Surface quasi-geostrophic equation

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(Created page with "$ \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 ...")
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Fractional diffusion is often added to the equation $$ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0.$$
Fractional diffusion is often added to the equation $$ \theta_t + u \cdot \nabla \theta + (-\Delta)^s \theta = 0.$$
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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 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]].
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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 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]].
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The key feature of the model is that the drift $u$ is a divergence free vector field obtained from 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.
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.

Revision as of 02:33, 19 May 2011

$ \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 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 obtained from 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.

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