+l\cdot \nabla a \sin (\lambda l\cdot x)+ (T^{(1)}_{\lambda l}a) \cos (\lambda l\cdot x)+(T^{(2)}_{\lambda l}a) \sin (\lambda l\cdot x), \end{aligned}$$

where

$$\begin{aligned} \widehat{T_{\lambda l}^{(1)}a}(k)= & {} \bigg (\frac{|\lambda l+k|+|\lambda l-k|}{2}-\lambda \bigg ){\hat{a}}(k),\\ \widehat{T_{\lambda l}^{(2)}a}(k)= & {} i\bigg (\frac{|\lambda l+k|-|\lambda l-k|}{2}-l\cdot k\bigg ){\hat{a}}(k). \end{aligned}$$

Finally, in order to control the operators in Lemma A.2, we rely on the following.

Lemma A.3

[12, Lemma 3.3] Assume $b_0:{{\mathbb {T}}}^2\rightarrow {{\mathbb {R}}}$ with ${{\,\textrm{supp}\,}}({\hat{b}}_0)\subset \sup \{|k|\leqslant \mu \}$ and $10\leqslant \mu \leqslant \frac{1}{2}\lambda$. Then

$$\begin{aligned} \Vert T_{\lambda l}^{(1)}b_0\Vert _{L^\infty }\lesssim \lambda ^{-1}\mu ^2\Vert b_0\Vert _{L^\infty }, \\ \Vert T_{\lambda l}^{(2)}b_0\Vert _{L^\infty }\lesssim \lambda ^{-2}\mu ^3\Vert b_0\Vert _{L^\infty }. \end{aligned}$$