On the Splash Singularity for the free-boundary problem of the viscous and non-resistive incompressible magnetohydrodynamic equations in 3D

TL;DR

This paper proves the finite-time splash singularity can occur in the free-boundary viscous, non-resistive incompressible MHD equations in 3D, extending previous results to magnetic fluids with free boundaries.

Contribution

It demonstrates the existence of splash singularities in 3D viscous non-resistive MHD free-boundary problems, generalizing prior work on viscous surface waves to magnetic fluids.

Findings

Finite-time splash singularity proven for 3D viscous non-resistive MHD.

Results extend previous work from surface waves to magnetic fluids.

Arguments applicable to any space dimension d ≥ 2.

Abstract

In this paper, the existence of finite-time splash singularity is proved for the free-boundary problem of the viscous and non-resistive incompressible magnetohydrodynamic (MHD) equations in $ \mathbb{R}^{3}$, based on a construction of a sequence of initial data alongside delicate estimates of the solutions. The result and analysis in this paper generalize those by Coutand and Shkoller in [14, Ann. Inst. H. Poincar\'{e} C Anal. Non Lin\'{e}aire, 2019] from the viscous surface waves to the viscous conducting fluids with magnetic effects for which non-trivial magnetic fields may present on the free boundary. The arguments in this paper also hold for any space dimension $d\ge 2$.