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Inviscid models generalizing the two-dimensional Euler and the surface quasi-geostrophic equations. (English) Zbl 1266.76010

Summary: Any classical solution of the two-dimensional incompressible Euler equation is global in time. However, it remains an outstanding open problem whether classical solutions of the surface quasi-geostrophic (SQG) equation preserve their regularity for all time. This paper studies solutions of a family of active scalar equations in which each component \(u_{j}\) of the velocity field \(u\) is determined by the scalar \(\theta \) through \({u_j =\mathcal{R}\Lambda^{-1}P(\Lambda) \theta}\), where \({\mathcal{R}}\) is a Riesz transform and \(\Lambda = (-\Delta )^{1/2}\). The two-dimensional Euler vorticity equation corresponds to the special case \(P(\Lambda ) = I\) while the SQG equation corresponds to the case \(P(\Lambda ) = \Lambda \). We develop tools to bound \({\|\nabla u||_{L^\infty}}\) for a general class of operators \(P\) and establish the global regularity for the Loglog-Euler equation for which \(P(\Lambda ) = (\log (I + \log (I - \Delta)))^{\gamma }\) with \(0 \leqq \gamma \leqq 1\). In addition, a regularity criterion for the model corresponding to \(P(\Lambda ) = \Lambda ^{\beta}\) with \(0 \leqq \beta \leqq 1\) is also obtained.

MSC:

76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35Q31 Euler equations
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