An inviscid regularization for the surface quasi-geostrophic equation.

*(English)*Zbl 1149.35018Summary: 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.

\[ \frac{\partial\theta}{\partial t}+\text{div}({\mathbf v}\theta)=0, \qquad (-\Delta)^{1/2}\psi=\theta, \]

\[ \nabla^\perp\psi={\mathbf v}, \qquad \theta({\mathbf x},0)= \theta_0({\mathbf x})\quad\text{in }\Omega, \]

\[ \int_\Omega \theta\,d{\mathbf x}=0, \qquad \int_\Omega\psi\,d{\mathbf x}=0, \qquad \int_\Omega{\mathbf v}\,d{\mathbf x}=0. \]

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 |

##### Keywords:

singularity development; gradient blow up; Berdina-like models for turbulence; blowup in finite time##### Software:

SQG
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\textit{B. Khouider} and \textit{E. S. Titi}, Commun. Pure Appl. Math. 61, No. 10, 1331--1346 (2008; Zbl 1149.35018)

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