Abstract
In this paper we prove the uniqueness of weak solutions and the global-in-time existence of smooth solutions of the 2D generalized MHD system with fractional diffusion with power. MSC:35Q30, 76D03, 76D09.
Highlights
In this paper, we consider the following D generalized MHD system with < α ≤ [ ]: div u = div b =, ( . )∂tu + (u · ∇)u + ∇ π + |b| + (– )αu = b · ∇b,∂tb + u · ∇b – b · ∇u – b =,(u, b)(t = ) = (u, b ).Here, u is the fluid velocity field, π is the pressure and b is the magnetic field.Very recently, Ji [ ] used the Fourier series analysis motivated in [ ] to prove the global-in-time existence of smooth solutions of problem ( . )-( . ) when
Testing ( . ) by δb and using ( . ) and ( . ), we find that
Applying s to ( . ), testing by sb and using ( . ), we find that sb dx +
Summary
1 Introduction In this paper, we consider the following D generalized MHD system with < α ≤ [ ]: div u = div b = , Ji [ ] used the Fourier series analysis motivated in [ ] to prove the global-in-time existence of smooth solutions of problem Ji [ ] pointed out that his result did not seem to come directly from the method like energy estimates. We use the standard energy method to deal with the case α
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