Surface quasi-geostrophic equation

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The surface quasi-geostrophic (SQG) equation consists of an evolution equation for a scalar function $\theta: \R^+ \times \R^2 \to \R$. In the inviscid case the equation is $$ \theta_t + u \cdot \nabla \theta = 0,$$ where $u = R^\perp \theta$ and $R$ stands for the Riesz transform.

Fractional diffusion is often added to the equation $$ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0.$$

The equation is used as a toy model for the 3D Euler equation and Navier-Stokes. The main question is to determine whether the Cauchy problem is well posed in the classical sense. In the inviscid case, it is a major open problem as well as in the supercritical diffusive case when $s<1/2$. It is believed that inviscid SQG equation presents a similar difficulty as 3D Euler equation in spite of being a scalar model in two dimensions . The same comparison can be made between the supercritical SQG equation and Navier-Stokes.

The key feature of the model is that the drift $u$ is a divergence free vector field related to the solution $\theta$ by a zeroth order singular integral operator.

For the diffusive case, the well posedness of the equation follows from perturbative techniques in the subcritical case ($s>1/2$). In the critical case the proof is more delicate and can be shown using three essentially different methods. In the sueprcritical regime ($s<1/2$) only partial results are known.

Global weak solutions, as well as classical solutions locally in time, are known to exist globally for the full range of $s \in [0,1]$ .

Conserved quantities

The following simple a priori estimates are satisfied by solutions (in order from strongest -locally- to weakest).

• Maximum principle

The supremum of $\theta$ occurs at time zero: $||\theta(t,.)||_{L^\infty} \leq ||\theta(0,.)||_{L^\infty}$.

• Conservation of energy.

A classical solution $u$ satisfies the energy equality $$ \int_{\R^2} \theta(0,x)^2 \ dx = \int_{\R^2} \theta(t,x)^2 \ dx + \int_0^t \int_{\R^2} |(-\Delta)^{s/2}\theta(r,x)|^2 \ dx \ dr.$$

In the case of weak solutions, only the energy inequality is available $$ \int_{\R^2} \theta(0,x)^2 \ dx \geq \int_{\R^2} \theta(t,x)^2 \ dx + \int_0^t \int_{\R^2} |(-\Delta)^{s/2}\theta(r,x)|^2 \ dx \ dr.$$

• $H^{-1/2}$ estimate

The $H^{-1/2}$ norm of $\theta$ does not increase in time.

$$ \int_{\R^2} |(-\Delta)^{-1/4} \theta(0,x)|^2 \ dx = \int_{\R^2} |(-\Delta)^{-1/4}\theta(t,x)|^2 \ dx + \int_0^t \int_{\R^2} |(-\Delta)^{s/2-1/4}\theta(r,x)|^2 \ dx \ dr.$$

Scaling and criticality

If $\theta$ solves the equation, so does the rescaled solution $\theta_r(t,x) = r^{2s-1} \theta(r^{2s} t,rx)$.

The $L^\infty$ norm is invariant by the scaling of the equation if $s=1/2$. This observation makes $s=1/2$ the critical exponent for the strongest a priori estimate available. For larger values of $s$, the diffusion dominates the drift in small scales and the equation is well posed. For larger values of $s$, the drift might be dominant at small scales.

Well posedness results

Sub-critical case: $s>1/2$

The equation is well posed globally. The proof can be done with perturbation methods using only soft functional analysis or Fourier analysis.

Critical case: $s=1/2$

The equation is well posed globally. There are four known proofs.

• Evolution of a modulus of continuity : An explicit modulus of continuity which is comparable to Lipschitz in small scales but growth logarithmically in large scales is shown to be preserved by the flow. The method is vaguely comparable to Ishii-Lions.
• De Giorgi approach : From the $L^\infty$ modulus of continuity, it is concluded that $u$ stays bounded in $BMO$. A variation to the parabolic De Giorgi-Nash-Moser can be carried out to obtain Holder continuity of $\theta$. The result does not use the relations $u = R^\perp \theta$, but only that $u$ is a divergence-free vector field in BMO. Therefore, it is actually a regularity result for arbitrary drift-diffusion equations.
• Dual flow method : Also from the information that $u$ is $BMO$ and divergence free, it can be shown that the solution $\theta$ becomes Holder continuous by studying the dual flow and characterizing Holder functions in terms of how they integrate against simple test functions. This is a regularity result for general drift-diffusion equations as well.
• Nonlinear maximum principle : By studying the evolution of $|\nabla \theta|^2$ and using a nonlinear lower bound on the fractional Laplacian when evaluated at extrema, one may prove that a solution of SQG which has only (sufficiently) small jumps, then it is in fact smooth. By looking at the evolution of finite differences $\delta_h \theta(x) = \theta(x+h) - \theta(x)$, one may measure the evolution of these small jumps, and prove that if the initial data has only small jumps, then so does the corresponding solution of SQG. Combined, the above statements imply global regularity.

Supercritical case: $s<1/2$

The global well posedness of the equation is an open problem. Some partial results are known:

• Existence of solutions locally in time.
• Existence of global weak solutions.
• Global smooth solution if the initial data is sufficiently small.
• Smoothness of weak solutions for sufficiently large time.

Inviscid case

The global well posedness of the equation is an open problem. Some partial results are known:

• Existence of smooth solutions locally in time^{[citation needed]}
• Existence of weak solutions globally in time^{[citation needed]}