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Micropolar equations, modeling micropolar uid ows, consist of coupled equations obeyed by the evolution of the velocity u and that of the microrotation w. This paper focuses on the two-dimensional micropolar equations with the fractional dissipation (-Δ)αu and (-Δ)βw, where 0 < α,β < 1. The goal here is the global regularity of the fractional micropolar equations with minimal fractional dissipation. Recent eαorts have resolved the two borderline cases α = 1, β = 0 and α = 0, β = 1. However, the situation for the general critical case α + β = 1 with 0 < α < 1 is far more complex and the global regularity appears to be out of reach. When the dissipation is split among the equations, the dissipation is no longer as efficient as in the borderline cases and diαerent ranges of α and β require diαerent estimates and tools. We aim at the subcritical case α + β > 1 and divide α ∊ (0, 1) into βve sub-intervals to seek the best estimates so that we can impose the minimal requirements on α and β. The proof of the global regularity relies on the introduction of combined quantities, sharp lower bounds for the fractional dissipation and delicate upper bounds for the nonlinearity and associated commutators. [ABSTRACT FROM AUTHOR]

This paper examines the global regularity problem on the two-dimensional (2D) incompressible Boussinesq equations with fractional horizontal dissipation and thermal diffusion. The goal is to establish the global existence and regularity for the Boussinesq equations with minimal dissipation and thermal diffusion. By working with this general 1D fractional Laplacian dissipation, we are no longer constrained to the standard partial dissipation and this study will help understand the issue on how much dissipation is necessary for the global regularity. Due to the nonlocality of these 1D fractional operators, some of the standard energy estimate techniques such as integration by parts no longer apply and new tools including several anisotropic embedding and interpolation inequalities involving fractional derivatives are derived. These tools allow us to obtain very sharp upper bounds for the nonlinearities. [ABSTRACT FROM AUTHOR]

In this paper, we consider the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation and thermal diffusion. Attention is focused on the subcritical case when the velocity dissipation dominates. More precisely, we establish the global regularity result of the 2D Boussinesq equations in a new range of fractional powers of the Laplacian, namely [ABSTRACT FROM AUTHOR]

As a continuation of the previous work [48] , in this paper we focus on the Cauchy problem of the two-dimensional (2D) incompressible Boussinesq equations with fractional Laplacian dissipation. We give an elementary proof of the global regularity of the smooth solutions of the 2D Boussinesq equations with a new range of fractional powers of the Laplacian. The argument is based on the nonlinear lower bounds for the fractional Laplacian established in [13] . Consequently, this result significantly improves the recent works [13,45,48] . [ABSTRACT FROM AUTHOR]

In this paper, we are interested in the study of the Cauchy problem to the two-dimensional (2D) incompressible Boussinesq equations with fractional dissipation. By making use of the nonlinear lower bounds for the fractional Laplacian established in Constantin and Vicol (2012), we establish the global regularity of the smooth solutions of the 2D Boussinesq equations with a new range of fractional powers of the Laplacian. This result significantly improves the recent works of Constantin and Vicol (2012) and Yang et al. (2014). [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we prove the global regularity to the Cauchy problem of 3D incompressible magnetohydrodynamic- (MHD- ) model with fractional diffusion. We prove that if the exponents and satisfy with , then the 3D MHD- model with fractional diffusion admits a unique global smooth solution. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the global regularity of the two-dimensional generalized incompressible magnetohydrodynamics system with velocity dissipation exponent and magnetic diffusion exponent based on the improved integrability in terms of vorticity. This result is a further improvement of very recent works (Jiu and Zhao (2014) and Yamazaki (2013)) which require and . Therefore our result is more closer to the resolution of the well-known global regularity issue on the 2D magnetohydrodynamics system . [Copyright &y& Elsevier]

The two-dimensional (2D) incompressible Boussinesq equations model geophysical fluids and play an important role in the study of the Raleigh–Bernard convection. Mathematically this 2D system retains some key features of the 3D Navier–Stokes and Euler equations such as the vortex stretching mechanism. The issue of whether the 2D Boussinesq equations always possess global (in time) classical solutions can be difficult when there is only partial dissipation or no dissipation at all. This paper obtains the global regularity for two partial dissipation cases and proves several global a priori bounds for two other prominent partial dissipation cases. These results take us one step closer to a complete resolution of the global regularity issue for all the partial dissipation cases involving the 2D Boussinesq equations. [ABSTRACT FROM AUTHOR]

Abstract: The two-dimensional (2D) incompressible Euler equations have been thoroughly investigated and the resolution of the global (in time) existence and uniqueness issue is currently in a satisfactory status. In contrast, the global regularity problem concerning the 2D inviscid Boussinesq equations remains widely open. In an attempt to understand this problem, we examine the damped 2D Boussinesq equations and study how damping affects the regularity of solutions. Since the damping effect is insufficient in overcoming the difficulty due to the “vortex stretching”, we seek unique global small solutions and the efforts have been mainly devoted to minimizing the smallness assumption. By positioning the solutions in a suitable functional setting (more precisely, the homogeneous Besov space ), we are able to obtain a unique global solution under a minimal smallness assumption. [Copyright &y& Elsevier]