Abstract:
This paper is a study of global attractors of abstract semilinear parabolic equations and their embeddings in finite-dimensional manifolds. As is well known, a sufficient condition for the existence of smooth (at least C1-smooth) finite-dimensional inertial manifolds containing a global attractor is the so-called spectral gap condition for the corresponding linear operator. There are also a number of examples showing that if there is no gap in the spectrum, then a C1-smooth inertial manifold may not exist. On the other hand, since an attractor usually has finite fractal dimension, by Mañé's theorem it projects bijectively and Hölder-homeomorphically into a finite-dimensional generic plane if its dimension is large enough. It is shown here that if there are no gaps in the spectrum, then there exist attractors that cannot be embedded in any Lipschitz or even log-Lipschitz finite-dimensional manifold. Thus, if there are no gaps in the spectrum, then in the general case the inverse Mañé projection of the attractor cannot be expected to be Lipschitz or log-Lipschitz. Furthermore, examples of attractors with finite Hausdorff and infinite fractal dimension are constructed in the class of non-linearities of finite smoothness.
Bibliography: 35 titles.
Keywords:
global attractors, inertial manifolds, Mañé projections, regularity.
Citation:
A. Eden, S. V. Zelik, V. K. Kalantarov, “Counterexamples to regularity of Mañé projections in the theory of attractors”, Russian Math. Surveys, 68:2 (2013), 199–226
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\paper Counterexamples to regularity of Ma\~n\'e projections in the theory of attractors
\jour Russian Math. Surveys
\yr 2013
\vol 68
\issue 2
\pages 199--226
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This publication is cited in the following 17 articles:
Anna Kostianko, Sergey Zelik, “Smooth extensions for inertial manifolds of semilinear parabolic equations”, Analysis & PDE, 17:2 (2024), 499
Russian Math. Surveys, 78:4 (2023), 635–777
Varga Kalantarov, Anna Kostianko, Sergey Zelik, “Determining functionals and finite-dimensional reduction for dissipative PDEs revisited”, Journal of Differential Equations, 345 (2023), 78
Kostianko A., Li X., Sun Ch., Zelik S., “Inertial Manifolds Via Spatial Averaging Revisited”, SIAM J. Math. Anal., 54:1 (2022), 268–305
Kostianko A., Zelik S., “Kwak Transform and Inertial Manifolds Revisited”, J. Dyn. Differ. Equ., 2021
Kostianko A., “Bi-Lipschitz Mane Projectors and Finite-Dimensional Reduction For Complex Ginzburg-Landau Equation”, Proc. R. Soc. A-Math. Phys. Eng. Sci., 476:2239 (2020), 20200144
Chepyzhov V.V., Kostianko A., Zelik S., “Inertial Manifolds For the Hyperbolic Relaxation of Semilinear Parabolic Equations”, Discrete Contin. Dyn. Syst.-Ser. B, 24:3, SI (2019), 1115–1142
A. Kostianko, S. Zelik, “Inertial manifolds for 1D reaction-diffusion-advection systems. Part II: periodic boundary conditions”, Commun. Pure Appl. Anal, 17:1 (2018), 285–317
A. Kostianko, “Inertial manifolds for the 3D modified-Leray-α model with periodic boundary conditions”, J. Dynam. Differential Equations, 30:1 (2018), 1–24
C. G. Gal, Ya. Guo, “Inertial manifolds for the hyperviscous Navier-Stokes equations”, J. Differ. Equ., 265:9 (2018), 4335–4374
A. Kostianko, S. Zelik, “Inertial manifolds for 1D reaction-diffusion-advection systems. Part I: Dirichlet and Neumann boundary conditions”, Commun. Pure Appl. Anal, 16:6 (2017), 2357–2376
J. C. Robinson, J. J. Sanchez-Gabites, “On finite-dimensional global attractors of homeomorphisms”, Bull. London Math. Soc., 48:3 (2016), 483–498
A. V. Romanov, “On the hyperbolicity properties of inertial manifolds of reaction-diffusion equations”, Dyn. Partial Differ. Equ., 13:3 (2016), 263–272
A. Kostianko, S. Zelik, “Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions”, Commun. Pure Appl. Anal., 14:5 (2015), 2069–2094
A. V. Romanov, “A Parabolic Equation with Nonlocal Diffusion without a Smooth Inertial Manifold”, Math. Notes, 96:4 (2014), 548–555
S. Zelik, “Inertial manifolds and finite-dimensional reduction for dissipative PDEs”, Proc. Roy. Soc. Edinburgh Sect. A, 144:6 (2014), 1245–1327
E. Pinto de Moura, J. C. Robinson, “Log-Lipschitz continuity of the vector field on the attractor of certain parabolic equations”, Dyn. Partial Differ. Equ., 11:3 (2014), 211–228
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