Nonexistence of Self-similar Singularities in the Ideal Magnetohydrodynamics

Abstract

In this paper we exclude the scenario of the apparition of finite time singularity in the form of self-similar singularities in the ideal magnetohydrodynamic equations, assuming suitable integrability conditions on the vorticity and the magnetic field. We also consider the more refined possibility of asymptotically self-similar singularities, where the local classical solution converges to the self-similar profile as we approach the possible time of singularity. The scenario of asymptotically self-similar singularity is also excluded under suitable conditions on the profile. In the two-dimensional magnetohydrodynamics the magnetic field evolution equations reduce to a divergence free transport equation for a scalar stream function. This helps us to improve the above nonexistence theorems on the self-similar singularities, in the sense that we require merely weaker integrability conditions on the profile in order to prove the results.