On the regularity of the axially symmetric solutions to the three‐dimensional MHD equations

We present the improved three‐dimensional axially symmetric incompressible magnetohydrodynamics (MHD) equations with nonzero swirl. We consider three kinds of smooth axially symmetric particular solutions to the MHD equations: (1) u^θ=0,B^r=B^z=0, (2) B^r=B^z=0, and (3) B^θ=0. In particular, we derive new regularity criteria for these three kinds of the three‐dimensional axially symmetric smooth solutions to the MHD equations. Our results also reveal some interesting dynamic behavior of the interaction by the angular vorticity field ω^θ and the angular current density field j^θ. Copyright © 2016 John Wiley & Sons, Ltd.     

Viscous and inviscid magneto-hydrodynamics equations

In this paper we study the classical problem in turbulence for the magneto-hydrodynamics (MHD) equations: whether the solutions (u ^(v),B ^(v)) of the viscous MHD equations tend to the solutions (u ⁽⁰⁾,B ^(v)) of the inviscid MHD equations as the Reynolds numbersRe, Rm → ∞. As a preparation we first derive bounds for ||(u ⁽⁰⁾,B ⁽⁰⁾(t)||H ^m) (m ≥3) in terms of deformation tensor related quantities (0.1) {ie251-1} We then show that asRe → ∞ andRm → ∞, the difference {ie-251-2} {ie-251-3} converges to zero uniformly int as long as the quantities in (0.1) remain finite. The convergence rates are explicit. Supported by the NSF grant DMS 9304580 at IAS.     

Convergence of an unconditionally stable semi-implicit scheme for linear incompressible MHD equations

Semi-implicit methods have been introduced by Harned et al. to solve magneto-hydrodynamic equations (MHD) with numerical schemes which are unconditionally stable with respect to fast and shear Alfven modes. They prove the stability of their scheme for linear ideal MHD equations with periodic boundary conditions, and with some technical assumptions. In this paper, we prove convergence of the numerical approximation (time discretization), under the same hypothesis, but looking for solutions on any regular bounded open set of R³ with appropriate boundary conditions, and introducing finite resistivity and viscosity.     

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.     

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.     

THE GLOBAL L~2 STABILITY OF SOLUTIONS TO THREE DIMENSIONAL MHD EQUATIONS

In this paper,we mainly study the global L2 stability for large solutions to the MHD equations in three-dimensional bounded or unbounded domains.Under suitable conditions of the large solutions,it is shown that the large solutions are stable.And we obtain the equivalent condition of this stability condition.Moreover,the global existence and the stability of two-dimensional MHD equations under three-dimensional perturbations are also established.     

A remark on the Beale-Kato-Majda criterion for the 3D MHD equations with zero kinematic viscosity

In this paper,we study the blow-up criterion of smooth solutions to the 3D magneto-hydrodynamic system in ˙ B 0 ∞,∞.We show that a smooth solution of the 3D MHD equations with zero kinematic viscosity in the whole space R 3 breaks down if and only if certain norm of the vorticity blows up at the same time.     

Energy Equality and Uniqueness of Weak Solutions to MHD Equations in L∞（0, T; L^n（Ω））

In this paper, we study the energy equality and the uniqueness of weak solutions to the MHD equations in the critical space L∞（0, T; L^n（Ω）. We prove that if the velocity u belongs to the critical space L∞（0, T; L^n（Ω）, the energy equality holds. On the basis of the energy equality, we further prove that the weak solution to the MHD equations is unique.     

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 eddy simulation for turbulent magnetohydrodynamic flows

We investigate the mathematical properties of a model for the simulation of large eddies in turbulent, electrically conducting, viscous, incompressible flows. We prove existence and uniqueness of solutions for the simplest (zeroth) closed MHD model (1.7), we show that its solutions converge to the solution of the MHD equations as the averaging radii converge to zero, and derive a bound on the modeling error. Furthermore, we show that the model preserves the properties of the 3D MHD equations: the kinetic energy and the magnetic helicity are conserved, while the cross helicity is approximately conserved and converges to the cross helicity of the MHD equations, and the model is proven to preserve the Alfvén waves, with the velocity converging to that of the MHD, as δ₁,δ₂ tend to zero. We perform computational tests that verify the accuracy of the method and compare the conserved quantities of the model to those of the averaged MHD.     

A regularity criterion for the 2D MHD system with zero magnetic diffusivity

This note proves a regularity criterion ∇b∈L¹(0,T;BMO(R²)) for the 2D MHD system with zero magnetic diffusivity.     

Second order unconditionally convergent and energy stable linearized scheme for MHD equations

In this paper, we propose an efficient numerical scheme for magnetohydrodynamics (MHD) equations. This scheme is based on a second order backward difference formula for time derivative terms, extrapolated treatments in linearization for nonlinear terms. Meanwhile, the mixed finite element method is used for spatial discretization. We present that the scheme is unconditionally convergent and energy stable with second order accuracy with respect to time step. The optimal L ² and H ¹ fully discrete error estimates for velocity, magnetic variable and pressure are also demonstrated. A series of numerical tests are carried out to confirm our theoretical results. In addition, the numerical experiments also show the proposed scheme outperforms the other classic second order schemes, such as Crank-Nicolson/Adams-Bashforth scheme, linearized Crank-Nicolson’s scheme and extrapolated Gear’s scheme, in solving high physical parameters MHD problems.     

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.     

A logarithmic regularity criterion for the 3D generalized MHD system

We prove a logarithmic regularity criterion for the 3D generalized magnetohydrodynamics (MHD) system with diffusion terms ?Δu and (?Δ)^βb, with . Copyright © 2015 John Wiley & Sons, Ltd.     

L 3,∞-solutions to the MHD equations

We prove that weak solutions to the MHD system are smooth provided that they belong to the so-called “critical” Ladyzhenskaya-Prodi-Serrin class L_3,∞. Besides the independent interest, this result disproves the hypothesis on existence of collapsing self-similar solutions to the MHD equations for which the generating profile belongs to the space L₃. Thus, we extend the results which were known before for the Navier-Stokes system to the case of the MHD equations. Bibliography: 14 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 336, 2006, pp. 112–132.     

PL~n∩PL~p空间中MHD方程组强解的存在唯一性及衰减性质

本文运用半群理论和Kato的方法,研究了MHD方程组在PL~n∩PL~p(1相似文献   

Long time <Emphasis Type="Italic">L</Emphasis><Superscript>1</Superscript>-behavior for the incompressible magneto-hydrodynamic equations in a half-space

The large time L ¹-behavior of the strong solution (including the first and second order spacial derivatives) to the incompressible magneto-hydrodynamic (MHD) equations is given in a half-space. The main tool employed in this article is a new weighted estimate for the Stokes flow in L ¹(R₊ ⁿ), such a study is of independent interest.     

Existence of solutions for the MHD-Leray-alpha equations and their relations to the MHD equations

In this paper we derive some new equations and we call them MHD-Leray-alpha equations which are similar to the MHD equations. We put forward the concept of weak and strong solutions for the new equations. Whether the 3-dimensional MHD equations have a unique weak solution is unknown, however, there is a unique weak solution for the 3-dimensional MHD-Leray-alpha equations. The global existence of strong solution and the Gevrey class regularity for the new equations are also obtained. Furthermore, we prove that the solutions of the MHD-Leray-alpha equations converge to the solution of the MHD equations in the weak sense as the parameter ε in the new equations converges to zero.     

A New Regularity Criterion in Terms of the Direction of the Velocity for the MHD Equations

In this paper we obtain a new regularity criterion for weak solutions to the 3D MHD equations. It is proved that if div( \fracu|u|) \mathrm{div}( \frac{u}{|u|}) belongs to L^\frac21-r( 0,T;[(X)\dot]_r( \mathbbR³) ) L^{\frac{2}{1-r}}( 0,T;\dot{X}_{r}( \mathbb{R}^{3}) ) with 0≤r≤1, then the weak solution actually is regular and unique.     

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.