Surface quasi-geostrophic equation
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$ \newcommand{\R}{\mathbb{R}} $
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^{[citation needed]}. 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 $s \in [0,1]$.
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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 equation. For smaller values of $s$, the diffusion is stronger than 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 several methods using only soft functional analysis or Fourier analysis.
Critical case: $s=1/2$
The equation is well posed globally. There are three 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 $\theta$.
- 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.
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:
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