We establish the local well-posedness for the viscous Degasperis-Procesi equation. We show that the blow-up phenomena occurs in finite time. Moreover, applying the energy identity, we obtain a global existence result in the energy space.
In this article, we prove the global existence of the three-dimensional inhomogeneous incompressible magneto-hydrodynamic system under the assumptions that the initial velocity field and the initial conductivity are small in the critical spaceḂ 1/2 2,1 (R 3 ).
MSC: 35Q35; 76D03
The purpose of this paper is to study the incompressible non-resistive MHD equations in R 3 . We establish the global well-posedness of classical solutions if the initial data is axially symmetric and the swirl components of the velocity and magnetic vorticity vanish. In particular, the special axially symmetric initial data can be arbitrarily large and satisfy low regularity assumptions.
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