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This article is cited in 14 scientific papers (total in 14 papers)
Finite-dimensional limiting dynamics for dissipative parabolic equations
A. V. Romanov
All-Russian Institute for Scientific and Technical Information of Russian Academy of Sciences
Abstract:
For a broad class of semilinear parabolic equations with compact attractor $\mathscr A$ in a Banach space $E$ the problem of a description of the limiting phase dynamics (the dynamics on $\mathscr A$) of a corresponding system of ordinary differential equations in $\mathbb R^N$ is solved in purely topological terms. It is established that the limiting dynamics for a parabolic equation is finite-dimensional if and only if its attractor can be embedded in a sufficiently smooth finite-dimensional submanifold $\mathscr M\subset E$. Some other criteria are obtained for the finite dimensionality of the limiting dynamics:
- a) the vector field of the equation satisfies a Lipschitz condition on $\mathscr A$;
- b) the phase semiflow extends on $\mathscr A$ to a Lipschitz flow;
- c) the attractor $\mathscr A$ has a finite-dimensional Lipschitz Cartesian structure.
It is also shown that the vector field of a semilinear parabolic equation is always Holder on the attractor.
DOI:
https://doi.org/10.4213/sm466
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English version:
Sbornik: Mathematics, 2000, 191:3, 415–429
Bibliographic databases:
   
UDC:
517.95
MSC: Primary 35K55, 58F12; Secondary 47H06, 58G11, 34D45, 35Q30
Received: 15.04.1998
Citation:
A. V. Romanov, “Finite-dimensional limiting dynamics for dissipative parabolic equations”, Mat. Sb., 191:3 (2000), 99–112; Sb. Math., 191:3 (2000), 415–429
Citation in format AMSBIB
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A. V. Romanov, “Finite-dimensional dynamics on attractors of non-linear parabolic equations”, Izv. Math., 65:5 (2001), 977–1001
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Rezounenko, A, “A sufficient condition for the existence of approximate inertial manifolds containing the global attractor”, Comptes Rendus Mathematique, 334:11 (2002), 1015
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Langa, JA, “Fractal dimension of a random invariant set”, Journal de Mathematiques Pures et Appliquees, 85:2 (2006), 269
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A. V. Romanov, “Effective finite parametrization in phase spaces of parabolic
equations”, Izv. Math., 70:5 (2006), 1015–1029
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de Moura E.P., Robinson J.C., Sanchez-Gabites J.J., “Embedding of Global Attractors and their Dynamics”, Proc Amer Math Soc, 139:10 (2011), 3497–3512
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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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J.C.. Robinson, “Attractors and Finite-Dimensional Behaviour in the 2D Navier–Stokes Equations”, ISRN Mathematical Analysis, 2013 (2013), 1
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Zelik S., “Inertial Manifolds and Finite-Dimensional Reduction For Dissipative PDEs”, Proc. R. Soc. Edinb. Sect. A-Math., 144:6 (2014), 1245–1327
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de Moura E.P., Robinson J.C., “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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Anna Kostianko, Sergey Zelik, “Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions”, CPAA, 14:5 (2015), 2069
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Robinson J.C., Sanchez-Gabites J.J., “On finite-dimensional global attractors of homeomorphisms”, Bull. London Math. Soc., 48:3 (2016), 483–498
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Sanchez-Gabites J.J., “Arcs, balls and spheres that cannot be attractors in $\mathbb {R}^3$”, Trans. Am. Math. Soc., 368:5 (2016), 3591–3627
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Kostianko A., Zelik S., “Inertial Manifolds For 1D Reaction-Diffusion-Advection Systems. Part i: Dirichlet and Neumann Boundary Conditions”, Commun. Pure Appl. Anal, 16:6 (2017), 2357–2376
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Kostianko A., Zelik S., “Nertial Manifolds For 1D Reaction-Diffusion-Advection Systems. Part II: Periodic Boundary Conditions”, Commun. Pure Appl. Anal, 17:1 (2018), 285–317
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