Abstract
The existence of a nonautonomous approximate inertial manifold is shown for problems of the formu′ + Au + N(t,u)=0, in whichA is a self-adjoint operator with compact resolvent in a Hilbert spaceH. The operatorN(t, u) = G(u) + F(t, u) is nonlinear withG a monotone gradient that is locally Lipschitz fromD(A 1/2) intoH, andF:ℝ+×H→H a Lipschitz perturbation that is Hölder continuous int. Weak solutions are shown to be uniformly locally Hölder continuous intoD(A) with equicontinuity in families of solutions with ¦u(0)¦ ⩽ r.A priori estimates of ¦Au(t)¦ are also verified and used in a skew-product flow to show there is a global attractor whose component elements form a equicontinuous family of solutions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Brezis, H. (1973).Operateurs Maximaux Monotones et Semi-Groupes de Contractions Dans les Espaces de Hilbert, North-Holland publaddr Amsterdam.
Dafermos, C. (1974). Almost periodic processes and almost periodic solutions of evolution equations,Dynamical Systems, proc. of University of Florida Int. Symp. (eds. Bednarek and Cesari), Academic Press, New York, pp. 43–57.
Dieudonné, J. (1969).Treatise on Analysis, Vol. I, Academic Press, New York.
Foias, C., Manley, O., and Temam, R. (1987). Sur l'interaction des petits et grands tourbillons dans des écoulements turbulents,C.R. Acad. Sci. Paris, Sér. I Math. 305, 497–500.
Gill, T., and Zachary, W. (1992). Dimensionality of invariant sets for non-autonomous processes,SIAM J. Math. Anal. 23, (5), 1204–1229.
Hale, J. (1988). Asymptotic Behavior of Dissipative Systems, Math. Surveys and Monographs 25, Amer. Math. Soc., Providence, Rhode Island.
Haraux, A. (1988). Attractors of asymptotically compact processes and applications to non-linear partial differential equations,Comm. P.D.E. 13, 1383–1414.
Jolly, M., Kevrekidis, I., and Titi, E. (1990). Approximate inertial manifolds for the Kuramoto-Sivashinsky equation: analysis and computations,Physica D44, 38–60.
Jones, D., and Titi, E. (1994). A remark on quasi-stationary approximate inertial manifolds for the Navier-Stokes equations,SIAM J. Math. Anal. 25, (3), 894–914.
Marion, M. (1989). Approximate inertial manifolds for reaction-diffusion equations in higher space dimensions,J. Dynamics and Differential Equations 1, (3), 245–267.
Pazy, A. (1983).Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York.
Raugel, G., and Sell, G. (1993a). Navier-Stokes equations on thin 3D domains. I: Global attractors and global regularity of solutions,J. Amer. Math. Soc. 6, (3), 503–568.
Raugel, G., and Sell, G. (1993b). Navier-Stokes equations on thin 3D domains. III: Global and local attractors,Turbulence in Fluid Flows: A Dynamical Systems Approach, The IMA volumes in Mathematics and its Applications, Vol.55 (eds. Sell-Foias-Temam), Springer-Verlag, New York.
Raugel, G., and Sell, G. (1994). Navier-Stokes equations on thin 3D domains. II: Global regularity of spatially periodic solutions,Nonlinear Partial Differential Equations and Their Applications, Pitman Research Notes in Math. Ser. No. 299, Collège de France Seminar, Vol. XI, Longman Scientific & Technical, Essex, England, pp. 205–247.
Sell, G. (1971).Topological Dynamics and Ordinary Differential Equations, Van Nostrand Reinhold, London.
Sell, G. (1993). References on Dynamical Systems, Updated version of AHPCRC Preprint 92–129.
Smiley, M. (1993). Global attractors and approximate inertial manifolds for nonautonomous dissipative equations,Applicable Analysis 50, 217–241.
Temam, R. (1988).Infinite Dimensional Dynamical Systems in Mechanics and Physics, Springer Verlag, New York.
Titi, E. (1990). Approximate inertial manifolds to the Navier-Stokes equations,J. Math. Anal. Appl. 149, 540–557.
Vainberg, M. (1973).Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations, John Wiley & Sons, New York.
Vishik, I. M. (1992).Asymptotic Behaviour of Solutions of Evolutionary Equations, Cambridge University Press, Cambridge, England.
Vuillermot, P. (1991). Almost-periodic attractors for a class of nonautonomous reaction-diffusion equations on ℝN. I. Global stabilization processes,J. Diff. Eqns. 94, (2), 228–253.
Vuillermot, P. (1992). Almost-periodic attractors for a class of nonautonomous reaction-diffusion equations on ℝN. II. Codimension-one stable manifolds,Diff. and Int. Eqns. 5, (3), 693–720.
Zaidman, S. (1985).Almost-Periodic Functions in Abstract Spaces, Research Notes in Math. Vol. 126, Pitman, Boston, Massachusetts.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Smiley, M.W. Regularity and asymptotic behavior of solutions of nonautonomous differential equations. J Dyn Diff Equat 7, 237–262 (1995). https://doi.org/10.1007/BF02219357
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02219357