Global existence for the MHD system in critical spaces

In this article, we show that the magneto-hydrodynamic system (MHD) in $\R^N$ with variable density, variable viscosity and variable conductivity has a local weak solution in the Besov space $\dot B^{\frac{N}{p_1}}_{p_1,1}(\R^N)\times\dot B^{\frac{N}{p_2}-1}_{p_2,1}(\R^N) \times\dot B^{\frac{N}{p_2}-1}_{p_2,1}(\R^N)$ for all $1<p_2<+\infty$ and some $1<p_1\leq\frac{2N}{3}$ if the initial density approaches a positive constant. Moreover, this solution is unique if we impose the restrictive condition $1<p_2\leq2N$. We prove also that the constructed solution exist globally in time if the initial data are small enough. In particular, this allows us to work in the frame of Besov space with negative regularity indices and this fact is particularly important when the initial data are strong oscillating.