analytic results related to magneto hydrodynamic turbulence abstract analytic results are provided for topics related to the magneto hydrodynamic turbulence scaling exponents of general structure functions geometric criteria for singularity formation and dissipative solutions

global existence and decay of smooth solution for the 2 d mhd equations without magnetic diffusion abstract we prove the global existence and the decay estimates of small smooth solution for the 2 d mhd equations without magnetic diffusion this confirms the numerical observation that the energy of the mhd equations is dissipated at a rate independent of the ohmic resistivity

an interior point method for bordered block diagonal linear programs this paper presents an interior point method for solving a bordered block diagonal linear program which consists of a number of disjoint blocks coupled by a total of p variables and constraints this structure includes the well known block angular and dual block angular structures as well as their special cases such as staircase problems generalized bounds and multicommodity flows when p is small relative to the total dimension of the problem the method achieves a substantial speedup relative to other general purpose methods

hausdorff measure of boundary singular points in the magnetohydrodynamic equations abstract we study boundary singular points of suitable weak solutions to the three dimensional incompressible magnetohydrodynamic equations by using the generalised hausdorff measure we estimate the size of boundary singular points and present the improved range of powers of logarithmic factors

explicit dimension estimates of attractors for the mhd equations in three dimensional space abstract in this article we estimate the dimension of functional invariant sets that arise from the mhd equations in terms of some physically relevant parameters we follow the analysis of constantin foias and temam for 3d navier stokes equations and derive comparable estimates

long term dissipativity of time stepping algorithms for an abstract evolution equation with applications to the incompressible mhd and navier stokes equations abstract this paper examines the long term dissipativity and unconditional non linear stability of time integration algorithms for the incompressible mhd equations the results apply as a particular case to the uncoupled incompressible navier stokes equations the analysis is performed in the setting defined by an abstract evolution equation possessing the same long term dissipative structure as the above physical systems a general class of time stepping schemes is analyzed in this context showing the non linear unconditional stability and long term dissipativity of particular members of this family of algorithms in particular algorithms showing these strong properties which additionally involve the solution of a linear problem at each time step are identified a particular linear algorithm is shown to exhibit optimal long term dissipative properties and to possess an algorithmic global attractor these abstract results apply rigorously to galerkin mixed finite element discretizations of the problem to this end it is shown that such galerkin type projection inherits the dissipative properties of the continuum problem computational aspects involved in the numerical implementation of these methods are addressed and illustrated in representative numerical simulations

irreducibility and exponential mixing of some stochastic hydrodynamical systems driven by pure jump noise in this paper we derive several results related to the long time behavior of a class of stochastic semilinear evolution equations in a separable hilbert space h d mathfrak u t rm a mathfrak u t rm b mathfrak u t mathfrak u t dt dl t quad mathfrak u 0 x in rm h here a is a positive self adjoint operator and b is a bilinear map and the driving noise l is basically a d rm a 1 2 valued levy process satisfying several technical assumptions by using a density transformation theorem type for levy measure we first prove a support theorem and an irreducibility property of the ornstein uhlenbeck processes associated to the nonlinear stochastic problem second by exploiting the previous results we establish the irreducibility of the nonlinear problem provided that for a certain gamma in 0 1 4 b is continuous on d rm a gamma times d rm a gamma with values in d rm a 1 2 using a coupling argument the exponential ergodicity is also proved under the stronger assumption that b is continuous on rm h times rm h while the latter condition is only satisfied by the nonlinearities of goy and sabra shell models the assumption under which the irreducibility property holds is verified by several hydrodynamical systems such as the 2d navier stokes magnetohydrodynamics equations the 3d leray varvec alpha model the goy and sabra shell models

generalized mhd equations abstract solutions of the d dimensional generalized mhd gmhd equations t u u u p b b u t b u b b u b are studied in this paper we pay special attention to the impact of the parameters and on the regularity of solutions our investigation is divided into three major cases 1 0 and 0 2 0 and 0 and 3 0 and 0 when 0 and 0 the gmhd equations with any 0 and 0 possess a global weak solution corresponding to any l2 initial data furthermore weak solutions associated with 1 2 d 4 and 1 2 d 4 are actually global classical solutions when their initial data are sufficiently smooth as a special consequence smooth solutions of the 3d gmhd equations with 5 4 and 5 4 do not develop finite time singularities the study of the gmhd equations with 0 and 0 is motivated by their potential applications in magnetic reconnection a local existence result of classical solutions and several global regularity conditions are established for this case these conditions are imposed on either the vorticity u or the current density j b but not both and are weaker than some of current existing ones when 0 and 0 the gmhd equations reduce to the ideal mhd equations it is shown here that the ideal mhd equations admit a unique local solution when the prescribed initial data is in a holder space cr with r 1

existence and uniqueness of steady solutions to the magnetohydrodynamic equations of compressible flows abstract we establish the existence and uniqueness of a strong solution to the steady magnetohydrodynamic equations for the compressible barotropic fluids in a bounded smooth domain with a perfectly conducting boundary under the assumption that the external force field is small

stability and convergence of semi implicit time stepping algorithm for stationary incompressible magnetohydrodynamics abstract for the stationary incompressible magnetohydrodynamics mhd equations we provide a new uniqueness assumption a0 and show the exponential stability of the solution then the semi implicit time stepping algorithm is used to solve the stationary mhd equations the algorithm is proved to be unconditionally stable the discrete velocity and magnetic field are bounded in l 0 l for any space and time step sizes the error estimates for the algorithm are deduced under the uniqueness conditions finally numerical experiments are performed to testify our theoretical analysis

length scale estimates for the 3d simplified bardina magnetohydrodynamic model we consider the mhd model known as double viscous simplified bardina mhd sbmhd on a 3d periodic box mhd stands for magnetohydrodynamic this system is a large eddy simulation model useful to approximate the turbulent behavior of an incompressible homogeneous magnetofluid because of the actual impossibility to handle the mhd model neither analytically nor via direct numerical simulation in a previous paper joint work with secchi the global existence of strong solutions to the sbmhd has been proved as well as the existence of a global attractor of finite fractal dimension upper bounds for such dimension are provided both in terms of the modified grashof number and the dissipation length associated to the mean rate of energy dissipation in this paper we commute the same bound in an estimate in terms of the modified reynolds number r this result is useful because the classical kolmogorov theory of turbulence is expressed using the reynolds number the global attractor estimate in terms of r is c

a note on the regularity criterion for 3d mhd equations in b 1 space abstract in this note we consider sufficient conditions for the regularity of leray hoph hopf solutions of the 3d incompressible magnetohydrodynamic equations via the velocity and magnetic fields in terms of b 1 spaces we prove that if u b belongs to the space l 2 0 t b 1 r 3 then the solution u b is regular this extends recent results contained in gala 2011 3

on an efficient second order backward difference newton scheme for mhd system abstract for the 2d 3d time dependent magnetohydrodynamics mhd system we propose a new second order backward difference formula newton scheme sbdfn which is a combination of second order backward difference approximation of the time derivative terms and newton treatment of the nonlinear terms meanwhile the finite element method is applied as spatial discretization firstly the optimal convergence of spatially semi discrete form is deduced secondly the stability and well posedness of sbdfn scheme are provided under c where c is independent of h based on these results optimal error estimates of the scheme in time are proved by negative norm technique finally numerical experiments are carried out to validate the theoretical analysis

global regularity to the 3d mhd equations with large initial data in bounded domains this paper considers the global regularity to the 3d incompressible mhd equations with large initial data in bounded domains let u and b denote the viscosity coefficient magnetic diffusivity velocity field and magnetic field respectively we construct new systems for u b and u b to overcome the difficulties caused by the large initial data it is shown that u b h1 is globally bounded as long as u0 b0 h1 1 or u0 b0 h1 1 is sufficiently small which indicates that the navier stokes equations can be regularized by the magnetic field

regularity criterion for 3d mhd fluid passing through the porous medium in terms of gradient pressure abstract the aim of the paper is to establish regularity criteria for the weak solution of fluid passing through the porous media in r 3 our results show that if p l with 2 3 3 1 then the weak solution is regular and unique if p l with 2 3 2 3 2 then the weak solution is regular and unique

strong solutions to the equations of a ferrofluid flow model abstract we study the differential system introduced by m i shliomis to describe the motion of a ferrofluid driven by an external magnetic field the system is a combination of the navier stokes equations the magnetization equation and the magnetostatic equations no regularizing term is added to the magnetization equation we prove the local in time existence of strong solutions to the system

global well posedness for the 3 d incompressible inhomogeneous mhd system in the critical besov spaces abstract in this paper we study the cauchy problem for the incompressible inhomogeneous magnetohydrodynamic mhd system in r 3 we obtain the global existence of a unique solution for this system without any small conditions imposed on the third component of the initial velocity field and magnetic field in the critical besov spaces our methods mainly rely upon the fourier frequency localization and bony s paraproduct decomposition

superconvergence and time evolution of discontinuous galerkin finite element solutions in this paper we study the convergence and time evolution of the error between the discontinuous galerkin dg finite element solution and the exact solution for conservation laws when upwind fluxes are used we prove that if we apply piecewise linear polynomials to a linear scalar equation the dg solution will be superconvergent towards a particular projection of the exact solution thus the error of the dg scheme will not grow for fine grids over a long time period we give numerical examples of p k polynomials with 1

global smooth solutions of mhd equations with large data abstract in this paper we establish the global existence of smooth solutions of the three dimensional mhd system for a class of large initial data both the initial velocity and magnetic field can be arbitrarily large in the critical norm

an efficient numerical scheme for a 3d spherical dynamo equation abstract we develop an efficient numerical scheme for the 3d mean field spherical dynamo equation the scheme is based on a semi implicit discretization in time and a spectral method in space based on the divergence free spherical harmonic functions a special semi implicit approach is proposed such that at each time step one only needs to solve a linear system with constant coefficients then using expansion in divergence free spherical harmonic functions in the transverse directions allows us to reduce the linear system at each time step to a sequence of one dimensional equations in the radial direction which can then be efficiently solved by using a spectral element method we show that the solution of fully discretized scheme remains bounded independent of the number of unknowns and present numerical results to validate our scheme

on approximate solutions of the equations of incompressible magnetohydrodynamics abstract inspired by an approach proposed previously for the incompressible navier stokes ns equations we present a general framework for the a posteriori analysis of the equations of incompressible magnetohydrodynamics mhd on a torus of arbitrary dimension d this setting involves a sobolev space of infinite order made of c vector fields with vanishing divergence and mean on the torus given any approximate solution of the mhd cauchy problem its a posteriori analysis with the method of the present work allows one to infer a lower bound on the time of existence of the exact solution and to bound from above the sobolev distance of any order between the exact and the approximate solution in certain cases the above mentioned lower bound on the time of existence is found to be infinite so one infers the global existence of the exact mhd solution we present some applications of this general scheme the most sophisticated one lives in dimension d 3 with the abc flow perturbed magnetically as an initial datum and uses for the cauchy problem a galerkin approximate solution in 124 fourier modes we illustrate the conclusions arising in this case from the a posteriori analysis of the galerkin approximant these include the derivation of global existence of the exact mhd solution with the abc datum when the dimensionless viscosity and resistivity are equal and stay above a critical value

a note on global existence of weak solutions to the compressible magnetohydrodynamic equations with coulomb force abstract in this note for the case of 9 33 12 4 3 we prove the existence of global in time finite energy weak solution of the equations of a two dimensional magnetohydrodynamics with coulomb force where denotes the adiabatic exponent the value 9 33 12 is the optimal lower bound of to establish global in time finite energy weak solution under current frame

remarks on the regularity criteria of generalized mhd and navier stokes systems we study the regularity criteria of the three dimensional generalized magnetohydrodynamics mhd and navier stokes systems in particular we show that the regularity criteria of the generalized mhd system may be reduced to depend only on two diagonal entries of the jacobian matrix of the velocity vector field or one vorticity component and one entry of the jacobian matrix of the velocity vector field

limiting case for the regularity criterion to the 3 d magneto hydrodynamics equations abstract we prove that any weak solution u b of three dimensional incompressible magneto hydrodynamics equations is regular if u l 0 t l 3 r 3 and b l 0 t vmo 1 r 3 the proof is based on the blow up analysis and backward uniqueness for the parabolic operator developed by escauriaza seregin sverak

decoupled schemes for unsteady mhd equations ii finite element spatial discretization and numerical implementation in this article a decoupled fully discrete scheme for solving 2d magnetohydrodynamics mhd equations is proposed the decoupled scheme is used for time discretization and the finite element method is used for spatial discretization firstly the almost unconditional stability t c of this scheme is established then optimal l2 and h1 error estimates of numerical solution are provided finally a numerical example is presented to confirm our theoretical results and show the high efficiency of the decoupled scheme

iterative methods in penalty finite element discretization for the steady mhd equations this paper characterizes one penalty finite element method for the incompressible mhd equations the method is an interesting combination of the classic iterative schemes stokes newton and oseen iterations with two different finite element pairs p1b p1 p1b and p1 p0 p1 moreover the rigorous analysis of stability and error estimate for the proposed methods are given finally the applicability and effectiveness of the presented schemes are illustrated in several numerical experiments

global regularity for the 212d incompressible hall mhd system with partial dissipation abstract in this paper we investigate the cauchy problem for the 2 1 2 dimensional incompressible hall magnetohydrodynamic equations with partial dissipation by using energy estimate we obtain the local existence of strong solution and the global in time existence of the strong solution

global regularity to the 3d incompressible mhd equations abstract in this paper we prove the global regularity to the 3d nonhomogeneous incompressible magnetohydrodynamic equations let 0 m 0 and h 0 be the initial density momentum and magnetic field respectively we establish a unique strong solution on r 3 0 t for any 0 t under the assumption that the viscosity coefficient 0 is sufficiently large or m 0 2 0 l 1 2 h 0 l 2 2 or u 0 l 2 2 h 0 h 1 2 is small enough moreover if the given data are more regular and satisfy an additional compatibility condition for the existence of strong solution then we show that the strong solution is indeed a classical one moreover the weak strong uniqueness of solutions is also obtained which generalizes the result in 13 to the case of vacuum

global existence and uniqueness theorem to 2 d incompressible non resistive mhd system with non equilibrium background magnetic field abstract in this article we consider the global existence and uniqueness of the solution to the 2d incompressible non resistive mhd system with non equilibrium background magnetic field our result implies that a strong enough non equilibrium background magnetic field will guarantee the stability of the nonlinear mhd system beside the classical energy method the interpolation inequalities and the algebraic structure of the equations coming from the incompressibility of the fluid are crucial in our arguments

remarks on the regularity criteria of three dimensional magnetohydrodynamics system in terms of two velocity field components we study the three dimensional magnetohydrodynamics system and obtain its regularity criteria in terms of only two velocity vector field components eliminating the condition on the third component completely the proof consists of a new decomposition of the four nonlinear terms of the system and estimating a component of the magnetic vector field in terms of the same component of the velocity vector field this result may be seen as a component reduction result of many previous works c he and z xin on the regularity of weak solutions to the magnetohydrodynamic equations j differ equ 213 2 234 254 2005 y zhou remarks on regularities for the 3d mhd equations discrete contin dyn syst 12 5 881 886 2005

stability of stationary solution for the compressible viscous magnetohydrodynamic equations with large potential force in bounded domain abstract in this paper we consider the 3 d compressible viscous magnetohydrodynamic mhd equations with some large potential force in bounded rigid vessel we firstly construct the non constant stationary solutions of the compressible viscous mhd equations under suitable constitutive assumptions next a critical energy identity is established to achieve a universal stability criterion of the stationary solution in this case the stationary solution is exponential stable for any large external potential force finally we show the well posedness of the initial boundary value problem for the compressible viscous mhd equations with the large potential force provided that the prescribed initial data is close to the stationary solution it implies that the set satisfying the stability criterion is not empty

some mathematical questions related to the mhd equations some questions relating to the large time behavior of the solutions of mhd equations for a viscous incompressible resistive fluid are investigated the physical system is briefly described and the functional setting of the equations a flow in a bounded domain or in whole space with a space periodicity property in all directions the main existence and uniqueness results for weak and strong solutions of the mhd equations are recalled regularity properties and bounds on the solutions to the equations which are valid for all time are established and the concept of functional invariant sets is introduced which is contained in the space of smooth functions if the data are sufficiently regular the squeezing property of the trajectories are stated and it is shown that any functional invariant set for the mhd equations and in particular any attractor has a finite haussdorf dimension the flow is found to be totally determined for large dimensions by a finite number of parameters 26 references