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Mat. Sb., 2000, Volume 191, Number 3, Pages 99–112 (Mi msb466)  

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.


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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
\by A.~V.~Romanov
\paper Finite-dimensional limiting dynamics for dissipative parabolic equations
\jour Mat. Sb.
\yr 2000
\vol 191
\issue 3
\pages 99--112
\jour Sb. Math.
\yr 2000
\vol 191
\issue 3
\pages 415--429

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    This publication is cited in the following articles:
    1. A. V. Romanov, “Finite-dimensional dynamics on attractors of non-linear parabolic equations”, Izv. Math., 65:5 (2001), 977–1001  mathnet  crossref  crossref  mathscinet  zmath
    2. Rezounenko, A, “A sufficient condition for the existence of approximate inertial manifolds containing the global attractor”, Comptes Rendus Mathematique, 334:11 (2002), 1015  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    3. Langa, JA, “Fractal dimension of a random invariant set”, Journal de Mathematiques Pures et Appliquees, 85:2 (2006), 269  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    4. A. V. Romanov, “Effective finite parametrization in phase spaces of parabolic equations”, Izv. Math., 70:5 (2006), 1015–1029  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    5. 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  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    6. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    7. J.C.. Robinson, “Attractors and Finite-Dimensional Behaviour in the 2D Navier–Stokes Equations”, ISRN Mathematical Analysis, 2013 (2013), 1  crossref  mathscinet
    8. Zelik S., “Inertial Manifolds and Finite-Dimensional Reduction For Dissipative PDEs”, Proc. R. Soc. Edinb. Sect. A-Math., 144:6 (2014), 1245–1327  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    9. 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  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    10. Anna Kostianko, Sergey Zelik, “Inertial manifolds for the 3D Cahn-Hilliard equations with periodic boundary conditions”, CPAA, 14:5 (2015), 2069  crossref  zmath  scopus  scopus  scopus
    11. Robinson J.C., Sanchez-Gabites J.J., “On finite-dimensional global attractors of homeomorphisms”, Bull. London Math. Soc., 48:3 (2016), 483–498  crossref  mathscinet  zmath  isi  scopus
    12. 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  crossref  mathscinet  zmath  isi  scopus
    13. 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  crossref  mathscinet  zmath  isi  scopus
    14. 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  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
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