Mixed Partial Dissipation Research Articles

Global solution of 3D MHD-Boussinesq system with mixed partial dissipation and thermal damping

Global solution of 3D MHD-Boussinesq system with mixed partial dissipation and thermal damping

Optimal decay for the 2D anisotropic Navier–Stokes equations with mixed partial dissipation

This paper studies the stability and large time behavior to the 2D anisotropic Navier–Stokes equations with mixed partial dissipation. We establish the uniform upper bounds and the global stability of solutions, and obtain the optimal decay properties to these global solutions and their higher order derivatives without any small assumptions on the initial data.

Stability of hydrostatic equilibrium in a uniform magnetic field with mixed partial dissipation

This paper intend to study the BMHD system in the whole space $ \mathbb{R}^{2} $ with vertical dissipation, horizontal magnetic dissipation and vertical temperature dissipation.Under the perturbation of hydrostaticequilibrium in a uniform magnetic field, we prove the nonlinear stability by introducing two additional functionals of the horizontalderivative of the velocity and the temperature.Simultaneously, the large-time behavior is also obtained.

Stability of the 3D MHD equations without vertical dissipation near an equilibrium

<abstract><p>Important progress has been made on the standard Laplacian case with mixed partial dissipation and diffusion. The stability problem of the 3D incompressible magnetohydrodynamic (MHD) equations without vertical dissipation but with the fractional velocity dissipation $ (-\Delta)^\alpha u $ and magnetic diffusion $ (-\Delta)^\beta b $ is unfortunately not often well understood for many ranges of fractional powers. This paper discovers that there are new phenomena with the case $ \alpha, \beta \leq 1 $. We establish that, if an initial datum ($ u_0, b_0 $) in the Sobolev space $ H^3(\mathbb{R}^3) $ is close enough to the equilibrium state, and we replace the terms $ (-\Delta)^\alpha u $ and $ (-\Delta)^\beta b $ by $ (-\Delta_h)^\alpha u $ and $ (-\Delta_h)^\beta b $, respectively, the resulting equations with $ \alpha, \beta \in(\frac{1}{2}, 1] $ then always lead to a steady solution, where $ \Delta_h = \partial_{x_1}^2+\partial_{x_2}^2 $.</p></abstract>

Sharp decay estimates for 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion

This paper focuses on the 3D incompressible magnetohydrodynamic (MHD) equations with mixed partial dissipation and magnetic diffusion. By using the energy methods, we obtain that this system possesses a global solution when the initial data is small in H3(R3). In addition, the optimal decay rates of this global solution and its first-order derivatives are established by bounding two time-weighted norms and applying a bootstrapping argument.

Stability of the 2D anisotropic Boussinesq equations with mixed partial dissipation

Stability is an important property of solutions to many fluid flow equations. In this paper, we focus on the stability problem for two-dimensional Boussinesq equations with only vertical viscosity and horizontal thermal diffusion near a stratified, hydrostatic equilibrium in the domain Ω=T×R, where T=[−12,12] is a 1D periodic box. We are mainly concerned with the H2-stability of global solutions when the initial data are closed to an equilibrium state and sufficiently small. Due to the lack of horizontal viscosity in velocity equations and vertical thermal diffusion in temperature equation, this problem seems to be not trivial and becomes particularly difficult. In order to achieve our main goal, we divide the velocity field and temperature into two parts: the horizontal average and the remaining oscillation part. Besides, we fully exploit the Lt2Lx2-norm of ∂xu2 to help us dominate the nonlinear parts. Our result shows that with a small initial perturbation, there exists a unique global solution to the stated Boussinesq equations. In addition, the perturbation remains small for all time. In particular, our result also demonstrates the stabilizing effect of internal gravity waves, which are induced by the stratified background hydrostatic state.

Global well-posedness for the 2D micropolar Bénard fluid system with mixed partial dissipation, angular viscosity and without thermal diffusivity

Global well-posedness for the 2D micropolar Bénard fluid system with mixed partial dissipation, angular viscosity and without thermal diffusivity

Global smooth large solutions to 2D magnetic Bénard systems with mixed partial dissipation

Global smooth large solutions to 2D magnetic Bénard systems with mixed partial dissipation

Global stability solution of the 2D MHD equations with mixed partial dissipation

<p style='text-indent:20px;'>This paper is devoted to understanding the global stability of perturbations near a background magnetic field of the 2D magnetohydrodynamic (MHD) equations with partial dissipation. We establish the global stability for the solutions of the nonlinear MHD system by the bootstrap argument.</p>

Stability near hydrostatic equilibrium to the three-dimensional Boussinesq equations with partial dissipation

This paper focuses on the stability problem for perturbations near a hydrostatic equilibrium associated with the three-dimensional Boussinesq equations with partial dissipation. We mainly study the global stability and large-time behavior of solutions to this system with only horizontal dissipation and obtain three results. The first establishes the global $$H^2$$ -stability, the second obtains the large-time asymptotic of the solutions, and the last assesses the asymptotic linear stability with explicit decay rates. Finally, we remark that these results can also be extended to the system with mixed partial dissipation.

Asymptotic stability and decay rates for the 2D magnetic Bénard fluid equations with mixed partial dissipation, magnetic diffusion and thermal diffusivity

We consider the asymptotic stability and large-time behavior for the 2D magnetic Bénard fluid equations with mixed partial dissipation, magnetic diffusion and thermal diffusivity. More precisely, we acquire the asymptotic stability and decay rate for the system on two perturbations near two steady state solutions (u˜,b˜,θ˜,p˜) and (uˇ,bˇ,θˇ,pˇ), respectively.

Stability of the 3D incompressible MHD equations with horizontal dissipation in periodic domain

<abstract><p>The stability problem on the magnetohydrodynamics (MHD) equations with partial or no dissipation is not well-understood. This paper focuses on the 3D incompressible MHD equations with mixed partial dissipation and magnetic diffusion. Our main result assesses the stability of perturbations near the steady solution given by a background magnetic field in periodic domain. The new stability result presented here is among few stability conclusions currently available for ideal or partially dissipated MHD equations.</p></abstract>

Stability of hydrostatic equilibrium for the 2D magnetic Bénard fluid equations with mixed partial dissipation, magnetic diffusion and thermal diffusivity

In mathematics and physics, the problem of the stability of perturbations near the hydrostatic balance is very important. Due to the classical tools designed for the fully dissipated systems are no longer apply, stability and global regularity problems on partially dissipated magnetic Benard fluid equations can be extremely challenging. This paper considers the stability problem on perturbations near the hydrostatic equilibrium for the 2D magnetic Benard fluid equations. We establish the global $$H^1$$ -stability of the 2D magnetic Benard fluid equations with mixed partial dissipation, magnetic diffusion and thermal diffusivity and affirm the global stability in the Sobolev space $$H^1$$ setting.

Global solutions of 3D incompressible MHD system with mixed partial dissipation and magnetic diffusion near an equilibrium

This paper focuses on the 3D incompressible magnetohydrodynamic (MHD) equations with mixed partial dissipation and magnetic diffusion. Our main result assesses the global stability of perturbations near the steady solution given by a background magnetic field. The stability problem on the MHD equations with partial or no dissipation has attracted considerable interests recently and there are substantial developments. The new stability result presented here is among the very few stability conclusions currently available for ideal or partially dissipated MHD equations. As a special consequence of the techniques introduced in this paper, we obtain the small data global well-posedness for the 3D incompressible Navier-Stokes equations without vertical dissipation.

Stability of perturbations near a background magnetic field of the 2D incompressible MHD equations with mixed partial dissipation

The stability and large-time behavior problem on some partially dissipated systems is not well-understood. The vorticity gradient of the 2D incompressible Euler equation can grow double exponentially in time while the same quantity to the 2D Navier-Stokes equation decays algebraically in time. However, the stability and large-time behavior of the vorticity gradients of the 2D Navier-Stokes equation with only vertical or horizontal dissipation appears to be unknown. This paper presents a global stability result on perturbations near a background magnetic field to the 2D incompressible magnetohydrodynamic (MHD) equations with vertical dissipation and horizontal magnetic diffusion. This stability result provides a significant example for the stabilizing effects of the magnetic field on electrically conducting fluids. In addition, we obtain an explicit decay rate for the solution of this nonlinear system.

Global existence of smooth solutions for three-dimensional magnetic Bénard system with mixed partial dissipation, magnetic diffusion and thermal diffusivity

This paper deals with the Cauchy problem to the 3D system of incompressible magnetic Bénard fluids. We prove that as the initial data satisfy ‖u0‖H1(R3)2+‖b0‖H1(R3)2+‖θ0‖H1(R3)2≤ε, where ε is a suitably small positive number, then the 3D magnetic Bénard system with mixed partial dissipation, magnetic diffusion and thermal diffusivity admit global smooth solutions.

Global existence of three-dimensional incompressible magneto-micropolar system with mixed partial dissipation, magnetic diffusion and angular viscosity

The magneto-micropolar fluid flows describe the motion of electrically conducting micropolar fluids in the presence of a magnetic field. The issue of whether the strong solution of magneto-micropolar equations in three-dimensional can exist globally in time with large initial data is still unknown. In this paper, we deal with the Cauchy problem of the three-dimensional magneto-micropolar system with mixed partial dissipation, magnetic diffusion and angular viscosity. More precisely, the global existence of smooth solutions to the three-dimensional incompressible magneto-micropolar fluid equations with mixed partial dissipation, magnetic diffusion and angular viscosity are obtained by energy method under the assumption that H1-norm of the initial data (u0,b0,w0) sufficiently small, namely ‖u0,b0,ω0‖H1(R3)2≤ε2, where ε is a sufficiently small positive number. This work follows the techniques in the paper of Cao and Wu (2011).

On two-dimensional incompressible magneto-micropolar system with mixed partial viscosity

The magneto-micropolar fluid flows describe the motion of electrically conducting micropolar fluids in the presence of a magnetic field. The issue of whether the two-dimensional magneto-micropolar equations always possess a global (in time) classical solution can be difficult when there is only partial dissipation or no dissipation at all. In this paper, we deal with the Cauchy problem of the two-dimensional magneto-micropolar problem with mixed partial viscosity. More precisely, the global existence and regularity of classical solutions to the two-dimensional incompressible magneto-micropolar equations with mixed partial dissipation, magnetic diffusion and angular viscosity are obtained. Moreover, some conditional regularity of strong solutions is obtained for two-dimensional magneto-micropolar problem with mixed partial viscosity. This work is inspired by the recent work (Regmi and Wu, 2016) by Regmi and Wu, and our results extend their results to other mixed partial viscosities cases and global existence and regularity are established.

Global regularity to the 2D incompressible MHD with mixed partial dissipation and magnetic diffusion in a bounded domain

This article considers the global regularity to the initial-boundary value problem for the 2D incompressible MHD with mixed partial dissipation and magnetic diffusion. To overcome the difficulty caused by the vanishing viscosities, we first establish the elliptic system for Ux and by, which are estimated by ∇ × ux and ∇ × by, respectively. Then, we establish the global estimates for ∇ × u and ∇ × b.

Global regularity of the [formula omitted] magneto-micropolar fluid flows with partial magnetic diffusion

In this paper, we studied the Cauchy problem of the magneto-micropolar fluid flows in 212 dimensions. If the initial data (u0,w0,b0)∈H2(R2), we obtain the global regularity of the 212D incompressible magneto-micropolar fluid flows with only horizontal magnetic diffusion or vertical magnetic diffusion. This work is inspired by the recent progresses made by Cao, etc., for 2D MHD equations with mixed partial dissipation and magnetic diffusion and Ji, etc., for 212D MHD system with mixed partial dissipation and magnetic diffusion.