Partial Magnetic Diffusion Research Articles

Large Time Behavior and Stability for Two-Dimensional Magneto-Micropolar Equations with Partial Dissipation

This paper is devoted to the stability and decay estimates of solutions to the two-dimensional magneto-micropolar fluid equations with partial dissipation. Firstly, focus on the 2D magneto-micropolar equation with only velocity dissipation and partial magnetic diffusion, we obtain the global existence of solutions with small initial in H^s({mathbb {R}}^2)(s>1), and by fully exploiting the special structure of the system and using the Fourier splitting methods, we establish the large time decay rates of solutions. Secondly, when the magnetic field has partial dissipation, we show the global existence of solutions with small initial data in dot{B}^0_{2,1}({mathbb {R}}^2). In addition, we explore the decay rates of these global solutions are correspondingly established in dot{B}^m_{2,1}({mathbb {R}}^2) with 0 le m le s, when the initial data belongs to the negative Sobolev space dot{H}^{-l}({mathbb {R}}^2) (for each 0 le l <1).

The 2D magnetohydrodynamic system with Euler‐like velocity equation and partial magnetic diffusion

AbstractWhen the velocity equation of the incompressible 2D magnetohydrodynamic (MHD) system is inviscid, the global well‐posedness and stability problem in the whole space case remains an extremely challenging open problem. Broadman, Lin, and Wu (SIAM J. Math. Anal. 52(5) (2020): 5001‐5035) were able to establish the global well‐posedness and stability near a background magnetic field when there is damping in one velocity component. Their work exploited the stabilizing effect of the background magnetic field. This paper presents new progress. We are able to prove the global well‐posedness and stability even when the magnetic diffusion is degenerate and only in the vertical direction. The velocity equation is still inviscid and has damping only in the vertical component. The proof of this new result overcomes two main difficulties, the potential rapid growth of the velocity due to the lack of dissipation or horizontal damping and the control of nonlinearity associated with the magnetic field. By discovering the key hidden smoothing effects and incorporating them in the construction of a two‐layered energy function, we are able to obtain uniform bounds on the solution in the H3‐norm when the initial perturbation is near the background magnetic field. In addition, we prove that certain Lebesgue and Sobolev norms of the solution approach zero as time approaches infinity.

Global regularity and time decay for the 2D magneto-micropolar system with fractional dissipation and partial magnetic diffusion

Global regularity and time decay for the 2D magneto-micropolar system with fractional dissipation and partial magnetic diffusion

Stochastic MHD equations with fractional kinematic dissipation and partial magnetic diffusion in [formula omitted

This paper focuses on a system of the 2-D MHD equations with fractional kinematic dissipation and partial magnetic diffusion under random perturbations. We first establish the local existence, uniqueness and blow-up criterion of pathwise classical solution to the corresponding SPDE with general multiplicative noise in R2. Moreover, when the noise is non-autonomous and linear, we further establish global existence and large-time behavior of pathwise solutions by exploiting the structure of the equations and some estimates on Girsanov type stochastic processes.

Global regularity problem of two‐dimensional magnetic Bénard fluid equations

In the paper, we devote to broadening the current global regularity results for the two‐dimensional magnetic Bénard fluid equations. We study three cases: (i) fractional Laplacian dissipation (‐ Δ)αu, partial magnetic diffusion , and Laplacian thermal diffusivity Δθ; (ii) partial fractional dissipation , partial magnetic diffusion , and Laplacian thermal diffusivity Δθ; (iii) partial fractional magnetic diffusion , Laplacian thermal diffusivity Δθ, and without Laplacian dissipation Δu (i.e., ), and establish the global regularity for each cases.

Global regularity and decay estimates for 2D magneto-micropolar equations with partial dissipation

This paper examines the global regularity problem and decay estimates for two classes of two-dimensional (2D) magneto-micropolar equations with partial dissipation. By fully exploiting the special structure of the system and using the maximal regularity property of the 1D heat operator, we establish the global existence of classical solution for 2D magneto-micropolar equations with only velocity dissipation and partial magnetic diffusion. In addition, we obtain the global classical solution for small initial data and decay estimates of solution to 2D magneto-micropolar equations with only microrotational dissipation and magnetic diffusion.

Remarks on the global regularity and time decay of the 2D MHD equations with partial dissipation

This paper focuses on a system of the two‐dimensional (2D) magnetohydrodynamic (MHD) equations with the partial kinematic dissipation (∂yyu1,∂xxu2) and the partial magnetic diffusion (∂yyb1,∂xxb2). Based on the basic energy estimates only, we are able to show that this system always possesses a unique global smooth solution when the initial data are sufficiently smooth. Moreover, we obtain optimal large‐time decay rates of both solutions and their higher order derivatives by developing the classic Fourier splitting methods together with the auxiliary decay estimates of the first derivative of solutions and induction technique.

Global regularity for the 2D magneto-micropolar equations with partial and fractional dissipation

This paper studies two cases of global regularity problems on the 2D magneto-micropolar equations with partial magnetic diffusion and fractional dissipation. For the first case the velocity field is ideal, the micro-rotational velocity is with Laplacian dissipation and the magnetic field has fractional partial diffusion (−∂22βb1,−∂11βb2) with β>1. In the second case, the velocity has a fractional Laplacian dissipation (−Δ)αu with any α>0, the micro-rotational velocity is with Laplacian dissipation and the magnetic field has partial diffusion (−∂22b1,−∂11b2). In two cases the global well-posedness of classical solutions is proved in this paper.

Global Regularity and Time Decay for the 2D Magnetohydrodynamic Equations with Fractional Dissipation and Partial Magnetic Diffusion

This paper focuses on a system of the 2D magnetohydrodynamic (MHD) equations with the kinematic dissipation given by the fractional operator $$(-\Delta )^\alpha $$ and the magnetic diffusion by partial Laplacian. We are able to show that this system with any $$\alpha >0$$ always possesses a unique global smooth solution when the initial data is sufficiently smooth. In addition, we make a detailed study on the large-time behavior of these smooth solutions and obtain optimal large-time decay rates. Since the magnetic diffusion is only partial here, some classical tools such as the maximal regularity property for the 2D heat operator can no longer be applied. A key observation on the structure of the MHD equations allows us to get around the difficulties due to the lack of full Laplacian magnetic diffusion. The results presented here are the sharpest on the global regularity problem for the 2D MHD equations with only partial magnetic diffusion.

Global Regularity for the 2D MHD Equations with Partial Hyper-resistivity

AbstractThis article establishes the global existence and regularity for a system of the two-dimensional (2D) magnetohydrodynamic (MHD) equations with only directional hyper-resistivity. More precisely, the equation of $b_1$ (the horizontal component of the magnetic field) involves only vertical hyperdiffusion (given by $\Lambda_2^{2\beta} b_1$) while the equation of $b_2$ (the vertical component) has only horizontal hyperdiffusion (given by $\Lambda_1^{2\beta} b_2$), where $\Lambda_1$ and $\Lambda_2$ are directional Fourier multiplier operators with the symbols being $|\xi_1|$ and $|\xi_2|$, respectively. We prove that, for $\beta&amp;gt;1$, this system always possesses a unique global-in-time classical solution when the initial data is sufficiently smooth. The model concerned here is rooted in the MHD equations with only magnetic diffusion, which play a significant role in the study of magnetic reconnection and magnetic turbulence. In certain physical regimes and under suitable scaling, the magnetic diffusion becomes partial (given by part of the Laplacian operator). There have been considerable recent developments on the fundamental issue of whether classical solutions of these equations remain smooth for all time. The papers of Cao–Wu–Yuan [8] and of Jiu–Zhao [26] obtained the global regularity when the magnetic diffusion is given by the full fractional Laplacian $(-\Delta)^\beta$ with $\beta&amp;gt;1$. The main result presented in this article requires only directional fractional diffusion and yet we prove the regularization in all directions. The proof makes use of a key observation on the structure of the nonlinearity in the MHD equations and technical tools on Fourier multiplier operators such as the Hörmander–Mikhlin multiplier theorem. The result presented here appears to be the sharpest for the 2D MHD equations with partial magnetic diffusion.