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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 \(\Omega= [0,1]^2\subset\mathbb R^2\),
\[ \frac{\partial\theta}{\partial t}+\text{div}({\mathbf v}\theta)=0, \qquad (-\Delta)^{1/2}\psi=\theta, \]
\[ \nabla^\perp\psi={\mathbf v}, \qquad \theta({\mathbf x},0)= \theta_0({\mathbf x})\quad\text{in }\Omega, \]
\[ \int_\Omega \theta\,d{\mathbf x}=0, \qquad \int_\Omega\psi\,d{\mathbf x}=0, \qquad \int_\Omega{\mathbf v}\,d{\mathbf x}=0. \]
Here \(\Delta\) is the horizontal Laplacian operator and \((-\Delta)^{1/2}\) is the pseudodifferential operator defined in the Fourier space \(\widehat{(-\Delta)^{1/2}u({\mathbf k})}= |{\mathbf k}|\widehat{u}({\mathbf k})\).
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 to PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
86A05 Hydrology, hydrography, oceanography

Software:

SQG
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References:

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