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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 Ω=[0,1] 2 2 ,

θ t+div(𝐯θ)=0,(-Δ) 1/2 ψ=θ,
ψ=𝐯,θ(𝐱,0)=θ 0 (𝐱)inΩ,
Ω θd𝐱=0, Ω ψd𝐱=0, Ω 𝐯d𝐱=0·

Here Δ is the horizontal Laplacian operator and (-Δ) 1/2 is the pseudodifferential operator defined in the Fourier space (-Δ) 1/2 u(𝐤) ^=|𝐤|u ^(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:
35B65Smoothness and regularity of solutions of PDE
35B05Oscillation, zeros of solutions, mean value theorems, etc. (PDE)
35B40Asymptotic behavior of solutions of PDE
86A05Hydrology, hydrography, oceanography
Software:
SQG