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In this paper, we study the inhomogeneous boundary value problem for the steady MHD system of a viscous incompressible fluid in an arbitrary bounded multiply connected domain. We prove the existence of generalized solutions of the steady MHD system with some smallness conditions on the boundary values. [ABSTRACT FROM AUTHOR]

In this paper, we consider the classical solutions to a model of two-dimensional incompressible inviscid Boussinesq–Bénard equations. Notice that, in the case when the source term of temperature equation in this model is the second component of velocity u 2 or no source term, there is no global-in-time existence result for the general initial data. Here, if the source term is only chosen as Δ u 2 , then we can obtain the global well-posedness, inviscid limit and some exponential decay estimates. Our key observation is the nice symmetrical structure hidden in the corresponding system, which plays an extremely important role in the global well-posedness studied here. [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]

This paper is dedicated to the Oldroyd-B model with fractional dissipation (−Δ) α τ for any α > 0. We establish the global smooth solutions to the Oldroyd-B model in the corotational case with arbitrarily small fractional powers of the Laplacian in two spatial dimensions. Moreover, in the Appendix, we provide some a priori estimates to the Oldroyd-B model in the critical case, which may be useful and of interest for future improvement. Therefore, our result is closer to the resolution of the well-known global regularity issue on the critical 2D Oldroyd-B model. Copyright © 2016 John Wiley & Sons, Ltd. [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]

In this paper, we consider the $$n$$ -dimensional ( $$n\ge 2$$ ) damped models of incompressible fluid mechanics in Besov spaces and establish the global (in time) regularity of classical solutions provided that the initial data are suitable small. [ABSTRACT FROM AUTHOR]

The hydrostatic equilibrium is a prominent topic in fluid dynamics and astrophysics. Understanding the stability of perturbations near the hydrostatic equilibrium of the Boussinesq system helps gain insight into certain weather phenomena. The 2D Boussinesq system focused here is anisotropic and involves only horizontal dissipation and horizontal thermal diffusion. Due to the lack of the vertical dissipation, the stability and precise large-time behavior problem is difficult. When the spatial domain is R 2 , the stability problem in a Sobolev setting remains open. When the spatial domain is T × R , this paper solves the stability problem and specifies the precise large-time behavior of the perturbation. By decomposing the velocity u and temperature θ into the horizontal average (u ¯ , θ ¯) and the corresponding oscillation (u ~ , θ ~) , and deriving various anisotropic inequalities, we are able to establish the global stability in the Sobolev space H 2 . In addition, we prove that the oscillation (u ~ , θ ~) decays exponentially to zero in H 1 and (u , θ) converges to (u ¯ , θ ¯) . This result reflects the stratification phenomenon of buoyancy-driven fluids. [ABSTRACT FROM AUTHOR]