Global existence for the MHD system in critical spaces

Abstract

In this article, we show that the magneto-hydrodynamic system (MHD) in ${\mathbb{R}}^N$ with variable density, variable viscosity and variable conductivity has a local weak solution in the Besov space ${\dot{B}}_{p_1,1}^{\frac{N}{p_1}}({\mathbb{R}}^N)\times {\dot{B}}_{p_2,1}^{\frac{N}{p_2}-1}({\mathbb{R}}^N)\times {\dot{B}}_{p_2,1}^{\frac{N}{p_2}-1}({\mathbb{R}}^N)$ for all $1<p_2<+\infty$ and some $1<p_1\le \frac{2N}{3}$ if the initial density approaches a positive constant. Moreover, this solution is unique if we impose the restrictive condition $1<p_2\le 2N$. 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.

31 pages, to appear in Proceedings of the Royal Society of Edinburgh. Section A. Mathematics

Cited by in corpus (3)

• On Local Strong Solutions to the Cauchy Problem of Two-Dimensional Density-Dependent Magnetohydrodynamic Equations with Vacuum
• Global Existence and Large Time Asymptotic Behavior of Strong Solutions to the Cauchy Problem of 2D Density-Dependent Magnetohydrodynamic Equations with Vacuum
• On the Decay and Stability of Global Solutions to the 3D Inhomogeneous MHD system