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 $s \in [0,1]$ }}