Long time behavior of solutions to 3D generalized MHD equations

2020 ◽  
Vol 32 (4) ◽  
pp. 977-993
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Fractional Laplacian ◽  
Mhd Equations ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Algebraic Decay ◽  
Generalized Mhd Equations ◽  
Long Time ◽  
Whole Space ◽  
Mhd System ◽  
Incompressible Mhd

AbstractIn this paper, we consider the long time behavior of solutions for 3D incompressible MHD equations with fractional Laplacian. Firstly, in a periodic bounded domain, we prove the existence of a global attractor. The analysis reveals a relation between the Laplacian exponent and the regularity of the spaces of velocity and magnetic fields. Finally, in the whole space {\mathbb{R}^{3}}, we establish the sharp algebraic decay rate of solutions to the generalized MHD system provided that the parameters satisfy {\alpha,\beta\in(0,2]}.

scholarly journals On the Dynamics of Controlled Magnetohydrodynamic Systems

2008 ◽  
Vol 13 (3) ◽  
pp. 351-377 ◽  
Author(s):  
S. S. Ravindran
Keyword(s):  
Optimal Control ◽  
Optimal Control Problem ◽  
Control Problem ◽  
Optimal Solution ◽  
Mhd Equations ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Existence Of A Solution ◽  
Decay Properties ◽  
Long Time

In this paper we study the long time behavior of solutions for an optimal control problem associated with the viscous incompressible electrically conducting fluid modeled by the magnetohydrodynamic (MHD) equations in a bounded two dimensional domain through the adjustment of distributed controls. We first construct a quasi-optimal solution for the MHD systems which possesses exponential decay in time. We then derive some preliminary estimates for the long-time behavior of all admissible solutions of the MHD systems. Next we prove the existence of a solution for the optimal control problem for both finite and infinite time intervals. Finally, we establish the long-time decay properties of the solutions for the optimal control problem.

scholarly journals LONG TIME BEHAVIOR OF STOCHASTIC MHD EQUATIONS PERTURBED BY MULTIPLICATIVE NOISES

2016 ◽  
Vol 6 (4) ◽  
pp. 1081-1104 ◽  
Author(s):  
Hongyong Cui ◽  
◽  
Yangrong Li ◽  
Jinyan Yin
Keyword(s):  
Mhd Equations ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Multiplicative Noises ◽  
Long Time

scholarly journals Global Existence and Long-Time Behavior of Solutions to the Full Compressible Euler Equations with Damping and Heat Conduction in ℝ 3

2021 ◽  
Vol 2021 ◽  
pp. 1-13
Author(s):  
Yunshun Wu ◽  
Yong Wang ◽  
Rong Shen
Keyword(s):  
Heat Conduction ◽  
Euler Equations ◽  
Three Dimensional ◽  
Compressible Euler Equations ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Algebraic Decay ◽  
Compressible Euler ◽  
The Cauchy Problem ◽  
Long Time

We study the Cauchy problem of the three-dimensional full compressible Euler equations with damping and heat conduction. We prove the existence and uniqueness of the global small H N N ≥ 3 solution; in particular, we only require that the H 4 norms of the initial data be small when N ≥ 5 . Moreover, we use a pure energy method to show that the global solution converges to the constant equilibrium state with an optimal algebraic decay rate as time goes to infinity.

scholarly journals On global dynamics of 2D convective Cahn–Hilliard equation

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Xiaopeng Zhao
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Global Dynamics ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Initial Value ◽  
Hilliard Equation ◽  
Long Time ◽  
Cahn Hilliard Equation ◽  
Initial Boundary Value

AbstractIn this paper, we study the long time behavior of solution for the initial-boundary value problem of convective Cahn–Hilliard equation in a 2D case. We show that the equation has a global attractor in $H^{4}(\Omega )$ H 4 ( Ω ) when the initial value belongs to $H^{1}(\Omega )$ H 1 ( Ω ) .

The coupled Cahn–Hilliard/Allen–Cahn system with dynamic boundary conditions

2021 ◽  
pp. 1-27
Author(s):  
Ahmad Makki ◽  
Alain Miranville ◽  
Madalina Petcu
Keyword(s):  
Fractal Dimension ◽  
Boundary Conditions ◽  
Well Posedness ◽  
Time Behavior ◽  
Long Time Behavior ◽  
Dynamic Boundary Conditions ◽  
Finite Dimensional ◽  
Dynamic Boundary ◽  
Long Time ◽  
Finite Fractal Dimension

In this article, we are interested in the study of the well-posedness as well as of the long time behavior, in terms of finite-dimensional attractors, of a coupled Allen–Cahn/Cahn–Hilliard system associated with dynamic boundary conditions. In particular, we prove the existence of the global attractor with finite fractal dimension.

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