Existence theorem and blow-up criterion of strong solutions to the two-fluid MHD equation in ${\mathbb R}^3$

Details Existence theorem and blow-up criterion of strong solutions to the two-fluid MHD equation in ${\mathbb R}^3$ Existence theorem and blow-up criterion of strong solutions to the two-fluid MHD equation in ${\mathbb R}^3$

2006-11-07

arxiv.org/abs/math/0611165v4

J. Differential Equations 239 (2007)251-271

Mathematics

Analysis of PDEs

18 pages

Scientific paper

10.1016/j.jde.2007.03.029

We first give the local well-posedness of strong solutions to the Cauchy problem of the 3D two-fluid MHD equations, then study the blow-up criterion of the strong solutions. By means of the Fourier frequency localization and Bony's paraproduct decomposition, it is proved that strong solution $(u,b)$ can be extended after $t=T$ if either $u\in L^q_T(\dot B^{0}_{p,\infty})$ with $\frac{2}{q}+\frac{3}{p}\le 1$ and $b\in L^1_T(\dot B^{0}_{\infty,\infty})$, or $(\omega, J)\in L^q_T(\dot B^{0}_{p,\infty})$ with $\frac{2}{q}+\frac{3}{p}\le 2$, where $\omega(t)=\na\times u $ denotes the vorticity of the velocity and $J=\na\times b$ the current density.

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