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We study the two-dimensional magnetic Benard problem with noise, white in time. We prove the well-posedness including the path-wise uniqueness of the generalized solution, and the existence of the unique invariant, and consequently ergodic, measure under random perturbation.

We study the regularity of the three-dimensional magnetohydrodynamics system, and obtain criteria in terms of one velocity field component and two magnetic field components. In contrast to the previous results such as [22], we have eliminated the condition on the third component of the magnetic field completely while preserving the same upper bound on the integrability condition.

We study the two-dimensional beta-generalized supercritical quasi-geostrophic equation, and in particular show that to obtain global regularity results, one needs to bound only its partial derivative. Results may be generalized to similar active scalars, e.g. the porous media equation.