Global regularity of generalized magnetic Benard problem

Abstract

We study the magnetic Bénard problem in two‐dimensional space with generalized dissipative and diffusive terms, namely, fractional Laplacians and logarithmic supercriticality. Firstly, we show that when the diffusive term for the magnetic field is a full Laplacian, the solution initiated from data sufficiently smooth preserves its regularity as long as the power of the fractional Laplacians for the dissipative term of the velocity field and the diffusive term of the temperature field adds up to 1. Secondly, we show that with zero dissipation for the velocity field and a full Laplacian for the diffusive term of the temperature field, the global regularity result also holds when the diffusive term for the magnetic field consists of the fractional Laplacian with its power strictly bigger than 1. Finally, we show that with no diffusion from the magnetic and the temperature fields, the global regularity result remains valid as long as the dissipation term for the velocity field has its strength at least at the logarithmically supercritical level. These results represent various extensions of previous work on both Boussinesq and magnetohydrodynamics systems. Copyright © 2016 John Wiley & Sons, Ltd.

Revisiting the hybrid quantum Monte Carlo method for Hubbard and electron-phonon models

“…The slight shift, e.g. α = 1.05, prevents the matrix to become ill conditioned and prevents a pivot breakdown of the CG 42 . The matrix defined in Eq.…”

“…The search for a better preconditioner is still ongoing and the only progress we know of can be found in Ref. 42, where progress for very strong interactions is reported, although we like to point out that their tests were not performed on real configurations from a full Monte Carlo simulation but on randomly generated configurations of the auxiliary field. The preconditioner we have chosen is well suited for interactions that are not too strong and especially fast to calculate because of the sparsity structure of our matrix.…”

Regularity Criteria on the 2D Anisotropic Magnetic Bénard Equations

“…The authors also obtained the global regularity as well as some conditional regularity of strong solutions of the problem with mixed partial viscosity, thus extending the existing result of the problem with the full dissipation. Likewise, the global regularity of generalized magnetic Bénard problem was studied by Y. Yamazaki in [28] by extending the existing results on Boussinesq equation and magneto-hydrodynamic equations. The author studied the problem with fractional Laplacian and logarithmic super criticality.…”

Second proof of the global regularity of the two-dimensional MHD system with full diffusion and arbitrary weak dissipation

“…[12,35]); however, this problem has remained open since, seemingly asking for a new idea. It is worth emphasizing that the resolution of this problem should lead immediately to analogous results for various related systems of equations such as magnetic Bénard problem, and possibly magneto-micropolar fluid system ( [33,34]). The purpose of this manuscript is to provide a second proof of the global well-posedness of the system (2a)-(2c) in case α > 0, β = 1 as follows.…”

“…by (34), Hölder's inequalities, Bernstein's inequality, (8) and Proposition 3.2; we also used the crucial hypothesis that q 2 (1 − α) > 2. Therefore, we obtain from (35), (36), (37) applied to (34),…”