Stability for the 2D anisotropic surface quasi-geostrophic equation with horizontal dissipation

Abstract

The stability of the surface quasi-geostrophic (SQG) equation with either horizontal or vertical dissipation is still an open problem. In this paper, we give a partial answer to this problem. More precisely, in the case when the domain is $\Omega =\mathbb{T}\times \mathbb{R}$, we obtain the stability by decomposing the solution into the horizontal averages and the oscillating parts.

Introduction

The surface quasi-geostrophic (SQG) equation assumes the form

$$\left\{\begin{array}{c}{\partial }_t\theta +\mathbf{u}\cdot \nabla \theta +\kappa {(-\Delta )}^{\alpha }\theta =0,\hfill \\ \mathbf{u}={\nabla }^{\perp }\psi ,-\Lambda \psi =\theta ,\hfill \end{array}\right.$$

where the potential temperature $\theta =\theta (x,t)$ is a scalar function of $x\in {\mathbb{R}}^2$, $t\ge 0$. $0<\alpha \le 1$, $\kappa \ge 0$ are parameters. The fractional operator ${(-\Delta )}^{\alpha }$ on ${\mathbb{R}}^2$ is defined through the Fourier transform,

$$\widehat{{(-\Delta )}^{\alpha }f}\left(\xi \right)={\left|\xi \right|}^{2\alpha }\widehat{f}\left(\xi \right),\widehat{f}\left(\xi \right)={\int }_{{\mathbb{R}}^2}{e}^{-ix\cdot \xi }f\left(x\right)dx.$$

$\Lambda ={(-\Delta )}^{\frac{1}{2}}$ is the Zygmund operator, $\psi$ is the stream function, and ${\nabla }^{\perp }=(-{\partial }_2,{\partial }_1)$, therefore the velocity vector $\mathbf{u}$ is automatically divergence free. It is easy to see $\mathbf{u}$ is determined by the Riesz transforms of $\theta$, which can be written as

$$\mathbf{u}=(-{\partial }_2{\Lambda }^{-1}\theta ,{\partial }_1{\Lambda }^{-1}\theta )=({\mathcal{R}}_2\theta ,-{\mathcal{R}}_1\theta )=-{\mathcal{R}}^{\perp }\theta .$$

The SQG equation arises from geophysics and has a strong physical background. If $\kappa =0$, then it reduces to the inviscid case, which shares some important features with the $3D$ Euler equations such as the vortex stretching mechanism. In fact, the SQG equation (1.1) resembles the Euler equations

$${\partial }_t\mathbf{v}+\mathbf{v}\cdot \nabla \mathbf{v}=-\nabla p,\nabla \cdot \mathbf{v}=0,$$

or, in terms of the vorticity $\nabla \times \mathbf{v}=\mathit{\omega }$,

$${\partial }_t\mathit{\omega }+\mathbf{v}\cdot \nabla \mathit{\omega }=\mathit{\omega }\cdot \nabla \mathbf{v}.$$

$\mathbf{W}={\nabla }^{\perp }\theta$ satisfies

$${\partial }_t\mathbf{W}+\mathbf{u}\cdot \nabla \mathbf{W}=\mathbf{W}\cdot \nabla \mathbf{u},$$

which is in the same form as the $3D$ vorticity equations.

There are two major cases of (1.1). One case is the inviscid SQG equation $(\kappa =0)$. In particular, it is derived from the general quasi-geostrophic approximations for atmospheric and oceanic fluid flow with small Rossby and Ekman numbers, see [1], [2]. The fundamental problem that whether the classical solution of the inviscid SQG equation corresponding to given initial data is global in time has been studied widely. The local existence results in various function spaces have been established (see, e.g., [1], [3], [4], [5]). Even though the inviscid SQG equation looks really simple, the global regularity problem for the general case still remains open. The other case is the dissipative SQG equation $(\kappa >0)$. The study of the dissipative SQG equation is further divided into three cases, the sub-critical case $(\alpha >\frac{1}{2})$, the critical case $(\alpha =\frac{1}{2})$ and the super-critical case $(\alpha <\frac{1}{2})$. For the sub-critical case, the solutions at several regularity levels, including the classical sense have been established satisfactorily [6], [7]. For the critical case, the global well-posedness issue has been solved successfully [8], [9], [10], [11]. For the super-critical case, it is shown in [4] that (1.1) with initial data has a unique local in time solution. For the global existence results under small data, there are many works [3], [12], [13], [14]. However, global existence for the super-critical case with large initial data is an open problem.

Recently, the case of “anisotropic viscosity” has been analyzed in various fluid mechanics models. The global regularity of the SQG equation with fractional horizontal dissipation and fractional vertical dissipation is obtained when the dissipation powers are restricted to a suitable range [15]. The SQG equation with only horizontal viscosity is given by

$$\left\{\begin{array}{c}{\partial }_t\theta +\mathbf{u}\cdot \nabla \theta -\kappa {\partial }_{11}\theta =0,(x,t)\in \Omega \times \mathbb{R},\hfill \\ \mathbf{u}={\nabla }^{\perp }\psi ,\Delta \psi =\Lambda \theta ,\hfill \\ \theta (x,0)={\theta }_0\left(x\right),\hfill \end{array}\right.$$

where $\kappa >0$. When $\Omega ={\mathbb{R}}^2$, the global regularity for solutions of (1.2) has been established [16]. However, the $H^2$ bound for $\theta$ grows double exponentially in time, which makes the stability problem a mystery. More precisely, the standard $L^2$ estimate for $\Delta \theta$ is

$$\frac{1}{2}\frac{d}{dt}{\Vert \Delta \theta \Vert }_{L^2}^2+{\Vert {\partial }_1\Delta \theta \Vert }_{L^2}^2=-\int \Delta \mathbf{u}\cdot \nabla \theta \Delta \theta dx-2\int \nabla \mathbf{u}\cdot \nabla \nabla \theta \Delta \theta dx.$$

The right hand side of it does not admit a time-integrable upper bound!

We nevertheless are able to prove a stability result when the spatial domain is

$$\Omega =\mathbb{T}\times \mathbb{R},$$

where $\mathbb{T}=[0,1]$ is a $1D$ periodic box. For functions defined on $\mathbb{T}\times \mathbb{R}$, we need to use a different definition of ${(-\Delta )}^{\alpha }$. For $f(x_1,x_2)\in L^1(\mathbb{T}\times \mathbb{R})$ and $(n_1,{\xi }_2)\in \mathbb{Z}\times \mathbb{R}$, the Fourier transform of $f$ reads

$$\widehat{f}(n_1,{\xi }_2)={\int }_{\mathbb{T}\times \mathbb{R}}{e}^{-i(n_1{x}_1+{\xi }_2{x}_2)}f(x_1,x_2)d{x}_{1}d{x}_2,$$

and the fractional operator ${(-\Delta )}^{\alpha }$ is defined by

$$\widehat{{(-\Delta )}^{\alpha }f}(n_1,{\xi }_2)={(n_1^2+{\xi }_2^2)}^{\alpha }\widehat{f}(n_1,{\xi }_2).$$

We will investigate the stability of perturbations near the trivial solution for the anisotropic SQG equations (1.2). With the domain $\mathbb{T}\times \mathbb{R}$, we can prove global existence of the solution in $H^2$. Further more, the solution admits a uniform upper bound in $H^2$, which means the stability of the perturbations near the trivial solution. In addition, we also obtain the large time behavior of the solution. Let us explain why such a domain $\Omega =\mathbb{T}\times \mathbb{R}$ is helpful in solving stability problems. Our main idea is that we divide the quantities into the horizontal averages and the oscillating parts. More precisely, for a function $f=f(x_1,x_2)$, which is integrable respect to $x_1$ on $\mathbb{T}$, we divide $f$ into its horizontal average $\overline{f}$ and its oscillation $\tilde{f}$, i.e. 

$$f=\overline{f}+\tilde{f},\overline{f}={\int }_{\mathbb{T}}f(x_1,x_2)d{x}_1.$$

The term $\overline{f}$ represents the zero-th horizontal Fourier mode of $f$. This decomposition has some useful properties. Such as

• $\overline{\tilde{f}}=0$, $\overline{{\partial }_{1}f}={\partial }_1\overline{f}=0$, $\overline{{\partial }_{2}f}={\partial }_2\overline{f}$, and $\widetilde{{\partial }_{1}f}={\partial }_1\tilde{f}$;

• A divergence free vector field $\mathbf{F}$ satisfies $\nabla \cdot \overline{\mathbf{F}}=0$, and $\nabla \cdot \tilde{\mathbf{F}}=0$;

• If $f\in L^2\left(\Omega \right)$, then $(\overline{f},\tilde{f})=0$, and ${\Vert f\Vert }_{L^2}^2={\Vert \overline{f}\Vert }_{L^2}^2+{\Vert \tilde{f}\Vert }_{L^2}^2$, where $(\cdot ,\cdot )$ denotes the $L^2$ inner product.

Another helpful property is that the oscillation part obeys the Poincaré type inequality,

$${\Vert \tilde{f}\Vert }_{L^2\left(\Omega \right)}\le C{\Vert {\partial }_1\tilde{f}\Vert }_{L^2\left(\Omega \right)}.$$

Splitting $u,\theta$ into $\overline{u},\overline{\theta }$ and $\tilde{u},\tilde{\theta }$ helps us to bound the nonlinear terms when doing the energy estimates.

In the rest of our paper, $\Omega$ always represents the domain $\mathbb{T}\times \mathbb{R}$. Our main result is stated as follows.

Theorem 1.1

Assume ${\theta }_0\in H^2\left(\Omega \right)$. Then there exists $\varepsilon >0$ such that, if

$${\Vert {\theta }_0\Vert }_{H^2}\le \varepsilon ,$$

then (1.2) has a unique global solution that remains uniformly bounded,

$${\Vert \theta \left(t\right)\Vert }_{H^2}^2+\kappa {\int }_0^t{\Vert {\partial }_1\theta \left(\tau \right)\Vert }_{H^2}^{2}d\tau \le C{\varepsilon }^2$$

for some pure constant $C>0$ and for all $t>0$. In particular, (1.5) implies the stability of any perturbation near the trivial solution.

Remark 1.2

By using the equation of $\tilde{\theta }$, we are also able to prove that $\tilde{\theta }$ decays exponentially to zero in time in the $H^1$ norm. Namely, if the initial data ${\theta }_0\in H^2\left(\Omega \right)$ satisfies (1.4) for sufficient small $\varepsilon$, then the oscillation part of the solution $\theta$ in Theorem 1.1 satisfies,

$${\Vert \tilde{\theta }\left(t\right)\Vert }_{H^1}\le {\Vert {\theta }_0\Vert }_{H^1}{e}^{-C_{1}t},$$

for some pure constant $C_1$. This means that ${\Vert \theta \left(t\right)\Vert }_{H^1}$ tends to ${\Vert \overline{\theta }\left(t\right)\Vert }_{H^1}$ in time.