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 $n\ge 3$ 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 $\dot{ B}^{ \frac np-1}_{ p,r}(\mr^n)$, $1\le p<\infty$ and $1\le r\le\infty$. 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 $\frac n{ 2p}+\frac2r>1$. 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 $2< p<\infty$ and $1\le r<\infty$.
Categories: math.AP