Some mathematical questions related to the MHD equations

Sermange, Michel; Témam, Roger

doi:10.1016/0167-7136(83)90286-x

Cited by 269 publications

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“…The given functions f = f (x, t) and g = g(x, t) represent external volume forces and the curl of external current applied to the fluid. We refer to [23], [16] and [33] for the mathematical theory for the MHD equations. Again, the above equations are a particular case of equation (2.1) for the following spaces and operators which satisfy (C1).…”

Section: D Magneto-hydrodynamic Equationsmentioning

confidence: 99%

Stochastic 2D Hydrodynamical Type Systems: Well Posedness and Large Deviations

Čhuešhov¹,

2009

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Abstract. We deal with a class of abstract nonlinear stochastic models, which covers many 2D hydrodynamical models including 2D Navier-Stokes equations, 2D MHD models and 2D magnetic Bénard problem and also some shell models of turbulence. We first prove the existence and uniqueness theorem for the class considered. Our main result is a Wentzell-Freidlin type large deviation principle for small multiplicative noise which we prove by weak convergence method.

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Section: D Magneto-hydrodynamic Equationsmentioning

confidence: 99%

Stochastic 2D Hydrodynamical Type Systems: Well Posedness and Large Deviations

Čhuešhov¹,

2009

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“…If ν > 0 and µ > 0, then there exists a unique global solution in time when n = 2, while for n = 3 the problem is still open, as discussed in [13].…”

Section: Introductionmentioning

confidence: 99%

Global existence for two regularized MHD models in three space-dimension

Catania¹,

Secchi²

2011

Port. Math.

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Abstract. The global existence of solutions for the 3D incompressible Euler equations is a major open problem. For the 3D inviscid MHD system, the global existence is an open problem as well. Our main concern in this paper is to understand which kind of regularization, of the form of α-regularization or partial viscous regularization, is capable to provide the global in time solvability for the 3D inviscid MHD system of equations. We consider two different regularized magnetohydrodynamic models for an incompressible fluid. In both cases, we provide a global existence result for the solution of the system. Mathematics Subject Classification (2000). Primary 35Q35; Secondary 76D03.

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“…Some regularity properties and large time behaviors of the solutions for a similar system, the MHD equations, are obtained in Sermange [10] and Temam [12]. More recently, we obtained in [2] the local in time existence and uniqueness of weak solutions of the system in L p with p > n. Motivated by Calderón's work on the Navier Stokes equations [1], we consider in this paper the initial value problem for the above system in the infinite cylinder S = (0, ∞) × R n with initial data v 0 , b 0 ∈ L p with p ≤ n.…”

Section: Introductionmentioning

confidence: 99%

Existence of global solution for a differential system with initial data in L^p

Bates

Chen

Wang

1999

International Journal of Mathematics and Mathematical Sciences

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Abstract. In this paper, we study the system governing flows in the magnetic field within the earth. The system is similar to the magnetohydrodynamic (MHD) equations. By establishing a new priori estimates and following Calderón's procedure for the Navier Stokes equations [1], we obtained, for initial data in space L p , the global in time existence and uniqueness of weak solution of the system subject to appropriate conditions.

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Some mathematical questions related to the MHD equations

Cited by 269 publications

References 0 publications

Stochastic 2D Hydrodynamical Type Systems: Well Posedness and Large Deviations

Stochastic 2D Hydrodynamical Type Systems: Well Posedness and Large Deviations

Global existence for two regularized MHD models in three space-dimension

Existence of global solution for a differential system with initial data in L^p

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Some mathematical questions related to the MHD equations

Cited by 269 publications

References 0 publications

Stochastic 2D Hydrodynamical Type Systems: Well Posedness and Large Deviations

Stochastic 2D Hydrodynamical Type Systems: Well Posedness and Large Deviations

Global existence for two regularized MHD models in three space-dimension

Existence of global solution for a differential system with initial data in Lp

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Existence of global solution for a differential system with initial data in L^p