65 research outputs found
The Beale-Kato-Majda criterion to the 3D Magneto-hydrodynamics equations
We study the blow-up criterion of smooth solutions to the 3D MHD equations.
By means of the Littlewood-Paley decomposition, we prove a Beale-Kato-Majda
type blow-up criterion of smooth solutions via the vorticity of velocity only,
i. e. \sup_{j\in\Z}\int_0^T\|\Delta_j(\na\times u)\|_\infty dt, where
is a frequency localization on .Comment: 12page
Some qualitative properties of the solutions of the Magnetohydrodynamic equations for nonlinear bipolar fluids
In this article we study the long-time behaviour of a system of nonlinear
Partial Differential Equations (PDEs) modelling the motion of incompressible,
isothermal and conducting modified bipolar fluids in presence of magnetic
field. We mainly prove the existence of a global attractor denoted by \A for
the nonlinear semigroup associated to the aforementioned systems of nonlinear
PDEs. We also show that this nonlinear semigroup is uniformly differentiable on
\A. This fact enables us to go further and prove that the attractor \A is
of finite-dimensional and we give an explicit bounds for its Hausdorff and
fractal dimensions.Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/s10440-014-9964-
Incompressible limit of the compressible magnetohydrodynamic equations with vanishing viscosity coefficients
This paper is concerned with the incompressible limit of the compressible
magnetohydrodynamic equations with vanishing viscosity coefficients and general
initial data in the whole space or 3). It is rigorously
showed that, as the Mach number, the shear viscosity coefficient and the
magnetic diffusion coefficient simultaneously go to zero, the weak solution of
the compressible magnetohydrodynamic equations converges to the strong solution
of the ideal incompressible magnetohydrodynamic equations as long as the latter
exists.Comment: 17pages. We have improved our paper according to the referees'
suggestion
On the regularity criterion of weak solution for the 3D viscous Magneto-hydrodynamics equations
We improve and extend some known regularity criterion of weak solution for
the 3D viscous Magneto-hydrodynamics equations by means of the Fourier
localization technique and Bony's para-product decomposition.Comment: 13page
Weak and strong solutions of equations of compressible magnetohydrodynamics
International audienceThis article proposes a review of the analysis of the system of magnetohydrodynamics (MHD). First, we give an account of the modelling asumptions. Then, the results of existence of weak solutions, using the notion of renormalized solutions. Then, existence of strong solutions in the neighbourhood of equilibrium states is reviewed, in particular with the method of Kawashima and Shizuta. Finally, the special case of dimension one is highlighted : the use of Lagrangian coordinates gives a simpler system, which is solved by standard techniques
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