Inviscid models generalizing the 2D Euler and the surface quasi-geostrophic equations

Abstract

Any classical solution of the 2D incompressible Euler equation is global in time. However, it remains an outstanding open problem whether classical solutions of the surface quasi-geostrophic (SQG) equation preserve their regularity for all time. This paper studies solutions of a family of active scalar equations in which each component $u_j$ of the velocity field $u$ is determined by the scalar $\theta$ through $u_j=\mathcal{R}{\Lambda }^{-1}P(\Lambda )\theta$ where $\mathcal{R}$ is a Riesz transform and $\Lambda =(-\Delta {)}^{1/2}$. The 2D Euler vorticity equation corresponds to the special case $P(\Lambda )=I$ while the SQG equation to the case $P(\Lambda )=\Lambda$. We develop tools to bound $\Vert \nabla u|{|}_{L^{\infty }}$ for a general class of operators $P$ and establish the global regularity for the Loglog-Euler equation for which $P(\Lambda )=(\log (I+\log (I-\Delta )){)}^{\gamma }$ with $0\le \gamma \le 1$. In addition, a regularity criterion for the model corresponding to $P(\Lambda )={\Lambda }^{\beta }$ with $0\le \beta \le 1$ is also obtained.

References in corpus (4)

• Global well-posedness for the critical 2D dissipative quasi-geostrophic equation
• A new Bernstein's Inequality and the 2D Dissipative Quasi-Geostrophic Equation
• Global Regularity for the Critical Dispersive Dissipative Surface Quasi-Geostrophic Equation
• Decay of weak solutions to the 2D dissipative quasi-geostrophic equation

Cited by in corpus (13)

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• Global well-posedness of slightly supercritical active scalar equations
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• Global well-posedness for a slightly supercritical surface quasi-geostrophic equation
• Global weak solutions for generalized SQG in bounded domains
• Strong illposedness for SQG in critical Sobolev spaces
• On the supercritically diffusive magneto-geostrophic equations
• On a multi-dimensional transport equation with nonlocal velocity
• Regularization by noise for the point vortex model of mSQG equations
• On 3D Hall-MHD equations with fractional Laplacians: global well-posedness
• Long time localization of modified surface quasi-geostrophic equations
• The $\alpha$-SQG patch problem is illposed in $C^{2,\beta }$ and $W^{2,p}$