On the blow-up criterion for the Hall-MHD problem with partial dissipation in R^3

We deal with the incompressible 3D Hall-magnetohydrodynamics with partial dissipation, a new blow-up criterion is obtained, based on which we also prove a new global solutions with small data.

On the Boussinesq Equations with Non-monotone Temperature Profiles

In this article, we consider the asymptotic stability of the two-dimensional Boussinesq equations with partial dissipation near a combination of Couette flow and temperature profiles T(y). As a first main result, we show that if $$T'$$ T ′ is of size at most $$\nu ^{1/3}$$ ν 1 / 3 in a suitable norm, then the linearized Boussinesq equations with only vertical dissipation of the velocity but not of the temperature are stable. Thus, mixing enhanced dissipation can suppress Rayleigh–Bénard instability in this linearized case. We further show that these results extend to the (forced) nonlinear equations with vertical dissipation in both temperature and velocity.