Invariant Measures and Global Well Posedness for the SQG Equation

Abstract

We construct an invariant measure $\mu$ for the Surface Quasi-Geostrophic (SQG) equation and show that almost all functions in the support of $\mu$ are initial conditions of global, unique solutions of SQG that depend continuously on the initial data. In addition, we show that the support of $\mu$ is infinite dimensional, meaning that it is not locally a subset of any compact set with finite Hausdorff dimension. Also, there are global solutions that have arbitrarily large initial condition. The measure a $\mu$ is obtained via fluctuation–dissipation method, that is, as a limit of invariant measures for stochastic SQG with a carefully chosen dissipation and random forcing.

Appendices

Appendix A: Some Facts on the Fluctuation–Dissipation Approach for Finite-Dimensional Hamiltonian Systems

In this section we elaborate on the question that was raised in the introduction: Do the constructed invariant measure $\mu$ for is (1.1) concentrates on the equilibria? Although we proved that the support of $\mu$ is infinite dimensional, it also known that the set of equilibria is also infinite dimensional; any solution of the equation

$$\begin{aligned} (-\Delta )^{\frac{1}{2}} \Phi = F(\Phi ) \end{aligned}$$

is an equilibrium of (1.1). Since every equilibrium is trivially a global solution, there is a possibility that $\mu$ concentrates on the set of equilibria, and we did not construct any new solution. As mentioned above, we don’t have a definite answer to this question, however we provide an example of a general system for which the measure arising from fluctuation dissipation method is not supported on equilibria.

Since the SQG equation has a Hamiltonian structure, we will focus only on the Hamiltonian systems. There are several trivial examples in which the equilibria form a discrete set, and therefore are of measure zero, for instance the cubic defocusing Schrödinger equation with only one equilibrium. The example closest to SQG is 2D Euler equation, which has infinite dimensional manifold of equilibria with similar structure. However, whether the invariant measures for 2D Euler equation concentrate on equilibria is an open question, hence regularizing the problem might not help.

Let us turn our attention to finite dimensional systems. Consider a 2n-dimensional Hamiltonian system

$$\begin{aligned} \dot{x}=-\partial _yH(x,y),\quad \dot{y}=\partial _xH(x,y), \end{aligned}$$

(A.1)

where $H:\mathbb {R}^n\times \mathbb {R}^n\rightarrow \mathbb {R}$ is a smooth Hamiltonian function. It is well known that $f(H)\mathrm{d}x\mathrm{d}y$ is an invariant measure for the system, for any integrable smooth function f. We consider now the fluctuation–dissipation model

$$\begin{aligned} \mathrm{d}x&=(-\partial _yH(x,y)-\alpha \partial _xH(x,y))\mathrm{d}t + \sqrt{2\alpha }\mathrm{d}\beta _1,\nonumber \\ \mathrm{d}y&= (\partial _xH(x,y)-\alpha \partial _yH(x,y))\mathrm{d}t + \sqrt{2\alpha }\mathrm{d}\beta _2, \end{aligned}$$

(A.2)

where $\beta _1, \beta _2$ are independent Brownian motions. Then, $e^{-H(x,y)}$ is a density of an invariant measure for (A.2), since $e^{-H(x,y)}$ is solution of the Fokker-Plank equation

$$\begin{aligned} \mathcal {L}\rho =\alpha \Delta \rho -\nabla \cdot \left[ (\partial _yH(x,y)+\alpha \partial _xH(x,y),-\partial _xH(x,y)+\alpha \partial _yH(x,y))^T\rho \right] =0. \end{aligned}$$

Thus $\mu (\mathrm{d}x\mathrm{d}y)=T^{-1}e^{-H(x,y)}\mathrm{d}x\mathrm{d}y$ is an invariant probability measure of (A.2), were we denote $T=\int _{\mathbb {R}^n\times \mathbb {R}^n} e^{-H(x,y)}\mathrm{d}x\mathrm{d}y$ to be a partition function (normalization). Note that T is finite if H has appropriate increase at infinity. Observe that $\mu$ does not depend on $\alpha$, thus by passing $\alpha \rightarrow 0$, we see that $\mu$ is an invariant measure of (A.1).

If H is constant on the unit ball of $\mathbb {R}^n\times \mathbb {R}^n$, then any point in that ball is an equilibrium of (A.1), and therefore we have an open set of equilibria. On the other hand, $\mu$ has positive density everywhere and in particular its support coincides with the whole space. There might be a possibility to apply this reasoning to infinite dimensional systems, but there are serious difficulties with coercivity of the dissipation. We leave this question open.

Appendix B: Itô Formula

For the reader’s convenience, we recall Itô’s formula in infinite dimensions, which is used several times in the proofs of the main results. We say that the equation (1.2) has the Itô property on the triple $(H^{s-1},H^s,H^{s+1})$ if

• (1) for some $T> 0$, (1.2) has a unique solution on [0, T) for any data in $H^s$;

• (2) the process $h{:}{=}-\alpha (\Delta ^2\theta -\nabla (|\nabla \theta |^2\nabla \theta ))-\mathbf {u}\cdot \nabla \theta$ is $\mathcal {F}_t$-adapted and

  $$\begin{aligned} \mathbb {P}\left( \int _0^t(\Vert \theta (r)\Vert _{s+1}^2+\Vert h(r)\Vert _{s-1}^2)dr< \infty , \ \ \forall \ t>0\right) =1,\ \ \sum _{m>0} a_m^{2}\lambda _m^s<\infty . \end{aligned}$$

We have the following version of Itô’s lemma proved in [43, Section A.7]:

Theorem B.1

([43]) Let $F\in C^2(H^s,\mathbb {R})$ be a functional which is locally uniformly continuous, together with its first two derivatives, on $H^s$. Suppose that (1.2) satisfies the Itô property on $(H^{s-1},H^s,H^{s+1})$ and that F satisfies the following conditions:

• (1) There is a function $K:\mathbb {R}_+\rightarrow \mathbb {R}_+$ such that

  $$\begin{aligned} |F'(\theta ;v)|\leqq K(\Vert \theta \Vert _{s})\Vert \theta \Vert _{{s+1}}\Vert v\Vert _{{s-1}},\ \ \ \theta \in H^{s+1},\ \ v\in H^{s-1}. \end{aligned}$$

  (B.1)

• (2) For any sequence $\{w_k\}\subset H^{s+1}$ converging toward $w\in H^{s+1}$ and any $v\in H^{s-1}$, we have

  $$\begin{aligned} F'(w_k;v)\rightarrow F'(w;v),\ \ as\ \ k\rightarrow \infty . \end{aligned}$$

  (B.2)

• (3) The solution $\theta$ of (1.2) satisfies

  $$\begin{aligned} \sum _{m} a_m^2\mathbb {E}\int _0^t|F'(\theta ;e_m)|^2\mathrm{d}s <\infty \ \ \ \ for\ all \ t>0. \end{aligned}$$

  (B.3)

Then we have

$$\begin{aligned} F(\theta (t))&=F(\theta (0))\nonumber \\&\quad +\int _0^t\left( F'(\theta (s);h(s))+\frac{\alpha }{2}\sum _{m} a_m^2F''(\theta (s); e_m, e_m)\right) \mathrm{d}s \\&\quad +\sqrt{\alpha }\sum _{m} a_m\int _0^tF'(\theta (s);e_m)d W_m(s). \end{aligned}$$

In particular,

$$\begin{aligned} \mathbb {E}F(\theta (t))= & {} \mathbb {E}F(\theta (0))\\&+\int _0^t\mathbb {E}\left( F'(\theta (s); h(s))+\frac{\alpha }{2}\sum _{m} a_m^2F''(\theta (s); e_m, e_m)\right) \mathrm{d}s. \end{aligned}$$

If one omits (B.3), then we have the formula (B.1) where t is replaced by the stopping time $t\wedge \tau _n$, with

$$\begin{aligned} \tau _n=\inf \{t\geqq 0,\ \Vert \theta (t)\Vert _{s}> n\}, \ \ n\geqq 0, \end{aligned}$$

with the convention $\inf \emptyset =+\infty .$

Appendix C: Embedding $L^2H^2 \cap W^{1,\frac{4}{3}}W^{-1, \frac{4}{3}} \hookrightarrow CH^{-\delta }$

Although the parabolic embedding $L^2H^2 \cap W^{1,\frac{4}{3}}W^{-1, \frac{4}{3}} \hookrightarrow CH^{-\delta }$ follows from standard arguments we were not able to locate the proof in the literature. Hence, we outline the main steps in this appendix.

By [2, Theorem 5.2], we have, for any $\theta > \frac{2}{3}$, that

$$\begin{aligned} L^2H^2 \cap W^{1,\frac{4}{3}}W^{-1, \frac{4}{3}} \hookrightarrow C (H^2, W^{-1, \frac{4}{3}})_{\theta , p_\theta } \,, \end{aligned}$$

where $(H^2, W^{-1, \frac{4}{3}})_{\theta , p_\theta }$ is the real interpolation space and $p_\theta$ satisfies

$$\begin{aligned} \frac{1}{p_\theta } = \frac{1-\theta }{2} + \frac{\theta }{\frac{4}{3}}. \end{aligned}$$

However, by [2, (3.5)], for any $\varepsilon \in (0, 1)$ one has

$$\begin{aligned}&(H^2, W^{-1, \frac{4}{3}})_{\theta , p_\theta } \hookrightarrow (H^{2 - \varepsilon }, W^{-1 - \varepsilon , \frac{4}{3}})_{\theta , p_\theta } = (B^{2-\varepsilon }_{2, 2}, B^{-1- \varepsilon }_{\frac{4}{3}, \frac{4}{3}})_{\theta , p_\theta } \,, \end{aligned}$$

where $B^{s}_{p, q}$ is a Besov space. From [5, Theorem 6.4.5, (3)] and [2, (3.5)] if follows that

$$\begin{aligned}&(B^{2-\varepsilon }_{2, 2}, B^{-1- \varepsilon }_{\frac{4}{3}, \frac{4}{3}})_{\theta , p_\theta } = B^{(-3+\varepsilon )\theta + (2 - \varepsilon )}_{\frac{4}{2 + \theta }, \frac{4}{2 + \theta }} = W^{(-3+\varepsilon )\theta + (2 - \varepsilon ), \frac{4}{2 + \theta }} \,. \end{aligned}$$

Finally, by Sobolev embeddings,

$$\begin{aligned} W^{(-3+\varepsilon )\theta + (2 - \varepsilon ), \frac{4}{2 + \theta }} \hookrightarrow W^{-\delta , 2} \,, \end{aligned}$$

where $\delta \geqq \frac{\theta }{2} + (-2 + \varepsilon ) + (3 - \varepsilon )\theta$. Since $\theta > \frac{2}{3}$ and $\varepsilon > 0$ can be chosen arbitrarily close to $\frac{2}{3}$ and 0, respectively, one obtains that

$$\begin{aligned} L^2H^2 \cap W^{1,\frac{4}{3}}W^{-1, \frac{4}{3}} \hookrightarrow C W^{-\delta , 2} \end{aligned}$$

for any $\delta > \frac{1}{3}$, as desired.