半离散Kuramoto-Sivashinsky方程的全局吸引子
Global Attractor for Semi-Discrete Kuramo-to-Sivashinsky Equation
摘要:
本文研究在上
具有周期边界条件的半离散Kuramoto-Sivashinsky型方程解的长时间行为。首先利用Crank-Nicolson格式对其进行离散，然后证明了该方程在
和
上紧的全局吸引子的存在。
Abstract: We study the long-time behaviour with a periodical boundary condition of semi-discrete Kuramoto- Sivashinsky equation in 
First, we use the Crank-Nicolson scheme to discrete this equation to prove that such a semi-discrete equation possesses a global arrtactor in 
, then we also show that this global attractor is actually a compact set of 
and
.
参考文献
|
[1]
|
K. L. Adams, J. R. King, R. H. Tew. Beyond-all-orders effects in muliple-scales asymptonics: Travelling-wave equations to the Kuramoto- Sivashinsky equation. Journal of Engineering Mathematics, 2003, 45(3-4) 197-226.
|
|
[2]
|
E. Cerpa, A. Mercado. Local exact controllability to the trajectories of the 1-D Kuramoto-Sivashinsky equation. Journal of Differential Equations, 2011, 250(4): 2024-2044.
|
|
[3]
|
S. Publjevic. Boundary model predictive control of Kuramoto-Sivashinsky equation with input and state constraints. Computers & Chemical Engineering, 2011, 34(10): 1655-1661.
|
|
[4]
|
L. Molinet. Local disspativity in L2 for the Kuramoto-Sivashinsky equation in spatial dimension 2. Journal of Dynamics and Differential Equations, 2000, 12(3): 533-556.
|
|
[5]
|
C. I. Byrnes, D. S. Gilliam, C. Hu and V. I. Shubov. Zero dynamics boundary control for regulation of the Kuramoto-Sivashinsky equation. Mathematical and Computer Modelling, 2010, 52(5-6): 875-891.
|
|
[6]
|
D. Wilczak. Chaos in the Kuramoto-Sivashinsky equation: A computer-assisted proof. Journal of Differential Equations, 2003, 194(2): 433- 459.
|
|
[7]
|
W. C. Troy. The existence of Steady of Kuramoto-Sivashinsky equation. Journal of Differential Equations, 1989, 82(2): 269-313.
|
|
[8]
|
N. Akroune. Regularity of the attractor for a weakly damped nonlinear Schrodinger equation on . Applied Mathematics Letters, 1999, 12(1): 45-48.
|
|
[9]
|
C. S. Zhu. Attractors of the nonlinear Schrodinger equation. Communications in Mathematical Analysis, 2008, 4(2): 67-75.
|
|
[10]
|
O. Goubet. Global attractor for weakly damped nonlinear Schrodinger equations in . Nonlinear Analysis Theory, 2009, 71(2): 317-320.
|
|
[11]
|
J. M. Ghidaglia, R. Temam. Attractors for damped nonlinear hyperbolic equations. Journal de Mathématiques Pures et Appliquées, 1987, 66: 273-319.
|
|
[12]
|
E. Ezzoug, W. Kechiche and E. Zahrouni. Finite dimensional global attractor for a semi-discrete nonlinear Schrodinger equation with a point defect. Applied Mathematics and Computation, 2011, 217(19): 7818-7830.
|
|
[13]
|
M. Abounouh. Global attractor for a time discretization of damped forced KdV equation, to appear.
|