Nonexistence of self-similar singularities in the viscous magnetohydrodynamics with zero resistivity

Details Nonexistence of self-similar singularities in the viscous magnetohydrodynamics with zero resistivity Nonexistence of self-similar singularities in the viscous magnetohydrodynamics with zero resistivity

2007-03-28

arxiv.org/abs/math/0703830v2

Mathematics

Analysis of PDEs

16 pages

Scientific paper

We are concerned on the possibility of finite time singularity in a partially viscous magnetohydrodynamic equations in $\Bbb R^n$, $n=2,3$, namely the MHD with positive viscosity and zero resistivity. In the special case of zero magnetic field the system reduces to the Navier-Stokes equations in $\Bbb R^n$. In this paper we exclude the scenario of finite time singularity in the form of self-similarity, under suitable integrability conditions on the velocity and the magnetic field. We also prove the nonexistence of asymptotically self-similar singularity. This provides us information on the behavior of solutions near possible singularity of general type as described in Corollary 1.1 below.

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