Energy dissipation for weak solutions of incompressible MHD equations

In this article, we mainly study the local equation of energy for weak solutions of 3D MHD equations. We define a dissipation term D(u, B) that stems from an eventual lack of smoothness in the solution, and then obtain a local equation of energy for weak solutions of 3D MHD equations. Finally, we consider the 2D case at the end of this article.     

Existence of solutions and Gevrey class regularity for Leray-alpha equations

In this paper we consider some equations similar to Navier-Stokes equations, the three-dimensional Leray-alpha equations with space periodic boundary conditions. We establish the regularity of the equations by using the classical Faedo-Galerkin method. Our argument shows that there exist an unique weak solution and an unique strong solution for all the time for the Leray-alpha equations, furthermore, the strong solutions are analytic in time with values in the Gevrey class of functions (for the space variable). The relations between the Leray-alpha equations and the Navier-Stokes equations are also considered.     

Weak and strong solutions for the incompressible Navier-Stokes equations with damping

In this paper, we show that the Cauchy problem of the Navier-Stokes equations with damping α|u|^β−1u(α>0) has global weak solutions for any β?1, global strong solution for any β?7/2 and that the strong solution is unique for any 7/2?β?5.     

Existence of weak solutions to the three-dimensional steady compressible magnetohydrodynamic equations

The purpose of this paper is to prove the existence of a spatially periodic weak solution to the steady compressible isentropic MHD equations in R3 for any specific heat ratio γ 1.The proof is based on the weighted estimates of both pressure and kinetic energy for the approximate system which result in some higher integrability of the density,and the method of weak convergence.According to the author's knowledge,it is the first result that treats in three dimensions the existence of weak solutions to the steady compressible MHD equations with γ 1.     

Regularity criteria for the three-dimensional MHD equations

In this paper,we consider regularity criteria for solutions to the 3D MHD equations with incompressible conditions.By using some classical inequalities,we obtain the regularity of strong solutions to the three-dimensional MHD equations under certain sufficient conditions in terms of one component of the velocity field and the magnetic field respectively.     

Differential equations in spaces of compact operators

We study a system of differential equations in C(H), the space of all compact operators on a separable complex Hilbert space, H. The systems considered are infinite-dimensional generalizations of mathematical models of learning, implementable as artificial neural networks. In this new setting, in addition to the usual questions of existence and uniqueness of solutions, we discuss issues which are operator theoretic in nature. Under some restrictions on the initial condition, we explicitly solve the system and represent the solution in terms of the spectral representation of the initial condition. We also discuss the stability of those solutions, and describe the weak, strong, and uniform limit sets in terms of their respective spectral properties.     

Remark on the regularity for weak solutions to the magnetohydrodynamic equations

We study the regularity criteria for weak solutions to the incompressible magnetohydrodynamic (MHD) equations. Some regularity criteria, which are related only with u+B or u?B, are obtained for weak solutions to the MHD equations. Copyright © 2008 John Wiley & Sons, Ltd.     

Global analysis for strong solutions to the equations of a ferrofluid flow model

In this paper we are concerned with the differential system proposed by Shliomis to describe the motion of an incompressible ferrofluid submitted to an external magnetic field. The system consists of the Navier-Stokes equations, the magnetization equations and the magnetostatic equations. No regularizing term is added to the magnetization equations. We prove the local existence of unique strong solution for the Cauchy problem and establish a finite time blow-up criterion of strong solutions. Under the smallness assumption of the initial data and the external magnetic field, we prove the global existence of strong solutions and derive a decay rate of such small solutions in L²-norm.     

Partial compactness for Landau-Lifshitz maxwell equation in two-dimension

We study the partial regularity of weak solutions to the 2-dimensional LandauLifshitz equations coupled with time dependent Maxwell equations by Ginzburg-Landau type approximation. Outside an energy concentration set of locally finite 2-dimensional parabolic Hausdorff measure, we prove the uniform local C ∞ bounds for the approaching solutions and then extract a subsequence converging to a global weak solution of the Landau-Lifshitz-Maxwell equations which are smooth away from finitely many points.     

Regularity criteria for suitable weak solutions of the Navier-Stokes equations near the boundary

We present some new regularity criteria for “suitable weak solutions” of the Navier-Stokes equations near the boundary in dimension three. We prove that suitable weak solutions are Hölder continuous up to the boundary provided that the scaled mixed norm with 3/p+2/q?2, 2<q?∞, (p,q)≠(3/2,∞) is small near the boundary. Our methods yield new results in the interior case as well. Partial regularity of weak solutions is also analyzed under some additional integral conditions.     

Decay properties of solutions to the incompressible magnetohydrodynamics equations in a half space

We consider the asymptotic behavior of the strong solution to the incompressible magnetohydrodynamics (MHD) equations in a half space. The L^r‐decay rates of the strong solution and its derivatives with respect to space variables and time variable, including the L¹ and L ^∞ decay rates of its first order derivatives with respect to space variables, are derived by using L^q ? L^r estimates of the Stokes semigroup and employing a decomposition for the nonlinear terms in MHD equations. In addition, if the given initial data lie in a suitable weighted space, we obtain more rapid decay rates than observed in general. Similar results are known for incompressible Navier–Stokes equations in a half space under same assumption. Copyright © 2012 John Wiley & Sons, Ltd.     

Large time behavior for the incompressible magnetohydrodynamic equations in half‐spaces

The weighted L^r‐asymptotic behavior of the strong solution and its first‐order spacial derivatives to the incompressible magnetohydrodynamic (MHD) equations is established in a half‐space. Further, the L^∞‐decay rates of the second‐order spatial derivatives of the strong solution are derived by using the Stokes solution formula and employing a decomposition for the nonlinear terms in MHD equations. Copyright © 2014 John Wiley & Sons, Ltd.     

A class of large solutions to the 3D incompressible MHD and Euler equations with damping

In this paper, we derive the global existence of smooth solutions of the 3 D incompressible Euler equations with damping for a class of laxge initial data, whose Sobolev norms H~s can be arbitrarily large for any s ≥ 0. The approach is through studying the quantity representing the difference between the vorticity and velocity. And also, we construct a family of large solutions for MHD equations with damping.     

Almost automorphic solutions for some evolution equations through the minimizing for some subvariant functional, applications to heat and wave equations with nonlinearities

In this work, we study the existence of bounded and almost automorphic solutions for evolution equations in Banach spaces. We suppose that the linear part is the infinitesimal generator of a compact C₀-semigroup of bounded linear operators and the nonlinear part is an almost automorphic function with respect to the second argument. We give sufficient conditions ensuring the existence of an almost automorphic solution when there is at least one bounded solution on R⁺. We use the subvariant functional method to show that every K-minimizing mild solution is compact almost automorphic. Applications are provided for both heat and wave equations with nonlinearities in several functional spaces.     

On global attractors of the 3D Navier-Stokes equations

In view of the possibility that the 3D Navier-Stokes equations (NSE) might not always have regular solutions, we introduce an abstract framework for studying the asymptotic behavior of multi-valued dissipative evolutionary systems with respect to two topologies—weak and strong. Each such system possesses a global attractor in the weak topology, but not necessarily in the strong. In case the latter exists and is weakly closed, it coincides with the weak global attractor. We give a sufficient condition for the existence of the strong global attractor, which is verified for the 3D NSE when all solutions on the weak global attractor are strongly continuous. We also introduce and study a two-parameter family of models for the Navier-Stokes equations, with similar properties and open problems. These models always possess weak global attractors, but on some of them every solution blows up (in a norm stronger than the standard energy one) in finite time.     

Global existence and zero α limit of the 3D incompressible two‐fluid MHD equations

In this paper, we establish global existence of strong solutions to the 3D incompressible two‐fluid MHD equations with small initial data. In addition, the explicit convergence rate of strong solutions from the two‐fluid MHD equations to the Hall‐MHD equations is obtained as . Copyright © 2017 John Wiley & Sons, Ltd.     

Remarks on equations of Benjamin-Bona-Mahony type

We investigate an initial-boundary value problem for equations of Benjamin-Bona-Mahony (BBM) type in two different physical situations. In the first, the mixed problem is considered on a cylinder domain Q of Rn×Rt. In the second one, the mixed problem is studied inside of an increasing noncylindrical domain of Rn×Rt. In both situations we show the existence of a unique nonlocal solution. In cylindrical case it is proved the existence of weak and strong solutions, regularity of strong solutions, and in noncylindrical case weak solutions. One of the goals of this paper is to show that the noncylindrical problem is well-posed by using the penalty method idealized by Lions [J.L. Lions, Une remarque sur les problèmes d'évolution non linéaires dans des domaines non cylindriques, Rev. Roumaine Math. Pures Appl. 9 (1964) 11-18].     

磁流体方程的时间解析性和时间周期解

考查了周期边界条件下的磁流体方程,证明了它的解关于时间是解析的,由此得到了磁流体方程的解的向后惟一性.对于周期解,证明了当周期小于某个常数时,周期的弱解是强解,进一步地这样的强解是定常解.     

Structure of the set of stationary solutions of phase-lock equations in superconductivity

In previous article [M. Zhan, Phase-lock equations and its connections to Ginzburg–Landau equations of superconductivity, J. Nonlinear Anal. 42 (2000) 1063–1075], we introduced a system of equations (phase-lock equations) to model the superconductivity phenomena. We investigated its connection to Ginzburg–Landau equations and proved the existence and uniqueness of both weak and strong solutions. In this article, we study the steady-state problem associated with the phase-lock equations. We prove that the steady-state problem has multiple solutions and show that the solution set enjoys some structural properties as proved by Foias and Teman for the Navier–Stokes equations in [C. Foias, R. Teman, Structure of the set of stationary solutions of the Navier–Stokes equations, Commun. Pure Appl. Math. XXX (1977) 149–164].     

Non blow-up criterion for the 3-D Magneto-hydrodynamics equations in the limiting case

In this paper, we prove that suitable weak solution(u, b) of the 3-D MHD equations can be extended beyond T if u∈L~∞(0,T; L~3(R~3)) and the horizontal components b_h of the magnetic field satisfies the well-known Ladyzhenskaya-Prodi-Serrin condition, which improves the corresponding regularity criterion by Mahalov-Nicolaenko-Shilkin.