December  2005, 4(4): 839-860. doi: 10.3934/cpaa.2005.4.839

The Boussinesq system:dynamics on the center manifold

1. 

Departamento de Matemática, Instituto Superior Técnico, 1049-001 Lisboa

Received  February 2005 Revised  May 2005 Published  September 2005

In this paper we analyze the behavior of small amplitude solutions of the variant of the classical Boussinesq system given by

$ \partial_t u = -\partial_x v - \alpha \partial_{x x x}v - \partial_x(u v), \quad \partial_t v = - \partial_x u - v \partial_x v,$

For $\alpha \leq 1$, this equation is ill-posed and most initial conditions do not lead to solutions. Nevertheless, we show that, for almost every $\alpha$, it admits solutions that are quasiperiodic in time. The proof uses the fact that the equation leaves invariant a smooth center manifold and for the restriction of the Boussinesq system to the center manifold, uses arguments of classical perturbation theory by considering the Hamiltonian formulation of the problem and studying the Birkhoff normal form.

Citation: Claudia Valls. The Boussinesq system:dynamics on the center manifold. Communications on Pure and Applied Analysis, 2005, 4 (4) : 839-860. doi: 10.3934/cpaa.2005.4.839
[1]

Jerry L. Bona, Zoran Grujić, Henrik Kalisch. A KdV-type Boussinesq system: From the energy level to analytic spaces. Discrete and Continuous Dynamical Systems, 2010, 26 (4) : 1121-1139. doi: 10.3934/dcds.2010.26.1121

[2]

Katarzyna Grabowska. Lagrangian and Hamiltonian formalism in Field Theory: A simple model. Journal of Geometric Mechanics, 2010, 2 (4) : 375-395. doi: 10.3934/jgm.2010.2.375

[3]

Marcel Oliver, Sergiy Vasylkevych. Hamiltonian formalism for models of rotating shallow water in semigeostrophic scaling. Discrete and Continuous Dynamical Systems, 2011, 31 (3) : 827-846. doi: 10.3934/dcds.2011.31.827

[4]

Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5449-5463. doi: 10.3934/dcdsb.2020353

[5]

Claudia Valls. Stability of some waves in the Boussinesq system. Communications on Pure and Applied Analysis, 2006, 5 (4) : 929-939. doi: 10.3934/cpaa.2006.5.929

[6]

A. Doubov, Enrique Fernández-Cara, Manuel González-Burgos, J. H. Ortega. A geometric inverse problem for the Boussinesq system. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1213-1238. doi: 10.3934/dcdsb.2006.6.1213

[7]

Mimi Dai, Han Liu. Low modes regularity criterion for a chemotaxis-Navier-Stokes system. Communications on Pure and Applied Analysis, 2020, 19 (5) : 2713-2735. doi: 10.3934/cpaa.2020118

[8]

Wei Wang. Closed trajectories on symmetric convex Hamiltonian energy surfaces. Discrete and Continuous Dynamical Systems, 2012, 32 (2) : 679-701. doi: 10.3934/dcds.2012.32.679

[9]

Min Chen. Numerical investigation of a two-dimensional Boussinesq system. Discrete and Continuous Dynamical Systems, 2009, 23 (4) : 1169-1190. doi: 10.3934/dcds.2009.23.1169

[10]

Dominique Blanchard, Nicolas Bruyère, Olivier Guibé. Existence and uniqueness of the solution of a Boussinesq system with nonlinear dissipation. Communications on Pure and Applied Analysis, 2013, 12 (5) : 2213-2227. doi: 10.3934/cpaa.2013.12.2213

[11]

V. Barbu. Periodic solutions to unbounded Hamiltonian system. Discrete and Continuous Dynamical Systems, 1995, 1 (2) : 277-283. doi: 10.3934/dcds.1995.1.277

[12]

Răzvan M. Tudoran, Anania Gîrban. On the Hamiltonian dynamics and geometry of the Rabinovich system. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 789-823. doi: 10.3934/dcdsb.2011.15.789

[13]

Morched Boughariou. Closed orbits of Hamiltonian systems on non-compact prescribed energy surfaces. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 603-616. doi: 10.3934/dcds.2003.9.603

[14]

Mitsuru Shibayama. Periodic solutions for a prescribed-energy problem of singular Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2017, 37 (5) : 2705-2715. doi: 10.3934/dcds.2017116

[15]

Liang Ding, Rongrong Tian, Jinlong Wei. Nonconstant periodic solutions with any fixed energy for singular Hamiltonian systems. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1617-1625. doi: 10.3934/dcdsb.2018222

[16]

Jochen Schmid. Stabilization of port-Hamiltonian systems with discontinuous energy densities. Evolution Equations and Control Theory, 2022  doi: 10.3934/eect.2021063

[17]

Margherita Nolasco. Breathing modes for the Schrödinger-Poisson system with a multiple--well external potential. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1411-1419. doi: 10.3934/cpaa.2010.9.1411

[18]

Kazumasa Fujiwara, Shuji Machihara, Tohru Ozawa. On a system of semirelativistic equations in the energy space. Communications on Pure and Applied Analysis, 2015, 14 (4) : 1343-1355. doi: 10.3934/cpaa.2015.14.1343

[19]

Tianwen Luo, Tao Tao, Liqun Zhang. Finite energy weak solutions of 2d Boussinesq equations with diffusive temperature. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3737-3765. doi: 10.3934/dcds.2019230

[20]

Jian Zhang, Wen Zhang, Xiaoliang Xie. Existence and concentration of semiclassical solutions for Hamiltonian elliptic system. Communications on Pure and Applied Analysis, 2016, 15 (2) : 599-622. doi: 10.3934/cpaa.2016.15.599

2021 Impact Factor: 1.273

Metrics

  • PDF downloads (104)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]