Global Regularity for the 2D Magneto-Micropolar Equations with Partial Dissipation

This paper studies the global existence and regularity of classical solutions to the 2D incompressible magneto-micropolar equations with partial dissipation. The magneto-micropolar equations model the motion of electrically conducting micropolar fluids in the presence of a magnetic field. When there is only partial dissipation, the global regularity problem can be quite difficult. We are able to single out three special partial dissipation cases and establish the global regularity for each case. As special consequences, the 2D Navier-Stokes equations, the 2D magnetohydrodynamic equations, and the 2D micropolar equations with several types of partial dissipation always possess global classical solutions. The proofs of our main results rely on anisotropic Sobolev type inequalities and suitable combination and cancellation of terms.     

Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion

Whether or not classical solutions of the 2D incompressible MHD equations without full dissipation and magnetic diffusion can develop finite-time singularities is a difficult issue. A major result of this paper establishes the global regularity of classical solutions for the MHD equations with mixed partial dissipation and magnetic diffusion. In addition, the global existence, conditional regularity and uniqueness of a weak solution is obtained for the 2D MHD equations with only magnetic diffusion.     

Global regularity for d-dimensional micropolar equations with fractional dissipation

This paper is devoted to the global in time existence of classical solutions to the d-Dimensional (dD) micropolar equations with fractional dissipation. Micropolar equations model a class of fluids with nonsymmetric stress tensor such as fluids consisting of particles suspended in a viscous medium. It remains unknown whether or not smooth solutions of the classical 3D micropolar equations can develop finite-time singularities. The purpose here is to explore the global regularity of solutions for dD micropolar equations under the smallest amount of dissipation. We establish the global regularity for two important fractional dissipation cases. Direct energy estimates are not sufficient to obtain the desired global a priori bounds in each case. To overcome the difficulties, we employ the Besov space techniques.     

A study on the global regularity for a model of the 3D axisymmetric Navier-Stokes equations

This paper investigates the global regularity issue concerning a model equation proposed by Hou and Lei (2008) [9] to understand the stabilizing effects of the nonlinear terms in the 3D axisymmetric Navier-Stokes and Euler equations. We establish the global regularity of a generalized version of their model with a fractional Laplacian when the fractional power satisfies an explicit condition. This condition is exactly the same as in the case of the 3D generalized Navier-Stokes equations and is due to the balance between a more regular nonlinearity and a less effective (five-dimensional) Laplacian.     

The 2D magneto‐micropolar equations with partial dissipation

We study the global existence and regularity of classical solutions to the 2D incompressible magneto‐micropolar equations with partial dissipation. We establish the global regularity for one partial dissipation case. The proofs of our main results rely on anisotropic Sobolev type inequalities and suitable combination and cancellation of terms.     

Global well-posedness for a class of 2D Boussinesq systems with fractional dissipation

The incompressible Boussinesq equations not only have many applications in modeling fluids and geophysical fluids but also are mathematically important. The well-posedness and related problem on the Boussinesq equations have recently attracted considerable interest. This paper examines the global regularity issue on the 2D Boussinesq equations with fractional Laplacian dissipation and thermal diffusion. Attention is focused on the case when the thermal diffusion dominates. We establish the global well-posedness for the 2D Boussinesq equations with a new range of fractional powers of the Laplacian.     

Nonlinear maximum principles for dissipative linear nonlocal operators and applications

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.     

Global regularity of the 2D magnetohydrodynamic equations with fractional anisotropic dissipation

This paper is dedicated to establishing the global regularity for the two dimensional magnetohydrodynamic equations with fractional anisotropic dissipation when the fractional powers are restricted to some certain ranges. In addition, the global regularity results for the two dimensional magnetohydrodynamic equations with partial dissipation are also obtained. Consequently, these results bring us more closer to the resolution of the global regularity problem on the two dimensional magnetohydrodynamic equations with standard Laplacian magnetic diffusion.     

Global regularity for 3D magneto-hydrodynamics equations with only horizontal dissipation

We investigate the Cauchy problem for the 3D magneto-hydrodynamics equations with only horizontal dissipation for the small initial data. With the help of the dissipation in the horizontal direction and the structure of the system, we analyze the properties of the decay of the solution and apply these decay properties to get the global regularity of the solution. In the process, we mainly use the frequency decomposition in Green's function method and energy method.     

Regularity criteria for the 3D generalized MHD equations in terms of vorticity

We prove regularity criteria for the 3D generalized MHD equations. These criteria impose assumptions on the vorticity only. In addition, we also prove a result of global existence for smooth solution under some special conditions.     

On the global existence and small dissipation limit for generalized dissipative Zakharov system

This paper deals with the existence and uniqueness of the global solutions to the initial boundary value problem for a generalized Zakharov system with direct self‐interaction of the dispersive waves and weak dissipation in the nondispersive subsystem. We prove the global existence of the generalized solution to the problem by a priori estimates and Galerkin method. We also establish the regularity of the global generalized solution and the existence and uniqueness of the global classical solution. Moreover, we obtain the convergence of the solutions of the generalized Zakharov system with dissipation as the dissipative coefficient approaches zero.     

A regularity criterion for 2D MHD flows with horizontal dissipation and horizontal magnetic diffusion

This paper concerns the conditional global regularity of incompressible MHD equations with horizontal dissipation and horizontal magnetic diffusion in two dimension. When only horizontal dissipation and horizontal magnetic diffusion are present, there is no control on the vertical derivatives of velocity field and magnetic field, which is the main difficulty to establish the global regularity. In this paper, we establish a global regularity criterion in terms of one entry of the velocity gradient tensor or one entry of the magnetic field gradient tensor, which extends the recent work (Fan and Ozawa, 2014).     

Non-blow-up of the 3D ideal magnetohydrodynamics equations for a class of three-dimensional initial data in cylindrical domains

The non blow-up of the 3D ideal incompressible magnetohydrodynamics (MHD) equations is proved for a class of three-dimensional initial data characterized by uniformly large vorticity and magnetic field in bounded cylindrical domains. There are no conditional assumptions on properties of solutions at later times, nor are the global solutions close to some 2D manifold. The approach of proving regularity is based on investigation of fast, singular, oscillating limits and nonlinear averaging methods in the context of almost periodic functions. We establish the global regularity of the 3D limit resonant MHD equations without any restrictions on the size of the 3D initial data. After establishing the strong convergence to the limit resonant equations, we bootstrap this into the regularity on arbitrarily large time intervals for solutions of the 3D MHD equations with weakly-aligned uniformly large vorticity and magnetic field at t = 0. Bibliography: 36 titles. Dedicated to the memory of O. A. Ladyzhenskaya Published in Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 203–219.     

On the global regularity of <Emphasis Type="Italic">N</Emphasis>-dimensional generalized Boussinesq system

We study the N-dimensional Boussinesq system with dissipation and diffusion generalized in terms of fractional Laplacians. In particular, we show that given the critical dissipation, a solution pair remains smooth for all time even with zero diffusivity. In the supercritical case, we obtain component reduction results of regularity criteria and smallness conditions for the global regularity in dimensions two and three.     

Regularity for 3D Navier-Stokes Equations Based on Energy Distribution at Wavenumber Bands

We present some regularity criteria for the Leray-Hopf weak solutions to the Cauchy problem for 3D Navier-Stokes equations based on energy distribution at wavenumber bands. We show sufficient conditions for regularity based on the ratios of energy dissipation rates at sufficiently high wavenumber bands and neighboring medium wavenumber bands. Moreover, we give a regularity criterion based on relations between energy at high and low wavenumber bands.     

Local existence of strong solutions to the generalized Boussinesq equations

This paper deals with the local existence and uniqueness of strong solutions for the generalized Boussinesq equations with fractional dissipation. As a corollary, we establish some regularity criteria to guarantee smoothness of solutions.     

Finite dimensionality of the global attractors for von Karman equations with nonlinear interior dissipation

In this paper we study the global attractors for von Karman equations with nonlinear interior dissipation. We prove regularity and then establish finite dimensionality of the global attractors without assuming large values for the damping parameter.     

Finite dimensionality and regularity of attractors for a 2-D semilinear wave equation with nonlinear dissipation

We consider a semilinear wave equation, defined on a two-dimensional bounded domain Ω, with a nonlinear dissipation. Our main result is that the flow generated by the model is attracted by a finite dimensional global attractor. In addition, this attractor has additional regularity properties that depend on regularity properties of nonlinear functions in the equation. To our knowledge this is a first result of this type in the context of higher dimensional wave equations.     

Gevrey regularity for a class of dissipative equations with applications to decay

In this paper, following the techniques of Foias and Temam, we establish Gevrey class regularity of solutions to a class of dissipative equations with a general quadratic nonlinearity and a general dissipation including fractional Laplacian. The initial data is taken to be in Besov type spaces defined via “caloric extension”. We apply our result to the Navier–Stokes equations, the surface quasi-geostrophic equations, the Kuramoto–Sivashinsky equation and the barotropic quasi-geostrophic equation. Consideration of initial data in critical regularity spaces allow us to obtain generalizations of existing results on the higher order temporal decay of solutions to the Navier–Stokes equations. In the 3D case, we extend the class of initial data where such decay holds while in 2D we provide a new class for such decay. Similar decay result, and uniform analyticity band on the attractor, is also proven for the sub-critical 2D surface quasi-geostrophic equation.     

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).