Asymptotic smoothing and attractors for the generalized Benjamin-Bona-Mahony equation on R 3

By M. Stanislavova, Atanas Stefanov, Bixiang Wang. 2005

58 citations recorded by the source corpus.

Abstract We study the asymptotic behavior of the solutions of the Benjamin–Bona–Mahony equation defined on R 3 . We first provide a sufficient condition to verify the asymptotic compactness of an evolution equation defined in an unbounded domain, which involves the Littlewood–Paley projection operators. We then prove the existence of an attractor for the Benjamin–Bona–Mahony equation in the phase space H 1 ( R 3 ) by showing the solutions are point dissipative and asymptotic compact. Finally, we establish the regularity of the attractor and show that the attractor is bounded in H 2 ( R 3 ) .

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/6062f498a33a6d4b3e4e0ca7f0ea08235a7cfe10

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In Applied Mathematics

1. Varga K. Kalantarov
2. Bixiang Wang
3. Edriss S. Titi
4. Darryl D. Holm
5. Vlad Vicol
6. Jinkai Li
7. John D. Gibbon

In Control and Systems Engineering

1. Varga K. Kalantarov
2. Bixiang Wang
3. Edriss S. Titi
4. Kening Lu
5. Shengfan Zhou
6. Anhui Gu
7. Xiaohu Wang
8. Peter W. Bates

In Mathematical Physics

1. Varga K. Kalantarov
2. Bixiang Wang
3. Edriss S. Titi
4. Jinkai Li
5. Vlad Vicol

Citations in the corpus

• Global attractors and determining modes for the 3D Navier-Stokes-Voight equations (2007, background): “In [42] it was shown the existence of the global attractor for GBBM equation in H(1)(R(3)).”
• Random Attractors for the Stochastic FitzHugh-Nagumo System on Unbounded Domains (2008, methodology): “mstances, the tail-estimates method must be used to deal with the problem caused by the unboundedness of domains. This approach was developed in [30] for deterministic parabolic equations and used in [1, 2, 4, 19, 22, 24, 27, 28]. Recently, the authors of [9] extended the tail-estimates method to th”
• Random Attractors for the Stochastic Benjamin-Bona-Mahony Equation on Unbounded Domains (2008, methodology): “In the deterministic case, the existence and uniqueness of solutions were studied in [6, 7, 13, 14, 15, 18, 24, 28, 29], and the global attractors were investigated in [5, 17, 19, 38, 42, 43, 44].”