CPAA
Global wellposedness for a transport equation with super-critial dissipation
Xumin Gu
Communications on Pure & Applied Analysis 2011, 10(2): 653-665 doi: 10.3934/cpaa.2011.10.653
We study a one-dimensional transport equation with non-local velocity and supercritial dissipation. Using the methods of modulus of continuity introduced in [1] and fractional Laplacian representaiton introduced in [2], we prove its global well-posedness for small periodic initial data in Holder spaces.
keywords: global regularity. Super-critical dissipation modulus of continuity fractional Laplacian
CPAA
Well-posedness of axially symmetric incompressible ideal magnetohydrodynamic equations with vacuum under the non-collinearity condition
Xumin Gu
Communications on Pure & Applied Analysis 2019, 18(2): 569-602 doi: 10.3934/cpaa.2019029

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.

keywords: MHD vacuum free boundary axially symmetric non-collinearity

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