A regularity criterion for the critical and supercritical dissipative quasi-geostrophic equations

In this note, by using a well-known commutator estimate, we give a new regularity criterion for the 2D dissipative quasi-geostrophic equations in the critical and supercritical cases.     

On regularity criterion for the dissipative quasi-geostrophic equations

In this paper we consider the 2D dissipative quasi-geostrophic equations and study the regularity criterion of the solutions. By means of a commutator estimate based on frequency localization and Bony's paraproduct decomposition, we obtain a regularity criterion

     

A remark on regularity criterion for the dissipative quasi-geostrophic equations

This paper concerns with a regularity criterion of solutions to the 2D dissipative quasi-geostrophic equations. Based on a logarithmic Sobolev inequality in Besov spaces, the absence of singularities of θ in [0,T] is derived for θ a solution on the interval [0,T) satisfying the condition

     

A remark on the logarithmically improved regularity criterion for the micropolar fluid equations in terms of the pressure

In this paper, we study the regularity criterion of weak solutions of the three dimensional micropolar fluid flows. It is proved that if the pressure satisfies where denotes the critical Besov space, then the weak solution (u,w) becomes a regular solution on (0,T]. This regularity criterion can be regarded as log in time improvements of the standard Serrin's criteria established before. Copyright © 2011 John Wiley & Sons, Ltd.     

Gevrey regularity for the supercritical quasi-geostrophic equation

In this paper, following the techniques of Foias and Temam, we establish suitable Gevrey class regularity of solutions to the supercritical quasi-geostrophic equations in the whole space, with initial data in “critical” Sobolev spaces. Moreover, the Gevrey class that we obtain is “near optimal” and as a corollary, we obtain temporal decay rates of higher order Sobolev norms of the solutions. Unlike the Navier–Stokes or the subcritical quasi-geostrophic equations, the low dissipation poses a difficulty in establishing Gevrey regularity. A new commutator estimate in Gevrey classes, involving the dyadic Littlewood–Paley operators, is established that allow us to exploit the cancellation properties of the equation and circumvent this difficulty.     

A logarithmically improved blow-up criterion for smooth solutions to the 3D micropolar fluid equations

In this paper, the Cauchy problem for the 3D micropolar fluid equations is investigated. A new logarithmically improved blow-up criterion for the 3D micropolar fluid equations in an appropriate homogeneous Besov space is established.     

Logarithmically improved regularity criterion for the 3D generalized magneto-hydrodynamic equations

This article proves the logarithmically improved Serrin's criterion for solutions of the 3D generalized magneto-hydrodynamic equations in terms of the gradient of the velocity field, which can be regarded as improvement of results in [10] (Luo Y W. On the regularity of generalized MHD equations. J Math Anal Appl, 2010, 365: 806–808) and [18] (Zhang Z J. Remarks on the regularity criteria for generalized MHD equations. J Math Anal Appl, 2011, 375: 799–802).     

Global regularity for modified critical dissipative quasi-geostrophic equations

We consider the n-dimensional modified quasi-geostrophic (SQG) equations

t_∂θ + u . ∇θ + κΛαθ = 0,${\partial }_t\theta \hspace{0.17em}+\hspace{0.17em}u\hspace{0.17em}.\hspace{0.17em}\nabla \theta \hspace{0.17em}+\hspace{0.17em}\kappa {\Lambda }^{\alpha }\theta \hspace{0.17em}=\hspace{0.17em}0,$

u = Λα-1 R^⊥θ$u\hspace{0.17em}=\hspace{0.17em}{\Lambda }^{\alpha -1}\hspace{0.17em}{R}^{\perp }\theta$

with κ > 0, α ∈ (0,1] and θ₀ ∈ W1, ∞ (?ⁿ). In this paper, we establish a different proof for the global regularity of this system. The original proof was given by Constantin, Iyer, and Wu [5], who employed the approach of Besov space techniques to study the global existence and regularity of strong solutions to modified critical SQG equations for two dimensional case. The proof provided in this paper is based on the nonlinear maximum principle as well as the approach in Constantin and Vicol [2].     

Logarithmically improved regularity criterion for the harmonic heat flow and related equations

We use an interpolation inequality on Besov spaces to show a logarithmically improved regularity criterion for the harmonic heat flow, the Landau-Lifshitz equations, and the Landau-Lifshitz-Maxwell system.     

On the 2D critical and supercritical dissipative quasi-geostrophic equation in Besov spaces

We prove the local smoothing effect of the 2D critical and supercritical dissipative quasi-geostrophic equations in critical Besov spaces. As an application, a global well-posedness result is established by adapting a method in Kiselev, Nazarov, and Volberg (2007) [16] and an idea in Dong and Du (2008) [15] with suitable modifications. Moreover, we show that the unique solution obtained in Chen, Miao, and Zhang (2007) [11] is a classical solution. These generalize some previous results in Dong (2010) [13], Dong and Du (2008) [15]. The main ingredients of the proofs are two commutator estimates and the preservation of suitable modulus of continuity of the solutions.     

A blow-up criterion for 3D Boussinesq equations in Besov spaces

In this paper, we consider the 3D Boussinesq equations with the incompressibility condition. We obtain a regularity condition for the three-dimensional Boussinesq equations by means of the Littlewood-Paley theory and Bony’s paradifferential calculus.     

On the regularity criterion for three-dimensional micropolar fluid flows in Besov spaces

This paper studies the regularity criterion of weak solutions for three-dimensional (3D) micropolar fluid flows. When the velocity field satisfies for −1<r<1, then the weak solution (u,w) is regular on (0,T]. The methods are mainly based on the Fourier localization technique and Bony’s para-product decomposition.     

The critical Sobolev inequalities in Besov spaces and regularity criterion to some semi-linear evolution equations

We show the critical Sobolev inequalities in the Besov spaces with the logarithmic form such as Brezis-Gallouet-Wainger and Beale-Kato-Majda. As an application of those inequalities, the regularity problem under the critical condition to the Navier-Stokes equations, the Euler equations in and the gradient flow to the harmonic map to the sphere are discussed. Namely the Serrin-Ohyama type regularity criteria are improved in the terms of the Besov spaces. Received: 21 September 2000; in final form: 16 Feburary 2001 / Published online: 18 January 2002