170 research outputs found
On well-posedness of the Cauchy problem for MHD system in Besov spaces
This paper is devoted to the study of the Cauchy problem of incompressible
magneto-hydrodynamics system in framework of Besov spaces. In the case of
spatial dimension we establish the global well-posedness of the Cauchy
problem of incompressible magneto-hydrodynamics system for small data and the
local one for large data in Besov space \dot{B}^{\frac np-1}_{p,r}(\mr^n),
and . Meanwhile, we also prove the weak-strong
uniqueness of solutions with data in \dot{B}^{\frac np-1}_{p,r}(\mr^n)\cap
L^2(\mr^n) for . In case of , we establish the
global well-posedness of solutions for large initial data in homogeneous Besov
space \dot{B}^{\frac2p-1}_{p,r}(\mr^2) for and .Comment: 23page
Existence theorem and blow-up criterion of the strong solutions to the Magneto-micropolar fluid equations
In this paper we study the magneto-micropolar fluid equations in ,
prove the existence of the strong solution with initial data in for
, and set up its blow-up criterion. The tool we mainly use is
Littlewood-Paley decomposition, by which we obtain a Beale-Kato-Majda type
blow-up criterion for smooth solution which relies on the
vorticity of velocity only.Comment: 19page
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
Existence theorem and blow-up criterion of strong solutions to the two-fluid MHD equation in
We first give the local well-posedness of strong solutions to the Cauchy
problem of the 3D two-fluid MHD equations, then study the blow-up criterion of
the strong solutions. By means of the Fourier frequency localization and Bony's
paraproduct decomposition, it is proved that strong solution can be
extended after if either with
and , or
with , where \omega(t)=\na\times u denotes the vorticity of the velocity and
J=\na\times b the current density.Comment: 18 page
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-
Higher order commutator estimates and local existence for the non-resistive MHD equations and related models
This paper establishes the local-in-time existence and uniqueness of strong solutions in Hs for s > n/2 to the viscous, non-resistive magnetohydrodynamics (MHD) equations in Rn, n = 2, 3, as well as for a related model where the advection terms are removed from the velocity equation. The uniform bounds required for proving existence are established by means of a new estimate, which is a partial generalisation of the commutator estimate of Kato & Ponce (Comm. Pure Appl. Math. 41(7), 891â907, 1988)
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