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Time dependent center manifold in PDEs. (English) Zbl 07273493
Summary: We consider externally forced equations in an evolution form. Mathematically, these are skew systems driven by a finite dimensional dynamical system. Two very common cases included in our treatment are quasi-periodic forcing and forcing by a stochastic process. We allow that the evolution is a PDE and even that it is not well-posed and that it does not define a flow (not all initial conditions lead to a solution).
We first establish a general abstract theorem which, under suitable (spectral, non-degeneracy, smoothness, etc) assumptions, establishes the existence of a “time-dependent invariant manifold” (TDIM). These manifolds evolve with the forcing. They are such that the original equation is always tangent to a vector field in the manifold. Hence, for initial data in the TDIM, the original equation is equivalent to an ordinary differential equation. This allows us to define families of solutions of the full equation by studying the solutions of a finite dimensional system. Note that this strategy may apply even if the original equation is ill posed and does not admit solutions for arbitrary initial conditions (the TDIM selects initial conditions for which solutions exist). It also allows that the TDIM is infinite dimensional.
Secondly, we construct the center manifold for skew systems driven by the external forcing.
Thirdly, we present concrete applications of the abstract result to the differential equations whose linear operators are exponential trichotomy subject to quasi-periodic perturbations. The use of TDIM allows us to establish the existence of quasi-periodic solutions and to study the effect of resonances.
##### MSC:
 35B42 Inertial manifolds 35B15 Almost and pseudo-almost periodic solutions to PDEs 35R25 Ill-posed problems for PDEs 37L10 Normal forms, center manifold theory, bifurcation theory for infinite-dimensional dissipative dynamical systems 35J60 Nonlinear elliptic equations 47J06 Nonlinear ill-posed problems 37L25 Inertial manifolds and other invariant attracting sets of infinite-dimensional dissipative dynamical systems
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##### References:
 [1] R. Abraham and J. Robbin, Transversal Mappings and Flows, An appendix by Al Kelley, W. A. Benjamin, Inc., New York-Amsterdam, 1967. · Zbl 0171.44404 [2] A. Afendikov and A. Mielke, A spatial center manifold approach to a hydrodynamical problem with O(2) symmetry, in Dynamics, Bifurcation and Symmetry (Cargèse, 1993), vol. 437 of NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Kluwer Acad. Publ., Dordrecht, 1994, 1-10. · Zbl 0809.76027 [3] L. F. A. Arbogast, Du Calcul Des Derivations, Levraut, Strasbourg, 1800, Available freely from Google Books. [4] L. Arnold, Random Dynamical Systems, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 1998. · Zbl 0906.34001 [5] T. Bartsch; J. M. Moix; S. Kawai, Time-dependent transition state theory, Advance in Chemical Physis, 140, 189-238 (2008) [6] J. Bass, Les Fonctions Pseudo-aléatoires, Mémor. Sci. Math., Fasc. CLIII, Gauthier-Villars, Éditeur-Imprimeur-Libraire, Paris, 1962. · Zbl 0106.11705 [7] P. W. Bates, K. Lu and C. Zeng, Existence and persistence of invariant manifolds for semiflows in Banach space, Mem. Amer. Math. Soc., 135 (1998), ⅷ+129pp. · Zbl 1023.37013 [8] M. Berti, KAM theory for partial differential equations, Anal. Theory Appl., 35, 235-267 (2019) · Zbl 1438.37044 [9] P. Boxler, How to construct stochastic center manifolds on the level of vector fields, in Lyapunov Exponents (Oberwolfach, 1990), vol. 1486 of Lecture Notes in Math., Springer, Berlin, 1991,141-158. · Zbl 0741.34023 [10] M. J. Capiński; C. Simó, Computer assisted proof for normally hyperbolic invariant manifolds, Nonlinearity, 25, 1997-2026 (2012) · Zbl 1256.37020 [11] J. Carr, Applications of Centre Manifold Theory, vol. 35 of Applied Mathematical Sciences, Springer-Verlag, New York-Berlin, 1981. · Zbl 0464.58001 [12] N. Chafee; E. F. Infante, A bifurcation problem for a nonlinear partial differential equation of parabolic type, Applicable Anal., 4, 17-37 (1974/75) · Zbl 0296.35046 [13] H. Cheng; R. de la Llave, Stable manifolds to bounded solutions in possibly ill-posed PDEs., J. Differ. Equations, 268, 4830-4899 (2020) · Zbl 1448.35564 [14] H. Cheng and J. Si, Quasi-periodic solutions for the quasi-periodically forced cubic complex Ginzburg-Landau equation on $${\mathbb T}^d$$, J. Math. Phys., 54 (2013), 082702, 27pp. · Zbl 1297.35223 [15] C. Chicone; Y. Latushkin, Center manifolds for infinite-dimensional nonautonomous differential equations, J. Differential Equations, 141, 356-399 (1997) · Zbl 0992.34033 [16] S.-N. Chow, W. Liu and Y. Yi, Center manifolds for invariant sets, J. Differential Equations, 168 (2000), 355-385, Special issue in celebration of Jack K. Hale’s 70th birthday, Part 2 (Atlanta, GA/Lisbon, 1998). · Zbl 0972.34033 [17] S.-N. Chow; W. Liu; Y. Yi, Center manifolds for smooth invariant manifolds, Trans. Amer. Math. Soc., 352, 5179-5211 (2000) · Zbl 0953.34038 [18] D. R. Cox and H. D. Miller, The Theory of Stochastic Processes, John Wiley & Sons, Inc., New York, 1965. · Zbl 0149.12902 [19] S. L. Day, A Rigorous Numerical Method in Infinite Dimensions, ProQuest LLC, Ann Arbor, MI, 2003, Thesis (Ph.D.)-Georgia Institute of Technology. [20] R. de la Llave, Smooth conjugacy and S-R-B measures for uniformly and non-uniformly hyperbolic systems, Comm. Math. Phys., 150, 289-320 (1992) · Zbl 0770.58029 [21] R. de la Llave; J. M. Marco; R. Moriyón, Canonical perturbation theory of Anosov systems and regularity results for the Livšic cohomology equation, Ann. of Math. (2), 123, 537-611 (1986) · Zbl 0603.58016 [22] R. de la Llave; J. D. Mireles James, Connecting orbits for compact infinite dimensional maps: Computer assisted proofs of existence, SIAM J. Appl. Dyn. Syst., 15, 1268-1323 (2016) · Zbl 1343.37078 [23] R. de la Llave; R. Obaya, Regularity of the composition operator in spaces of Hölder functions, Discrete Contin. Dynam. Systems, 5, 157-184 (1999) · Zbl 0956.47029 [24] R. de la Llave, A smooth center manifold theorem which applies to some ill-posed partial differential equations with unbounded nonlinearities, J. Dynam. Differential Equations, 21, 371-415 (2009) · Zbl 1179.35340 [25] R. de la Llave; Y. Sire, An a posteriori kam theorem for whiskered tori in hamiltonian partial differential equations with applications to some ill-posed equations, Arch Rational Mech. Anal., 231, 971-1044 (2019) · Zbl 1407.37108 [26] R. de la Llave; A. Windsor, Livšic theorems for non-commutative groups including diffeomorphism groups and results on the existence of conformal structures for Anosov systems, Ergodic Theory Dynam. Systems, 30, 1055-1100 (2010) · Zbl 1230.37042 [27] J.-P. Eckmann and C. E. Wayne, Propagating fronts and the center manifold theorem, Comm. Math. Phys., 136 (1991), 285-307, http://projecteuclid.org/euclid.cmp/1104202352. · Zbl 0790.34045 [28] G. Faye; A. Scheel, Center manifolds without a phase space, Trans. Amer. Math. Soc., 370, 5843-5885 (2018) · Zbl 1406.37026 [29] J. A. Goldstein, Semigroups of Linear Operators & Applications, Dover Publications, Inc., Mineola, NY, 2017, Second edition of [MR0790497], Including transcriptions of five lectures from the 1989 workshop at Blaubeuren, Germany. · Zbl 1364.47005 [30] J. Hadamard, Sur le module maximum d’une fonction et de ses derives, Bull. Soc. Math. France, 42, 68-72 (1898) [31] J. K. Hale, Ordinary Differential Equations, 2nd edition, Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., 1980. · Zbl 0433.34003 [32] M. Haragus and G. Iooss, Local Bifurcations, Center Manifolds, and Normal Forms in Infinite-Dimensional Dynamical Systems, Universitext, Springer-Verlag London, Ltd., London; EDP Sciences, Les Ulis, 2011. · Zbl 1230.34002 [33] D. Henry, Geometric Theory of Semilinear Parabolic Equations, vol. 840 of Lecture Notes in Mathematics, Springer-Verlag, Berlin-New York, 1981. · Zbl 0456.35001 [34] D. A. Jones; S. Shkoller, Persistence of invariant manifolds for nonlinear PDEs, Stud. Appl. Math., 102, 27-67 (1999) · Zbl 1002.37037 [35] J.-L. Journé, A regularity lemma for functions of several variables, Rev. Mat. Iberoamericana, 4, 187-193 (1988) · Zbl 0699.58008 [36] K. Kirchgässner; J. Scheurle, On the bounded solutions of a semilinear elliptic equation in a strip, J. Differential Equations, 32, 119-148 (1979) · Zbl 0372.35034 [37] S. G. Krantz, Lipschitz spaces, smoothness of functions, and approximation theory, Exposition. Math., 1, 193-260 (1983) · Zbl 0518.46018 [38] S. Krantz, Function Theory of Several Complex Variables, AMS Chelsea Publishing, Providence, RI, 2001, Reprint of the 1992 edition. · Zbl 1087.32001 [39] O. E. Lanford III, Bifurcation of periodic solutions into invariant tori: The work of Ruelle and Takens, in Nonlinear Problems in the Physical Sciences and Biology: P roceedings of a Battelle Summer Institute (eds. I. Stakgold, D. D. Joseph and D. H. Sattinger), Springer-Verlag, Berlin, Lecture Notes in Mathematics, 322 (1973), 159-192. · Zbl 0272.34039 [40] J. Li; K. Lu; P. Bates, Normally hyperbolic invariant manifolds for random dynamical systems: Part Ⅰ-Persistence, Trans. Amer. Math. Soc., 365, 5933-5966 (2013) · Zbl 1291.37093 [41] A. Mielke, Reduction of quasilinear elliptic equations in cylindrical domains with applications, Math. Methods Appl. Sci., 10, 51-66 (1988) · Zbl 0647.35034 [42] A. Mielke, Hamiltonian and Lagrangian Flows on Center Manifolds, vol. 1489 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1991, With applications to elliptic variational problems. · Zbl 0747.58001 [43] A. Mielke, Essential manifolds for an elliptic problem in an infinite strip, J. Differential Equations, 110, 322-355 (1994) · Zbl 0814.35028 [44] A. Pazy, Semigroups of operators in Banach spaces, in Equadiff 82 (Würzburg, 1982), vol. 1017 of Lecture Notes in Math., Springer, Berlin, 1983,508-524. [45] O. Perron, Über Stabilität und asymptotisches Verhalten der Integrale von Differentialgleichungssystemen, Math. Z., 29, 129-160 (1929) · JFM 54.0456.04 [46] M. Reed and B. Simon, Methods of Modern Mathematical Physics. Ⅳ. Analysis of Operators, Academic Press [Harcourt Brace Jovanovich, Publishers], New York-London, 1978. · Zbl 0401.47001 [47] G. R. Sell and Y. You, Dynamics of Evolutionary Equations, vol. 143 of Applied Mathematical Sciences, Springer-Verlag, New York, 2002. · Zbl 1254.37002 [48] J. Sijbrand, Properties of center manifolds, Trans. Amer. Math. Soc., 289, 431-469 (1985) · Zbl 0577.34039 [49] M. E. Taylor, Partial Differential Equations Ⅲ. Nonlinear Equations, vol. 117 of Applied Mathematical Sciences, 2nd edition, Springer, New York, 2011. [50] N. G. van Kampen, Stochastic Processes in Physics and Chemistry, vol. 888 of Lecture Notes in Mathematics, North-Holland Publishing Co., Amsterdam-New York, 1981. · Zbl 0511.60038 [51] A. Vanderbauwhede and G. Iooss, Center manifold theory in infinite dimensions, in Dynamics Reported: Expositions in Dynamical Systems, vol. 1 of Dynam. Report. Expositions Dynam. Systems (N.S.), Springer, Berlin, 1992,125-163. · Zbl 0751.58025 [52] L. Zhang; R. de la Llave, Transition state theory with quasi-periodic forcing, Commun. Nonlinear Sci. Numer. Simul., 62, 229-243 (2018)
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