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27
On the Euler equations of incompressible fluids
 Bull.Amer. Math. Soc
, 2007
"... 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 illposed free surface problems such as ..."
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Cited by 62 (1 self)
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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 illposed free surface problems such as the RayleighTaylor and KelvinHelmholtz 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. 1.
Nonlinear maximum principles for dissipative linear nonlocal operators and applications
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Global wellposedness for an advectiondiffusion equation arising in magnetogeostrophic dynamics
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Regularity and blow up for active scalars
"... We review some recent results for a class of fluid mechanics equations called active scalars, with fractional dissipation. Our main examples are the surface quasigeostrophic equation, the Burgers equation, and the CordobaCordobaFontelos model. We discuss nonlocal maximum principle methods which al ..."
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Cited by 21 (1 self)
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We review some recent results for a class of fluid mechanics equations called active scalars, with fractional dissipation. Our main examples are the surface quasigeostrophic equation, the Burgers equation, and the CordobaCordobaFontelos model. We discuss nonlocal maximum principle methods which allow to prove existence of global regular solutions for the critical dissipation. We also recall what is known about the possibility of finite time blow up in the supercritical regime. 1
GENERALIZED SURFACE QUASIGEOSTROPHIC EQUATIONS WITH SINGULAR VELOCITIES
"... Abstract. This paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals. The first family is a generalized surface quasigeostrophic (SQG) equation with the velocity field u r ..."
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Cited by 19 (2 self)
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Abstract. This paper establishes several existence and uniqueness results for two families of active scalar equations with velocity fields determined by the scalars through very singular integrals. The first family is a generalized surface quasigeostrophic (SQG) equation with the velocity field u related to the scalar θ by u = ∇ ⊥ Λ β−2 θ, where 1 < β ≤ 2 and Λ = (−∆) 1/2 is the Zygmund operator. The borderline case β = 1 corresponds to the SQG equation and the situation is more singular for β> 1. We obtain the local existence and uniqueness of classical solutions, the global existence of weak solutions and the local existence of patch type solutions. The second family is a dissipative active scalar equation with u = ∇ ⊥ (log(I − ∆)) µ θ for µ> 0, which is at least logarithmically more singular than the velocity in the first family. We prove that this family with any fractional dissipation possesses a unique local smooth solution for any given smooth data. This result for the second family constitutes a first step towards resolving the global regularity issue recently proposed by K. Ohkitani [84]. 1.
Inviscid models generalizing the 2D Euler and the surface quasigeostrophic equations
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Regularity criteria for the dissipative quasigeostrophic equations in Hölder spaces
 Comm. Math. Phys
"... Abstract. We study regularity criteria for weak solutions of the dissipative quasigeostrophic equation (with dissipation (−∆) γ/2, 0 < γ ≤ 1). We show in this paper that if θ ∈ C((0, T); C1−γ), or θ ∈ Lr ((0, T); Cα) with α = 1−γ + γ r is a weak solution of the 2D quasigeostrophic equation, the ..."
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Cited by 13 (1 self)
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Abstract. We study regularity criteria for weak solutions of the dissipative quasigeostrophic equation (with dissipation (−∆) γ/2, 0 < γ ≤ 1). We show in this paper that if θ ∈ C((0, T); C1−γ), or θ ∈ Lr ((0, T); Cα) with α = 1−γ + γ r is a weak solution of the 2D quasigeostrophic equation, then θ is a classical solution in (0, T] × R2. This result improves our previous result in [18]. 1.
A REGULARITY CRITERION FOR THE DISSIPATIVE QUASIGEOSTROPHIC EQUATIONS
, 710
"... Abstract. We establish a regularity criterion for weak solutions of the dissipative quasigeostrophic equations in mixed timespace Besov spaces. 1. ..."
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Cited by 12 (0 self)
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Abstract. We establish a regularity criterion for weak solutions of the dissipative quasigeostrophic equations in mixed timespace Besov spaces. 1.
ON THE LOSS OF CONTINUITY FOR SUPERCRITICAL DRIFTDIFFUSION EQUATIONS
"... ABSTRACT. We show that there exist solutions of driftdiffusion equations in two dimensions with divergencefree supercritical drifts, that become discontinuous in finite time. We consider classical as well as fractional diffusion. However, in the case of classical diffusion and timeindependent dr ..."
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Cited by 11 (3 self)
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ABSTRACT. We show that there exist solutions of driftdiffusion equations in two dimensions with divergencefree supercritical drifts, that become discontinuous in finite time. We consider classical as well as fractional diffusion. However, in the case of classical diffusion and timeindependent drifts we prove that solutions satisfy a modulus of continuity depending only on the local L 1 norm of the drift, which is a supercritical quantity. 1.