Browse > Article
http://dx.doi.org/10.4134/JKMS.j200565

REMARKS ON THE EXISTENCE OF AN INERTIAL MANIFOLD  

Kwak, Minkyu (Department of Mathematics Chonnam National University)
Sun, Xiuxiu (Department of Mathematics Chonnam National University)
Publication Information
Journal of the Korean Mathematical Society / v.58, no.5, 2021 , pp. 1261-1277 More about this Journal
Abstract
An inertial manifold is often constructed as a graph of a function from low Fourier modes to high ones and one used to consider backward bounded (in time) solutions for that purpose. We here show that the proof of the uniqueness of such solutions is crucial in the existence theory of inertial manifolds. Avoiding contraction principle, we mainly apply the Arzela-Ascoli theorem and Laplace transform to prove their existence and uniqueness respectively. A non-self adjoint example is included, which is related to a differential system arising after Kwak transform for Navier-Stokes equations.
Keywords
Inertial manifolds; asymptotic behaviour of solutions; infinite dimensional dynamical system;
Citations & Related Records
Times Cited By KSCI : 1  (Citation Analysis)
연도 인용수 순위
1 R. Temam, Infinite-dimensional dynamical systems in mechanics and physics, second edition, Applied Mathematical Sciences, 68, Springer-Verlag, New York, 1997. https://doi.org/10.1007/978-1-4612-0645-3   DOI
2 S. Zelik, Inertial manifolds and finite-dimensional reduction for dissipative PDEs, Proc. Roy. Soc. Edinburgh Sect. A 144 (2014), no. 6, 1245-1327. https://doi.org/10.1017/S0308210513000073   DOI
3 M. Kwak, Fourier spanning dimension of attractors for two-dimensional Navier-Stokes equations, Nonlinear Anal. 25 (1995), no. 3, 217-222. https://doi.org/10.1016/0362-546X(94)00178-K   DOI
4 J.-S. Baek, H.-K. Ju, and M. Kwak, Invariant manifolds and inertial forms for parabolic partial differential equations, Indiana Univ. Math. J. 42 (1993), no. 3, 721-731. https://doi.org/10.1512/iumj.1993.42.42032   DOI
5 Z. B. Fang and M. Kwak, A finite dimensional property of a parabolic partial differential equation, J. Dynam. Differential Equations 17 (2005), no. 4, 845-855. https://doi.org/10.1007/s10884-005-8272-y   DOI
6 Z. B. Fang and M. Kwak, Negatively bounded solutions for a parabolic partial differential equation, Bull. Korean Math. Soc. 42 (2005), no. 4, 829-836. https://doi.org/10.4134/BKMS.2005.42.4.82   DOI
7 C. Foias, G. R. Sell, and R. Temam, Inertial manifolds for nonlinear evolutionary equations, J. Differential Equations 73 (1988), no. 2, 309-353. https://doi.org/10.1016/0022-0396(88)90110-6   DOI
8 N. T. Huy, Admissiblly inertial manifolds for a class of semi-linear evolution equations, J. Diff. Eqns 254 (2013), 2638-2660.   DOI
9 I. Kukavica, Fourier parameterization of attractors for dissipative equations in one space dimension, J. Dynam. Differential Equations 15 (2003), no. 2-3, 473-484. https://doi.org/10.1023/B:JODY.0000009744.13730.01   DOI
10 M. Kwak, Finite-dimensional inertial forms for the 2D Navier-Stokes equations, Indiana Univ. Math. J. 41 (1992), no. 4, 927-981. https://doi.org/10.1512/iumj.1992.41.41051   DOI
11 M. Kwak and B. Lkhagvasuren, The cone property for a class of parabolic equations, J. Korean Soc. Ind. Appl. Math. 21 (2017), no. 2, 81-87. https://doi.org/10.12941/jksiam.2017.21.081   DOI
12 A. Kostianko and S. Zelik, Kwak transform and inertial manifolds revisited, arXiv: 1911.00698v1 [math.AP] 2 Nov (2019)
13 A. Romanov, Three counter examples in the theory of inertial manifolds, Math. Notes 68 (2000), 415-429.   DOI
14 G. R. Sell and Y. You, Dynamics of evolutionary equations, Applied Mathematical Sciences, 143, Springer-Verlag, New York, 2002. https://doi.org/10.1007/978-1-4757-5037-9
15 X. Sun, An inertial manifold for a non-self adjoint system, Honam Math. J. (2020)