Global Regularity Of Classical Solutions Research Articles

Regularity criteria for the two-and-half-dimensional magnetic B\xe9nard system with partial dissipation, magnetic diffusion, and thermal diffusivity

This paper concentrates on the global regularity of classical solution to the 2frac{1}{2}D magnetic Bénard system with partial dissipation, magnetic diffusion, and thermal diffusivity (i.e., horizontal dissipation, horizontal magnetic diffusion, and horizontal thermal diffusivity; vertical dissipation, vertical magnetic diffusion, and vertical thermal diffusivity). For the 2frac{1}{2}D magnetic Bénard system with full dissipation, magnetic diffusion, and thermal diffusivity, the global existence and uniqueness can be obtained by the standard energy method. However, can the classical solution for the 2frac{1}{2}D incompressible magnetic Bénard system still keep its global regularity when losing some partial dissipation, magnetic diffusion, and thermal diffusivity terms? We will give a rigorous proof to the global regularity for the 2frac{1}{2}D magnetic Bénard system with horizontal and vertical dissipation, magnetic diffusion, and thermal diffusivity respectively in this paper. Furthermore, we also show that any possible finite time blow-up can be controlled by the L^{infty }-norm of the vertical velocity and magnetic components, not include the temperature component (see Theorems 1.1 and 1.2). The results extend the recent work by Cheng and Du (J. Math. Fluid Mech. 17:769–797, 2015), and generalize the recent works by Regmi (Math. Methods Appl. Sci. 40:1497–1504, 2017), and Ma and Zhang (Bound. Value Probl. 2018:79, 2018).

Global regularity of the 2D anisotropic magnetic Bénard system with vertical dissipation

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 θe2 as well as the Rayleigh–Bénard convection term u2. 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.

Global smooth solutions of the 2D MHD equations for a class of large data without magnetic diffusion

In this paper, we establish the global regularity of classical solutions for the two-dimensional MHD equations with only velocity diffusion for a class of large initial data. Both the initial velocity and magnetic field can be arbitrarily large in H s .

On global regularity of incompressible MHD equations in [formula omitted

This article considers the existence of global smooth solutions to the Cauchy problem in 3D incompressible magnetohydrodynamic (MHD) flows, and prove the global regularity of classical solutions for a family of large initial data with finite energy.

Global regularity for 2D magneto-micropolar equations with only micro-rotational velocity dissipation and magnetic diffusion

This paper studies the global regularity of classical solutions to 2D magneto-micropolar fluid equations with only micro-rotational velocity dissipation and magnetic diffusion. Here the micro-rotational velocity dissipation and magnetic diffusion are given by −ΔΩ and (−Δ)βb. Making use of several combined quantities, maximal regularity of heat operator and Littlewood–Paley decomposition theory, we establish a regularity criterion in terms of magnetic field for the case β=1 and the global regularity for β>1. The regularity criterion given here is also new even for the 2D magnetohydrodynamic equations. In addition, to prove these two main results, as preparation we establish a new global a priori estimate for magnetic field, namely Δb∈L∞(0,T;Lp(R2)) with p≥2 which also holds for the 2D magnetohydrodynamic equations as a particular case.

Global regularity for unsteady flow of MHD Sisko fluid in a porous medium

The aim of the present paper is to establish the global regularity of classical solution for unidirectional flow of magnetohydrodynamic (MHD) Sisko fluid. The fluid is taken between two rigid plates. An incompressible fluid saturates the porous medium. The main interest in this paper is to establish the global regularity of classical solutions when [Formula: see text], [Formula: see text], and [Formula: see text] are sufficiently small. In addition, uniqueness of the classical solution is also verified. Here BMO denotes the homogeneous space of bounded mean oscillations, V is the velocity, u is the x-component of velocity, and [Formula: see text] is the vorticity.

Global well-posedness for the [formula omitted]D incompressible magneto-micropolar fluid equations with mixed partial viscosity

In this paper, we prove the global regularity of classical solution to the magneto-micropolar fluid equations in 212 dimension with vertical dissipation, horizontal magnetic diffusion and horizontal vortex viscosity.

A regularity criterion for two‐and‐half‐dimensional magnetohydrodynamic equations with horizontal dissipation and horizontal magnetic diffusion

We study the global regularity of classical solution to two‐and‐half‐dimensional magnetohydrodynamic equations with horizontal dissipation and horizontal magnetic diffusion. We prove that any possible finite time blow‐up can be controlled by the L∞‐norm of the vertical components. Copyright © 2016 John Wiley & Sons, Ltd.

Global regularity for the 2D incompressible MHD-α system without full dissipation and magnetic diffusions

In this paper, we consider the global regularity for the 2D incompressible MHD-α system without full dissipation and magnetic diffusions. We establish the global regularity of classical solutions for the MHD-α system with mixed partial dissipation and magnetic diffusion (ν1=0, ν2>0, η1>0, η2=0) or horizontal dissipation and horizontal magnetic diffusion (ν1>0, ν2=0, η1>0, η2=0). As α→0, the solution converges to that of the corresponding MHD equations. We also obtain the convergence rate of the limit as α→0 in L2-norm.

The 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion

This paper studies the global regularity of classical solutions to the 2D incompressible magnetohydrodynamic (MHD) equations with horizontal dissipation and horizontal magnetic diffusion. It is shown here that the horizontal component of any solution admits a global (in time) bound in any Lebesgue space L2r with 1⩽r<∞ and the bound grows no faster than the order of rlogr as r increases. In addition, we establish a conditional global regularity in terms of the Lt2Lx∞-norm of the horizontal component and the global regularity of a slightly regularized version of the aforementioned MHD equations.

Global regularity for the 2D MHD equations with mixed partial dissipation and magnetic diffusion

Whether or not classical solutions of the 2D incompressible MHD equations without full dissipation and magnetic diffusion can develop finite-time singularities is a difficult issue. A major result of this paper establishes the global regularity of classical solutions for the MHD equations with mixed partial dissipation and magnetic diffusion. In addition, the global existence, conditional regularity and uniqueness of a weak solution is obtained for the 2D MHD equations with only magnetic diffusion.

The 2D Boussinesq equations with vertical viscosity and vertical diffusivity

This paper aims at the global regularity of classical solutions to the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion. We prove that the L r -norm of the vertical velocity v for any 1 < r < ∞ is globally bounded and that the L ∞ -norm of v controls any possible breakdown of classical solutions. In addition, we show that an extra thermal diffusion given by the fractional Laplace ( − Δ ) δ for δ > 0 would guarantee the global regularity of classical solutions.

On the Lagrangian dynamics of the axisymmetric 3D Euler equations

We study the dynamics along the particle trajectories for the 3D axisymmetric Euler equations. In particular, by rewriting the system of equations we find that there exists a complex Riccati type of structure in the system on the whole of R 3 , which generalizes substantially the previous results in [5] (D. Chae, On the blow-up problem for the axisymmetric 3D Euler equations, Nonlinearity 21 (2008) 2053–2060). Using this structure of equations, we deduce the new blow-up criterion that the radial increment of pressure is not consistent with the global regularity of classical solution. We also derive a much more refined version of the Lagrangian dynamics than that of [6] (D. Chae, On the Lagrangian dynamics for the 3D incompressible Euler equations, Comm. Math. Phys. 269 (2) (2007) 557–569) in the case of axisymmetry.

On global regularity of incompressible Navier-Stokes equations in $\mathbf R^3$

In this paper we prove the global regularity of classical solutions to the incompressible Navier-Stokes equations in $\mathbf R^3$ for a family of large initial data with finite energy.

Global regularity of the two-dimensional regularized MHD equations

In this paper, we consider the Cauchy problem of the two-dimensional regularized incompressible magnetohydrodynamics equations. The main objective of this paper is to establish the global regularity of classical solutions of the magnetohydrodynamics equations with the minimal dissipation. Consequently, our results significantly improve the previous works.