April  2005, 13(3): 553-560. doi: 10.3934/dcds.2005.13.553

On Fourier parametrization of global attractors for equations in one space dimension

1. 

Department of Mathematics, University of Southern California, Los Angeles, CA 90089

Received  July 2004 Revised  November 2004 Published  May 2005

For the dissipative equations of the form

$ u_{t}-u_{x x}+f(x,u,u_x)=0$

we prove that the global attractor can be parametrized by a finite number of Fourier modes and that the number of modes is algebraic in parameters. This improves our earlier result [15], where the number of required modes is exponential. The method extends to equations of order higher than two.

Citation: Igor Kukavica. On Fourier parametrization of global attractors for equations in one space dimension. Discrete & Continuous Dynamical Systems, 2005, 13 (3) : 553-560. doi: 10.3934/dcds.2005.13.553
[1]

D. Hilhorst, L. A. Peletier, A. I. Rotariu, G. Sivashinsky. Global attractor and inertial sets for a nonlocal Kuramoto-Sivashinsky equation. Discrete & Continuous Dynamical Systems, 2004, 10 (1&2) : 557-580. doi: 10.3934/dcds.2004.10.557

[2]

Manil T. Mohan. Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory". Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020105

[3]

Mostafa Abounouh, Olivier Goubet. Regularity of the attractor for kp1-Burgers equation: the periodic case. Communications on Pure & Applied Analysis, 2004, 3 (2) : 237-252. doi: 10.3934/cpaa.2004.3.237

[4]

Jesenko Vukadinovic. Global dissipativity and inertial manifolds for diffusive burgers equations with low-wavenumber instability. Discrete & Continuous Dynamical Systems, 2011, 29 (1) : 327-341. doi: 10.3934/dcds.2011.29.327

[5]

Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299

[6]

Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021015

[7]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5321-5335. doi: 10.3934/dcdsb.2020345

[8]

Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations & Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013

[9]

Chun-Hsiung Hsia, Xiaoming Wang. On a Burgers' type equation. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1121-1139. doi: 10.3934/dcdsb.2006.6.1121

[10]

Milena Stanislavova. On the global attractor for the damped Benjamin-Bona-Mahony equation. Conference Publications, 2005, 2005 (Special) : 824-832. doi: 10.3934/proc.2005.2005.824

[11]

Wided Kechiche. Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1233-1252. doi: 10.3934/cpaa.2017060

[12]

Zhijian Yang, Zhiming Liu. Global attractor for a strongly damped wave equation with fully supercritical nonlinearities. Discrete & Continuous Dynamical Systems, 2017, 37 (4) : 2181-2205. doi: 10.3934/dcds.2017094

[13]

Azer Khanmamedov, Sema Simsek. Existence of the global attractor for the plate equation with nonlocal nonlinearity in $ \mathbb{R} ^{n}$. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 151-172. doi: 10.3934/dcdsb.2016.21.151

[14]

Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107

[15]

Brahim Alouini. Global attractor for a one dimensional weakly damped half-wave equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2655-2670. doi: 10.3934/dcdss.2020410

[16]

Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 3027-3042. doi: 10.3934/dcdss.2021031

[17]

Manil T. Mohan, Arbaz Khan. On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3943-3988. doi: 10.3934/dcdsb.2020270

[18]

Panagiotis Stinis. A hybrid method for the inviscid Burgers equation. Discrete & Continuous Dynamical Systems, 2003, 9 (4) : 793-799. doi: 10.3934/dcds.2003.9.793

[19]

Nobu Kishimoto, Minjie Shan, Yoshio Tsutsumi. Global well-posedness and existence of the global attractor for the Kadomtsev-Petviashvili Ⅱ equation in the anisotropic Sobolev space. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1283-1307. doi: 10.3934/dcds.2020078

[20]

Sergey V. Bolotin, Piero Negrini. Global regularization for the $n$-center problem on a manifold. Discrete & Continuous Dynamical Systems, 2002, 8 (4) : 873-892. doi: 10.3934/dcds.2002.8.873

2020 Impact Factor: 1.392

Metrics

  • PDF downloads (42)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]