Well-posedness of axially symmetric incompressible ideal magnetohydrodynamic equations with vacuum under the non-collinearity condition

Xumin Gu

doi:10.3934/cpaa.2019029

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2019, Volume 18, Issue 2: 569-602. Doi: 10.3934/cpaa.2019029

This issue Previous Article On the one-dimensional continuity equation with a nearly incompressible vector field Next Article Well-posedness issues for some critical coupled non-linear Klein-Gordon equations

Well-posedness of axially symmetric incompressible ideal magnetohydrodynamic equations with vacuum under the non-collinearity condition

Xumin Gu

School of Mathematics, Shanghai University of Finance and Economics, Shanghai Center of Mathematical Sciences, China

Received: October 31, 2017

Revised: May 31, 2018

Published: October 2018

The author is supported by NSFC grant 11601305

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Abstract

We consider a free boundary problem for the axially symmetric incompressible ideal magnetohydrodynamic equations that describe the motion of the plasma in vacuum. Both the plasma magnetic field and vacuum magnetic field are tangent along the plasma-vacuum interface. Moreover, the vacuum magnetic field is composed in a non-simply connected domain and hence is non-trivial. Under the non-collinearity condition for the plasma and vacuum magnetic fields, we prove the local well-posedness of the problem in Sobolev spaces.

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Mathematics Subject Classification: Primary: 35L65, 35Q35; Secondary: 76B03, 76W05.

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