Regularity results on the Leray-alpha magnetohydrodynamics systems

We study certain generalized Leray-alpha magnetohydrodynamics systems. We show that the solution pairs of velocity and magnetic fields to this system in two-dimension preserve their initial regularity in two cases: dissipation logarithmically weaker than a full Laplacian and zero diffusion, zero dissipation and diffusion logarithmically weaker than a full Laplacian. These results extend previous results in Zhou and Fan (2011). Moreover, we show that for a certain three-dimensional Leray-alpha magnetohydrodynamics system, sufficient condition of regularity may be reduced to a horizontal gradient or a partial derivative in just one direction of the magnetic field, reducing components from the results in Fan and Ozawa (2009).     

A regularity criterion for two‐and‐half‐dimensional magnetohydrodynamic equations with horizontal dissipation and horizontal magnetic diffusion

We study the global regularity of classical solution to two‐and‐half‐dimensional magnetohydrodynamic equations with horizontal dissipation and horizontal magnetic diffusion. We prove that any possible finite time blow‐up can be controlled by the L^∞‐norm of the vertical components. Copyright © 2016 John Wiley & Sons, Ltd.     

Global regularity of generalized magnetic Benard problem

We study the magnetic Bénard problem in two‐dimensional space with generalized dissipative and diffusive terms, namely, fractional Laplacians and logarithmic supercriticality. Firstly, we show that when the diffusive term for the magnetic field is a full Laplacian, the solution initiated from data sufficiently smooth preserves its regularity as long as the power of the fractional Laplacians for the dissipative term of the velocity field and the diffusive term of the temperature field adds up to 1. Secondly, we show that with zero dissipation for the velocity field and a full Laplacian for the diffusive term of the temperature field, the global regularity result also holds when the diffusive term for the magnetic field consists of the fractional Laplacian with its power strictly bigger than 1. Finally, we show that with no diffusion from the magnetic and the temperature fields, the global regularity result remains valid as long as the dissipation term for the velocity field has its strength at least at the logarithmically supercritical level. These results represent various extensions of previous work on both Boussinesq and magnetohydrodynamics systems. Copyright © 2016 John Wiley & Sons, Ltd.     

The regularity criterion for 3D Navier–Stokes equations involving one velocity gradient component

In this article, we establish sufficient conditions for the regularity of solutions of Navier–Stokes equations based on one of the nine entries of the gradient tensor. We improve the recent results of C.S. Cao, E.S. Titi [C.S. Cao, E.S. Titi, Global regularity criterion for the 3D Navier–Stokes equations involving one entry of the velocity gradient tensor, Arch. Ration. Mech. Anal. 202 (2011) 919–932] and Y. Zhou, M. Pokorný [Y. Zhou, M. Pokorný, On the regularity of the solutions of the Navier–Stokes equations via one velocity component, Nonlinearity 23 (2010) 1097–1107].     

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 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 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 of 2D Leray-Alpha Regularized Tropical Climate Models

In this paper, we establish the global regularity of 2D leray-alpha regularized tropical climate models. The global strong solution to the system with a half Laplacian of the first baroclinic model of velocity $(\Lambda v)$ and thermal diffusion $(-\Delta\theta)$ or with only the dissipation of the barotropic mode $(-\Delta u)$ are obtained.     

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.     

Global smooth solutions of the 2D MHD equations for a class of large data without magnetic diffusion

In this paper, we establish the global regularity of classical solutions for the two-dimensional MHD equations with only velocity diffusion for a class of large initial data. Both the initial velocity and magnetic field can be arbitrarily large in $H^s$.     

The 2D magnetohydrodynamic system with Euler-like velocity equation and partial magnetic diffusion

When the velocity equation of the incompressible 2D magnetohydrodynamic (MHD) system is inviscid, the global well-posedness and stability problem in the whole space ${\mathbb{R}}^2$ case remains an extremely challenging open problem. Broadman, Lin, and Wu (SIAM J. Math. Anal. 52(5) (2020): 5001-5035) were able to establish the global well-posedness and stability near a background magnetic field when there is damping in one velocity component. Their work exploited the stabilizing effect of the background magnetic field. This paper presents new progress. We are able to prove the global well-posedness and stability even when the magnetic diffusion is degenerate and only in the vertical direction. The velocity equation is still inviscid and has damping only in the vertical component. The proof of this new result overcomes two main difficulties, the potential rapid growth of the velocity due to the lack of dissipation or horizontal damping and the control of nonlinearity associated with the magnetic field. By discovering the key hidden smoothing effects and incorporating them in the construction of a two-layered energy function, we are able to obtain uniform bounds on the solution in the H³-norm when the initial perturbation is near the background magnetic field. In addition, we prove that certain Lebesgue and Sobolev norms of the solution approach zero as time approaches infinity.     

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.     

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.     

A New Regularity Criterion for the 3D MHD Equations Involving Partial Components

In this paper, we establish a new regularity criterion for the 3D incompressible MHD equations involving partial components of the velocity gradient and magnetic fields.     

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.     

A new scaling invariant regularity criterion for the 3D MHD equations in terms of horizontal gradient of horizontal components

We prove a new scaling invariant regularity criterion for the 3D MHD equations via horizontal gradient of horizontal components of weak solutions. This result improves a recent work by Ni et al. (2012), in the sense that the assumption on the horizontal gradient of the vertical components is removed. As a byproduct, a scaling invariant regularity criterion involving vertical components of vorticity and current density is also obtained.     

Blowup mechanism for viscous compressible heat-conductive magnetohydrodynamic flows in three dimensions

We investigate initial-boundary-value problem for three-dimensional magnetohydrodynamic(MHD)system of compressible viscous heat-conductive flows and the three-dimensional full compressible Navier-Stokes equations. We establish a blowup criterion only in terms of the derivative of velocity field, similar to the Beale-Kato-Majda type criterion for compressible viscous barotropic flows by Huang et al.(2011). The results indicate that the nature of the blowup for compressible MHD models of viscous media is similar to the barotropic compressible Navier-Stokes equations and does not depend on further sophistication of the MHD model, in particular, it is independent of the temperature and magnetic field. It also reveals that the deformation tensor of the velocity field plays a more dominant role than the electromagnetic field and the temperature in regularity theory. Especially, the similar results also hold for compressible viscous heat-conductive Navier-Stokes flows,which extend the results established by Fan et al.(2010), and Huang and Li(2009). In addition, the viscous coefficients are only restricted by the physical conditions in this paper.     

Global regularity to the 2D incompressible MHD with mixed partial dissipation and magnetic diffusion in a bounded domain

This article considers the global regularity to the initial–boundary value problem for the 2D incompressible MHD with mixed partial dissipation and magnetic diffusion.To overcome the difficulty caused by the vanishing viscosities,we first establish the elliptic system for uxand by,which are estimated by▽×u_x and▽×b_y,respectively.Then,we establish the global estimates for▽×u and▽×b.     

Leray–Hopf solutions to a viscoelastoplastic fluid model with nonsmooth stress–strain relation

We consider a fluid model including viscoelastic and viscoplastic effects. The state is given by the fluid velocity and an internal stress tensor that is transported along the flow with the Zaremba–Jaumann derivative. Moreover, the stress tensor obeys a nonlinear and nonsmooth dissipation law as well as stress diffusion. We prove the existence of global-in-time weak solutions satisfying an energy inequality under general Dirichlet conditions for the velocity field and Neumann conditions for the stress tensor.     

Stochastic dynamo in critical situations

Based on the functional method of consecutive approximations, we consider the problem of magnetic field excitation (stochastic dynamo) by a random velocity field with a finite temporal correlation radius. In critical situations, in the first (diffusion) approximation, the Lyapunov characteristic parameter of the magnetic field energy vanishes. This implies the absence of structure formation (clustering) in realizations of the magnetic field in that approximation. Critical situations occur in problems of magnetic field diffusion in an equilibrium thermal and random pseudoequilibrium and acoustic (in the absence of dissipation) velocity fields. The sign of the Lyapunov characteristic parameter in the second-order approximation determines the possibility of clustering of the magnetic field energy. We show that energy clustering does not occur in a thermal velocity field. In the cases of pseudoequilibrium and acoustic velocity fields, clustering occurs with probability one, i.e., in almost every realization. We evaluate the characteristic time for clustering to be established.