Attractor of a semi-discrete Benjamin–Bona–Mahony equation on $\mathbb {R}^1$

Chaosheng Zhu¹

¹School of Mathematics and Statistics Southwest University 400715 Chongqing, P.R. China

Annales Polonici Mathematici, Tome 115 (2015) no. 3, pp. 219-234

Résumé

This paper is concerned with the study of the large time behavior and especially the regularity of the global attractor for the semi-discrete in time Crank–Nicolson scheme to discretize the Benjamin–Bona–Mahony equation on $\mathbb {R}^1$. Firstly, we prove that this semi-discrete equation provides a discrete infinite-dimensional dynamical system in $H^1(\mathbb {R}^1)$. Then we prove that this system possesses a global attractor $\mathcal {A}_\tau $ in $H^1(\mathbb {R}^1)$. In addition, we show that the global attractor $\mathcal {A}_\tau $ is regular, i.e., $\mathcal {A}_\tau $ is actually included, bounded and compact in $H^2(\mathbb {R}^1)$. Finally, we estimate the finite fractal dimensions of $\mathcal {A}_\tau $.

Détail

DOI : 10.4064/ap115-3-2

Keywords: paper concerned study large time behavior especially regularity global attractor semi discrete time crank nicolson scheme discretize benjamin bona mahony equation mathbb firstly prove semi discrete equation provides discrete infinite dimensional dynamical system mathbb prove system possesses global attractor mathcal tau mathbb addition global attractor mathcal tau regular mathcal tau actually included bounded compact mathbb finally estimate finite fractal dimensions mathcal tau