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      Inviscid models generalizing the 2D Euler and the surface quasi-geostrophic equations

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          Abstract

          Any classical solution of the 2D 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 2D Euler vorticity equation corresponds to the special case \(P(\Lambda)=I\) while the SQG equation 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 \(0\le \gamma\le 1\). In addition, a regularity criterion for the model corresponding to \(P(\Lambda)=\Lambda^\beta\) with \(0\le \beta\le 1\) is also obtained.

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          An extension problem related to the fractional Laplacian

          The operator square root of the Laplacian \((-\lap)^{1/2}\) can be obtained from the harmonic extension problem to the upper half space as the operator that maps the Dirichlet boundary condition to the Neumann condition. In this paper we obtain similar characterizations for general fractional powers of the Laplacian and other integro-differential operators. From those characterizations we derive some properties of these integro-differential equations from purely local arguments in the extension problems.
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            Formation of strong fronts in the 2-D quasigeostrophic thermal active scalar

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              Surface quasi-geostrophic dynamics

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                Author and article information

                Journal
                07 October 2010
                Article
                10.1007/s00205-011-0411-5
                1010.1506
                a9486c9b-cd37-416d-bbe7-d8f36f7dbb46

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

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                Custom metadata
                35Q35, 76D03
                math.AP

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