The fixed boundary value problems for the equations of ideal Magneto-Hydrodynamics with a perfectly conducting wall condition

Yanagisawa, Taku; Matsumura, Akitaka

doi:10.1007/bf02096793

Cited by 48 publications

(56 citation statements)

References 14 publications

(17 reference statements)

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“…From it one can easily obtain an existence theorem for the original system (1.1) with p instead of q. The result improves the existence result in H m * (Ω) of the author [9] and of Yanagisawa-Matsumura [12].…”

Section: Formulation Of the Problem Notations And Resultssupporting

confidence: 62%

“…The application to (1.1) was given in [9], where we proved the well-posedness in the sense of Hadamard. Problem (1.1) was also studied by Yanagisawa and Matsumura [12]. As regards the loss of regularity for the solutions of the MHD equations, Ohno-Shirota [4] prove that a mixed problem for the linearized MHD equations is ill-posed in H m (Ω) for m ≥ 2.…”

Section: Introductionmentioning

confidence: 98%

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An initial boundary value problem in ideal Magneto-Hydrodynamics

Secchi

2002

NoDEA, Nonlinear differ. equ. appl.

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We consider the initial-boundary value problem for the system of equations of ideal Magneto-Hydrodynamics with a perfectly conducting wall boundary condition. Because of the characteristic boundary, where a loss of regularity in the normal direction to the boundary may occur, the natural functional setting is provided by anisotropic Sobolev spaces, which take account of such singular behavior. We show the existence of the regular solution in the anisotropic Sobolev space H m * * (Ω) for m ≥ 6. The result improves previous results of the author and of Yanagisawa -Matsumura.2000 Mathematics Subject Classification: 35L50, 35L60, 35Q35, 76W05.

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Section: Formulation Of the Problem Notations And Resultssupporting

confidence: 62%

Section: Introductionmentioning

confidence: 98%

An initial boundary value problem in ideal Magneto-Hydrodynamics

Secchi

2002

NoDEA, Nonlinear differ. equ. appl.

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“…Reference [5]). Therefore, our boundary condition for the electric ÿeld E seems to be physically reasonable in the light of these considerations.…”

Section: Theorem 12mentioning

confidence: 96%

Global solutions of the non-linear problem describing Joule's heating in three space dimensions

Bień¹

2005

Math. Meth. Appl. Sci.

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SUMMARYThe existence of global-in-time weak solutions to the Joule problem modelling heating or cooling in a current and heat conductive medium is proved via the Faedo -Galerkin method. The existence proof entails some a priori estimates that together with the monotonicity and compactness methods make up a main tool to prove the desired result. Under appropriate hypotheses on the data, it will be shown the boundedness in L∞(Q T ) of the absolute temperature of the medium and of the t-derivative of this temperature, which is achieved by means of the Gagliardo -Nirenberg theorem, the Sobolev embedding theorem and the method of Stampacchia. The paper is some extension of our investigation initiated in (Math. Meth. Appl. Sci. 1998; 23:1275-1291 PRELIMINARIES AND STATEMENT OF THE RESULTThe main aim of this paper is to prove existence of global-in-time weak solutions to the initial-boundary value problem describing Joule's heating in a heat and current conductive medium in three space dimensions; this problem reads:

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“…The method and existence result is also applied to magneto-fluid dynamics, see Refs. [19,20]. Secchi [21] studied the non-homogeneous problem directly.…”

Section: Remarks On English Versionmentioning

confidence: 99%

Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boundary

Chen

2007

Front. Math. China

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This paper is devoted to initial boundary value problems for quasilinear symmetric hyperbolic systems in a domain with characteristic boundary. It extends the theory on linear symmetric hyperbolic systems established by Friedrichs to the nonlinear case. The concept on regular characteristics and dissipative boundary conditions are given for quasilinear hyperbolic systems. Under some assumptions, an existence theorem for such initial boundary value problems is obtained. The theorem can also be applied to the Euler system of compressible flow.Keywords symmetric hyperbolic systems, initial boundary value problems, characteristic boundary MSC 35L60, 35L50, 35L40Many problems in mathematical physics can be described by quasilinear symmetric hyperbolic systems. For instance, the system of gas dynamics, the system of shallow water equations and the system of magnetohydrodynamics can be reduced to such systems. In Refs. [1,2] the initial boundary value problems for quasilinear symmetric hyperbolic systems with non-characteristic boundary have been intensively discussed. However, when the boundary is characteristic, only semilinear systems are considered there. For the characteristic boundary case, a related work is given by Ebin [3], who proved the local existence of the boundary value problem for the system of fluid dynamics in a region with rigid wall under the assumptions, that the initial velocity is subsonic, and the initial density is near to a constant. In this paper we will discuss a more general case of the problems for quasilinear symmetric hyper- *

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The fixed boundary value problems for the equations of ideal Magneto-Hydrodynamics with a perfectly conducting wall condition

Cited by 48 publications

References 14 publications

An initial boundary value problem in ideal Magneto-Hydrodynamics

An initial boundary value problem in ideal Magneto-Hydrodynamics

Global solutions of the non-linear problem describing Joule's heating in three space dimensions

Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boundary

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