-
Previous Article
Long-time behavior for competition-diffusion systems via viscosity comparison
- DCDS Home
- This Issue
-
Next Article
Quasi-periodic solutions for completely resonant non-linear wave equations in 1D and 2D
On Fourier parametrization of global attractors for equations in one space dimension
1. | Department of Mathematics, University of Southern California, Los Angeles, CA 90089 |
$ 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.
[1] |
D. Hilhorst, L. A. Peletier, A. I. Rotariu, G. Sivashinsky. Global attractor and inertial sets for a nonlocal Kuramoto-Sivashinsky equation. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 557-580. doi: 10.3934/dcds.2004.10.557 |
[2] |
Andy Hammerlindl, Bernd Krauskopf, Gemma Mason, Hinke M. Osinga. Determining the global manifold structure of a continuous-time heterodimensional cycle. Journal of Computational Dynamics, 2022, 9 (3) : 393-419. doi: 10.3934/jcd.2022008 |
[3] |
Manil T. Mohan. Global attractors, exponential attractors and determining modes for the three dimensional Kelvin-Voigt fluids with "fading memory". Evolution Equations and Control Theory, 2022, 11 (1) : 125-167. doi: 10.3934/eect.2020105 |
[4] |
Mostafa Abounouh, Olivier Goubet. Regularity of the attractor for kp1-Burgers equation: the periodic case. Communications on Pure and Applied Analysis, 2004, 3 (2) : 237-252. doi: 10.3934/cpaa.2004.3.237 |
[5] |
Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299 |
[6] |
Jesenko Vukadinovic. Global dissipativity and inertial manifolds for diffusive burgers equations with low-wavenumber instability. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 327-341. doi: 10.3934/dcds.2011.29.327 |
[7] |
Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete and Continuous Dynamical Systems - B, 2021, 26 (10) : 5321-5335. doi: 10.3934/dcdsb.2020345 |
[8] |
Biyue Chen, Chunxiang Zhao, Chengkui Zhong. The global attractor for the wave equation with nonlocal strong damping. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6207-6228. doi: 10.3934/dcdsb.2021015 |
[9] |
Guy V. Norton, Robert D. Purrington. The Westervelt equation with a causal propagation operator coupled to the bioheat equation.. Evolution Equations and Control Theory, 2016, 5 (3) : 449-461. doi: 10.3934/eect.2016013 |
[10] |
Chun-Hsiung Hsia, Xiaoming Wang. On a Burgers' type equation. Discrete and Continuous Dynamical Systems - B, 2006, 6 (5) : 1121-1139. doi: 10.3934/dcdsb.2006.6.1121 |
[11] |
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 |
[12] |
Wided Kechiche. Regularity of the global attractor for a nonlinear Schrödinger equation with a point defect. Communications on Pure and Applied Analysis, 2017, 16 (4) : 1233-1252. doi: 10.3934/cpaa.2017060 |
[13] |
Zhijian Yang, Zhiming Liu. Global attractor for a strongly damped wave equation with fully supercritical nonlinearities. Discrete and Continuous Dynamical Systems, 2017, 37 (4) : 2181-2205. doi: 10.3934/dcds.2017094 |
[14] |
Azer Khanmamedov, Sema Simsek. Existence of the global attractor for the plate equation with nonlocal nonlinearity in $ \mathbb{R} ^{n}$. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 151-172. doi: 10.3934/dcdsb.2016.21.151 |
[15] |
Tomás Caraballo, Marta Herrera-Cobos, Pedro Marín-Rubio. Global attractor for a nonlocal p-Laplacian equation without uniqueness of solution. Discrete and Continuous Dynamical Systems - B, 2017, 22 (5) : 1801-1816. doi: 10.3934/dcdsb.2017107 |
[16] |
Brahim Alouini. Global attractor for a one dimensional weakly damped half-wave equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2655-2670. doi: 10.3934/dcdss.2020410 |
[17] |
Wided Kechiche. Global attractor for a nonlinear Schrödinger equation with a nonlinearity concentrated in one point. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 3027-3042. doi: 10.3934/dcdss.2021031 |
[18] |
Manil T. Mohan, Arbaz Khan. On the generalized Burgers-Huxley equation: Existence, uniqueness, regularity, global attractors and numerical studies. Discrete and Continuous Dynamical Systems - B, 2021, 26 (7) : 3943-3988. doi: 10.3934/dcdsb.2020270 |
[19] |
Panagiotis Stinis. A hybrid method for the inviscid Burgers equation. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 793-799. doi: 10.3934/dcds.2003.9.793 |
[20] |
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 and Continuous Dynamical Systems, 2020, 40 (3) : 1283-1307. doi: 10.3934/dcds.2020078 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]