Global well-posedness for the critical 2D dissipative quasi-geostrophic equation

Kiselev, Alexander; Nazarov, Fëdor; Volberg, Alexander

doi:10.1007/s00222-006-0020-3

Cited by 450 publications

(596 citation statements)

References 7 publications

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“…Recently, two remarkable and quite different proofs of global existence in the critical case have been obtained. In one ( [92]) the result is one of global persistence of regularity, based on a new and promising idea, a maximum modulus of continuity principle. The other proof ( [21]) uses harmonic extension to prove a gain of regularity of weak solutions, in the spirit of the De Giorgi, from L 2 to L ∞ , from L ∞ to Hölder continuous, and beyond.…”

Section: Weak Solutionsmentioning

confidence: 99%

On the Euler equations of incompressible fluids

Constantin

2007

Bull. Amer. Math. Soc.

210

212

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Abstract. Euler equations of incompressible fluids use and enrich many branches of mathematics, from integrable systems to geometric analysis. They present important open physical and mathematical problems. Examples include the stable statistical behavior of ill-posed free surface problems such as the Rayleigh-Taylor and Kelvin-Helmholtz instabilities. The paper describes some of the open problems related to the incompressible Euler equations, with emphasis on the blowup problem, the inviscid limit and anomalous dissipation. Some of the recent results on the quasigeostrophic model are also mentioned.

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Section: Weak Solutionsmentioning

confidence: 99%

On the Euler equations of incompressible fluids

Constantin

2007

Bull. Amer. Math. Soc.

210

212

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“…And there are applications in Fluid Mechanics, cf. [29,45] and the references therein. An extensive list of current applications is contained in the survey paper [37].…”

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confidence: 99%

Nonlinear Diffusion with Fractional Laplacian Operators

2012

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We describe two models of flow in porous media including nonlocal (longrange) diffusion effects. The first model is based on Darcy's law and the pressure is related to the density by an inverse fractional Laplacian operator. We prove existence of solutions that propagate with finite speed. The model has the very interesting property that mass preserving self-similar solutions can be found by solving an elliptic obstacle problem with fractional Laplacian for the pair pressure-density. We use entropy methods to show that they describe the asymptotic behaviour of a wide class of solutions. The second model is more in the spirit of fractional Laplacian flows, but nonlinear. Contrary to usual Porous Medium flows (PME in the sequel), it has infinite speed of propagation. Similarly to them, an L 1 -contraction semigroup is constructed and it depends continuously on the exponent of fractional derivation and the exponent of the nonlinearity. Nonlinear diffusion and fractional diffusionSince the work by Einstein [39] and Smoluchowski [62] at the beginning of the last century (cf. also Bachelier [9]), we possess an explantation of diffusion and Brownian motion in terms of the heat equation, and in particular of the Laplace operator. This explanation has had an enormous success both in Mathematics and Physics. In the decades that followed, the Laplace operator has been often replaced by more general types of so-called elliptic operators with variable coefficients, and later by nonlinear differential operators; a huge body of theory is now available, both for the evolution equations [49] and for the stationary states, described by elliptic equations of different kinds [50,42].

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“…Taking the gradient with respect to x for equation (1.1), we have It is noticed that the classical quasi-geostrophic equation takes the same form (cf. [6,5,8], etc. ): .…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Well-posedness of fully nonlinear and nonlocal critical parabolic equations

Zhang

2012

J. Evol. Equ.

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Abstract. In this paper we prove the existence of smooth solutions to fully nonlinear and nonlocal parabolic equations with critical index. The proof relies on the apriori Hölder estimate for advection fractional-diffusion equation established by Silvestre [11]. Introduction and main resultIn this paper we are interested in solving the following fully nonlinear and nonlocal parabolic equation:where F (t, x, u, w, q) where F denotes the Fourier's transform, S(R d ) is the Schwartz class of smooth real-valued rapidly decreasing functions.Recently, in the sense of viscosity solutions, fully nonlinear and nonlocal elliptic and parabolic equations have been extensively studied (cf. [4,10,3,9], etc.). In [4], Caffarelli and Silvestre studied the following type of nonlocal equation:where α ∈ (0, 2), i, j ranges in arbitrary sets, c i j ∈ R and b i j ∈ R d , the kernel a i j (y) satisfiesThis type of equation appears in the stochastic control problems. In [4], the extremal Pucci operators are used to characterize the ellipticity, and the ABP estimate, Harnack inequality and interior C 1,β -regularity were obtained. In [11], Silvestre studied the following nonlocal parabolic equation with critical index α = 1:and established C 1,β -regularity of viscosity solutions. In particular, the following first order Hamilton-Jacobi equation is covered by the above equation when H is Lipschitz continuous:In [9], Lara and Davila extended Silvestre's result to the more general case, and in particular, focused on the uniformity of regularity as α → 2.However, it is not known how to solve the fully nonlinear and nonlocal equation (1.1) in Sobolev spaces. Let us fix the main idea of the present paper for solving (1.1). Assume that F does not depend on u. Taking the gradient with respect to x for equation (1.1), we have We make the following observation:

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Global well-posedness for the critical 2D dissipative quasi-geostrophic equation

Abstract: We give an elementary proof of the global well-posedness for the critical 2D dissipative quasi-geostrophic equation. The argument is based on a non-local maximum principle involving appropriate moduli of continuity.

Cited by 450 publications

References 7 publications

On the Euler equations of incompressible fluids

On the Euler equations of incompressible fluids

Nonlinear Diffusion with Fractional Laplacian Operators

Well-posedness of fully nonlinear and nonlocal critical parabolic equations

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