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An inviscid regularization for the surface quasi-geostrophic equation. (English) Zbl 1149.35018

Summary: Inspired by recent developments in Berdina-like models for turbulence, we propose an inviscid regularization for the surface quasi-geostrophic (SQG) equations with periodic boundary conditions on a basic periodic square ${\Omega }={\left[0,1\right]}^{2}\subset {ℝ}^{2}$,

$\frac{\partial \theta }{\partial t}+\text{div}\left(𝐯\theta \right)=0,\phantom{\rule{2.em}{0ex}}{\left(-{\Delta }\right)}^{1/2}\psi =\theta ,$
${\nabla }^{\perp }\psi =𝐯,\phantom{\rule{2.em}{0ex}}\theta \left(𝐱,0\right)={\theta }_{0}\left(𝐱\right)\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}{\Omega },$
${\int }_{{\Omega }}\theta \phantom{\rule{0.166667em}{0ex}}d𝐱=0,\phantom{\rule{2.em}{0ex}}{\int }_{{\Omega }}\psi \phantom{\rule{0.166667em}{0ex}}d𝐱=0,\phantom{\rule{2.em}{0ex}}{\int }_{{\Omega }}𝐯\phantom{\rule{0.166667em}{0ex}}d𝐱=0·$

Here ${\Delta }$ is the horizontal Laplacian operator and ${\left(-{\Delta }\right)}^{1/2}$ is the pseudodifferential operator defined in the Fourier space $\stackrel{^}{{\left(-{\Delta }\right)}^{1/2}u\left(𝐤\right)}=|𝐤|\stackrel{^}{u}\left(𝐤\right)$.

We are particularly interested in the celebrated question of blowup in finite time of the solution gradient of the SQG equations. The new regularization yields a necessary and sufficient condition, satisfied by the regularized solution, when a regularization parameter tends to zero for the solution of the original SQG equations to develop a singularity in finite time. As opposed to the commonly used viscous regularization, the inviscid equations derived here conserve a modified energy. Therefore, the new regularization provides an attractive numerical procedure for finite-time blowup testing. In particular, we prove that, if the initial condition is smooth, then the regularized solution remains as smooth as the initial data for all times.

##### MSC:
 35B65 Smoothness and regularity of solutions of PDE 35B05 Oscillation, zeros of solutions, mean value theorems, etc. (PDE) 35B40 Asymptotic behavior of solutions of PDE 86A05 Hydrology, hydrography, oceanography
SQG