On well-posedness of the Cauchy problem for MHD system in Besov spaces
Abstract
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
n≥3
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 $\dot{B}^{\frac np-1}_{p,r}(\mr^n)$,
1≤p<∞
and
1≤r≤∞
. 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
n
2p
+
2
r
>1
. In case of
n=2
, 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
2<p<∞
and
1≤r<∞
.