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Abstract The present paper is dedicated to the global-in-time existence and uniqueness issue for the three-dimensional incompressible Boussinesq equations with fractional partial dissipation. [ABSTRACT FROM AUTHOR]

Abstract This paper is dedicated to establishing the global regularity for the two dimensional magnetohydrodynamic equations with fractional anisotropic dissipation when the fractional powers are restricted to some certain ranges. In addition, the global regularity results for the two dimensional magnetohydrodynamic equations with partial dissipation are also obtained. Consequently, these results bring us more closer to the resolution of the global regularity problem on the two dimensional magnetohydrodynamic equations with standard Laplacian magnetic diffusion. [ABSTRACT FROM AUTHOR]

This paper studies the Cauchy problem to the two-dimensional (2D) anisotropic magnetic Bénard system with partial dissipation. The system considered in this paper has only vertical dissipation. We establish here the global regularity of classical solutions to this system. In proving the global regularity of classical solutions, the main difficulties are the absence of the horizontal dissipation and the presence of the buoyancy force term θ e 2 as well as the Rayleigh–Bénard convection term u 2 . To overcome them, we establish two new Gronwall type inequalities, which may be of independent interest. In addition, the global regularity of the 2D magnetic Bénard system with vertical dissipation, vertical magnetic diffusion and horizontal thermal diffusivity also holds true. This settles the global regularity issue unsolved in the previous works. [ABSTRACT FROM AUTHOR]

In this note we establish the existence and uniqueness of global strong solutions to the Cauchy problem of the three-dimensional incompressible micropolar equations with a nonlinear damping term. [ABSTRACT FROM AUTHOR]

In this paper, we consider the two-dimensional dissipative surface quasi-geostrophic equation, and establish a logarithmically improved regularity criterion. Consequently, our result extends the regularity criterion result of Dong and Pavlovié (2009). In addition, a logarithmically improved regularity criterion to the three-dimensional Navier–Stokes equations is also derived by the same arguments. Therefore, this result extends and improves many previous works. [ABSTRACT FROM AUTHOR]

In this short paper, we consider the two-dimensional inviscid Boussinesq equations and establish an improved regularity criterion. [ABSTRACT FROM AUTHOR]

In this paper, we investigate the Cauchy problem for the two-dimensional (2D) incompressible Bénard equations. On the one hand, we prove global-in-time existence of smooth solutions to the 2D Bénard equations with critical dissipation. On the other hand, we establish several regularity criteria involving temperature for 2D Bénard equations with supercritical dissipation. [ABSTRACT FROM AUTHOR]

In this paper we investigate the regularity criterion for the local-in-time classical solution to the three-dimensional (3D) incompressible Hall-magnetohydrodynamic equations (Hall-MHD). It is proved that the control of the vorticity alone can ensure the smoothness of the solution. [ABSTRACT FROM AUTHOR]

In this paper, we consider the two-dimensional (2D) Boussinesq equations with partial dissipation. The issue of whether the 2D Boussinesq equations always possess global (in time) classical solutions can be difficult when there is only partial dissipation. Therefore, the goal of this paper is to establish some regularity criteria of the corresponding systems. [ABSTRACT FROM AUTHOR]

In this paper we study the three-dimensional (3D) incompressible Hall-magnetohydrodynamic (Hall-MHD) equations and obtain its regularity criterion only in terms of the vorticity field. As a consequence, it improves the known results. [ABSTRACT FROM AUTHOR]

In this paper we establish the regularity criteria of solutions to the three space dimensions generalized viscous Hall-magnetohydrodynamics. Moreover, we also prove that there exist classical solutions globally in time to the generalized viscous Hall-MHD if the initial data is small enough or the fractional Laplacian dissipation powers α and β are suitably large. These results are some improvements of very recent work (Chae and Lee, 2014) established by Chae and Lee. [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 investigate the regularity criterion for the local in time classical solution to the 3D incompressible magnetohydrodynamic equations. We prove that if ∇ × u belongs to L 2 ( 0 , T ; B ̇ ∞ , ∞ − 1 ) , then the local solution ( u , B ) can be extended beyond time T . As a consequence, this result extends several previous works. [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]

Abstract: In this paper, we consider the inviscid 3D Boussinesq system in Besov spaces and obtain the lower bound for the lifespan of solutions. [Copyright &y& Elsevier]

Abstract: This paper is devoted to a porous medium system subject to nonlocal boundary conditions and with nonlocal sources. We investigate the global existence and blow-up in finite time of a nonnegative solution by using a sub–super-solution method. Moreover, under appropriate hypotheses for the conditions, we obtain asymptotic properties of the solutions. [Copyright &y& Elsevier]

In this paper we establish the global regularity of the 2D SQG equation with variable viscosity term in non-divergence form. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to provide an alternative approach to the global regularity for the two-dimensional Euler–Boussinesq equations which couple the incompressible Euler equation for the velocity and a transport equation with fractional critical diffusion for the temperature. In contrast to the first proof of this result in [T. Hmidi, S. Keraani, and F. Rousset, Comm. Partial Differential Equations, 36 (2011), pp. 420–445] that took fully exploit of the hidden structure of the coupling system, the main argument in this manuscript is mainly based on the differentiability of the drift–diffusion equation. [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]