We establish new Liouville-type theorems for the two-dimensional stationary magneto-hydrodynamic incompressible system assuming that the velocity and magnetic field have bounded Dirichlet integral. The key tool in our proof is observing that the stream function associated to the magnetic field satisfies a simple drift–diffusion equation for which a maximum principle is available.

A parabolic toy-model for the incompressible Navier-Stokes system is considered. This model shares a lot of similar features with the incompressible model, including the energy inequality, the scaling symmetry, and it is also supercritical in 3D. A goal is to establish some regularity results for this toy-model in order to get, if possible, better insight to the standard Navier-Stokes system. A Caffarelli-Kohn-Nirenberg type result for the model is also proved in a direct manner. Finally, the absence of divergence-free constraint allows us to study this model in the radially symmetric setting for which full regularity is established. Bibliography: 16 titles.

In this paper, we present two constructions of forward self-similar solutions to the 3D incompressible Navier-Stokes system, as the singular limit of forward self-similar solutions to certain parabolic systems.

We prove a local energy identity for a class of distributional solutions, in L 2,∞ ∩ W 1,0 2 , of parabolic equations with divergence-free drift.