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CPAA

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.

CPAA

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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