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We prove the absence of anomalous dissipation of energy for long time averaged solutions of the forced critical surface quasi-geostrophic equation in two spatial dimensions. [ABSTRACT FROM AUTHOR]

We obtain a family of nonlinear maximum principles for linear dissipative nonlocal operators, that are general, robust, and versatile. We use these nonlinear bounds to provide transparent proofs of global regularity for critical SQG and critical d-dimensional Burgers equations. In addition we give applications of the nonlinear maximum principle to the global regularity of a slightly dissipative anti-symmetric perturbation of 2D incompressible Euler equations and generalized fractional dissipative 2D Boussinesq equations. [ABSTRACT FROM AUTHOR]

The global regularity problem concerning the inviscid SQG and the 3D Euler equations remains an outstanding open question. This paper presents several geometric observations on solutions of these equations. One observation stems from a relation between what we call Eulerian and Lagrangian deformations and reflects the alignment of the stretching directions of these deformations and the tangent direction of the level curves for the SQG equation. Various spatial symmetries in solutions to the 3D Euler equations are exploited. In addition, two observations on the curvature of the level curves of the SQG equation are also included. [ABSTRACT FROM AUTHOR]

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 of the velocity field u is determined by the scalar θ through $${u_j =\mathcal{R}\Lambda^{-1}P(\Lambda) \theta}$$ , where $${\mathcal{R}}$$ is a Riesz transform and Λ = (−Δ). The two-dimensional Euler vorticity equation corresponds to the special case P(Λ) = I while the SQG equation corresponds to the case P(Λ) = Λ. 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(Λ) = (log( I + log( I − Δ))) with 0 ≦ γ ≦ 1. In addition, a regularity criterion for the model corresponding to P(Λ) = Λ with 0 ≦ β ≦ 1 is also obtained. [ABSTRACT FROM AUTHOR]

Abstract: We examine the regularity of weak solutions of quasi-geostrophic (QG) type equations with supercritical () dissipation . This study is motivated by a recent work of Caffarelli and Vasseur, in which they study the global regularity issue for the critical () QG equation [L. Caffarelli, A. Vasseur, Drift diffusion equations with fractional diffusion and the quasi-geostrophic equation, arXiv: math.AP/0608447, 2006]. Their approach successively increases the regularity levels of Leray–Hopf weak solutions: from to , from to Hölder (, ), and from Hölder to classical solutions. In the supercritical case, Leray–Hopf weak solutions can still be shown to be , but it does not appear that their approach can be easily extended to establish the Hölder continuity of solutions. In order for their approach to work, we require the velocity to be in the Hölder space . Higher regularity starting from with can be established through Besov space techniques and will be presented elsewhere [P. Constantin, J. Wu, Regularity of Hölder continuous solutions of the supercritical quasi-geostrophic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, in press]. [Copyright &y& Elsevier]

Abstract: We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical () dissipation : If a Leray–Hopf weak solution is Hölder continuous with on the time interval , then it is actually a classical solution on . [Copyright &y& Elsevier]

Abstract: We will describe necessary and sufficient conditions for blowup and discuss weak solutions for the incompressible Euler equations. We will also describe a result concerning anomalous dissipation of energy. [Copyright &y& Elsevier]

In this work we derive lower bounds for the Hausdorff and fractal dimensions of the global attractor of the Sabra shell model of turbulence in different regimes of parameters. We show that for a particular choice of the forcing term and for sufficiently small viscosity term ν, the Sabra shell model has a global attractor of large Hausdorff and fractal dimensions proportional to log ν −1 for all values of the governing parameter ε, except for ε =1. The obtained lower bounds are sharp, matching the upper bounds for the dimension of the global attractor obtained in our previous work. Moreover, the complexity of the dynamics of the shell model increases as the viscosity ν tends to zero, and we describe a precise scenario of successive bifurcations for different parameters regimes. In the “three-dimensional” regime of parameters this scenario changes when the parameter ε becomes sufficiently close to 0 or to 1. We also show that in the “two-dimensional” regime of parameters, for a certain non-zero forcing term, the long-term dynamics of the model becomes trivial for every value of the viscosity. [ABSTRACT FROM AUTHOR]

We present an infinite-dimensional family of exact solutions of the incompressible threedimensional Euler equations. These solutions have infinite kinetic energy and blow up in finite time. [ABSTRACT FROM AUTHOR]