On well-posedness of the Cauchy problem for MHD system in Besov spaces

Details On well-posedness of the Cauchy problem for MHD system in Besov spaces On well-posedness of the Cauchy problem for MHD system in Besov spaces

2006-07-19

arxiv.org/abs/math/0607458v3

Math.Meth.Appl.Sci.32(2009)53-76

Mathematics

Analysis of PDEs

23pages

Scientific paper

10.1002/mma.1026

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 $1\le r<\infty$.

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