An inviscid regularization for the surface quasi‐geostrophic equation

Abstract

Inspired by recent developments in Berdina-like models for turbulence, we propose an inviscid regularization for the surface quasi-geostrophic (SQG) equations. 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 α 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 blow up 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. Moreover, much like the original problem, the inviscid regularization admits a maximum principle.

Global attractors and determining modes for the 3D Navier-Stokes-Voight equations

“…In this sense the system is similar to damped hyperbolic systems. We also remark that this type of inviscid regularization has been recently used for the two-dimensional surface quasi-geostropic model [28]. In particular, necessary and sufficient conditions for the formation of singularity were presented in terms of regularizing parameter.…”

On the Inviscid Limit for the 2D Non-dissipative Quasi-geostrophic Equations

“…Recently, these equations have been of great interest and there have been many results in theory [3,5] and numerical analysis [3,6]. Indeed, the motivation of such studies on the equations is not only from its physical interest in atmosphere science, but also its striking resemblance to three dimensional incompressible Euler equations [1].…”

“…One can regard the finite finite time blow-up problem of the two-dimensional quasi-geostrophic equations as a very good model of the corresponding problem of the threedimensional Euler equations [2]. B.Khouider and E. S.Titi [3] showed the inviscid regularization for the surface quasigeostrophic equations. Constantin, Majda, and Tabak [2] proved the local well-posedness of 2D quasi-geostropphic equation for the initial data θ 0 ∈ H m (m ≥ 3) and gave the blow-up criterion.…”

Pullback attractors for three-dimensional non-autonomous Navier–Stokes–Voigt equations

“…On other hand, some questions on the inviscid regularization have been recently used for the study of a 2D surface quasi-geostrophic model (cf. [12]). …”