Upper semicontinuity of attractors for approximations of semigroups and partial differential equations

Authors:
Jack K. Hale, Xiao-Biao Lin and GeneviÃ¨ve Raugel

Journal:
Math. Comp. **50** (1988), 89-123

MSC:
Primary 47H20; Secondary 35K55, 35K70, 65N30

DOI:
https://doi.org/10.1090/S0025-5718-1988-0917820-X

MathSciNet review:
917820

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Suppose a given evolutionary equation has a compact attractor and the evolutionary equation is approximated by a finite-dimensional system. Conditions are given to ensure the approximate system has a compact attractor which converges to the original one as the approximation is refined. Applications are given to parabolic and hyperbolic partial differential equations.

**[1]**A. V. Babin & M. I. Vishik, "Regular attractors of semigroups and evolution equations,"*J. Math. Pures Appl.*, v. 62, 1983, pp. 442-491. MR**735932 (85g:58058)****[2]**J. E. Billotti & J. P. LaSalle, "Periodic dissipative processes,"*Bull. Amer. Math. Soc.*, v. 77, 1971, pp. 1082-1088. MR**0284682 (44:1906)****[3]**J. H. Bramble, A. H. Schatz, V. Thomée & L. B. Wahlbin, "Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations,"*SIAM J. Numer. Anal.*, v. 14, 1977, pp. 218-241. MR**0448926 (56:7231)****[4]**P. Brenner, M. Crouzeix & V. Thomée, "Single step methods for inhomogeneous linear differential equations in Banach spaces,"*RAIRO Anal. Numér.*, v. 16, 1982, pp. 5-26. MR**648742 (83d:65268)****[5]**P. G. Ciarlet*The Finite Element Method for Elliptic Problems*, North-Holland, Amsterdam, 1978. MR**0520174 (58:25001)****[6]**P. Constantin, C. Foias & R. Témam, "On the large time Galerkin approximation of the Navier-Stokes equations,"*SIAM J. Numer. Anal.*, v. 21, 1984, pp. 615-634. MR**749361 (85i:65145)****[7]**G. Cooperman, -*Condensing Maps and Dissipative Systems*, Ph.D. Thesis, Brown University, Providence, RI, June 1978.**[8]**M. Crouzeix & V. Thomée, "On the discretization in time of semilinear parabolic equations with nonsmooth initial data,"*Math. Comp.*, v. 49, 1987, pp. 359-377. MR**906176 (89c:65102)****[9]**M. Crouzeix & V. Thomée, "The stability in and of the -projection onto finite element function spaces,"*Math. Comp.*, v. 48, 1987, pp. 521-532. MR**878688 (88f:41016)****[10]**H. Fujita & A. Mizutani, "On the finite element method for parabolic equations. I. Approximation of holomorphic semigroups,"*J. Math. Soc. Japan*, v. 28, 1976, pp. 749-771. MR**0428733 (55:1753)****[11]**J. K. Hale, "Some recent results on dissipative processes," in*Functional Differential Equations and Bifurcations*(Ize, ed.), Lecture Notes in Math., vol. 799, Springer-Verlag, Berlin and New York, 1980, pp. 152-172. MR**585487 (82b:34059)****[12]**J. K. Hale, "Asymptotic behavior and dynamics in infinite dimensions," in*Research Notes in Math.*, vol. 132, Pitman, 1985, pp. 1-42. MR**908897 (89b:58115)****[13]**J. K. Hale, J. P. LaSalle, & M. Slemrod "Theory of a general class of dissipative processes,"*J. Math. Anal. Appl.*, v. 39, 1972, pp. 177-191. MR**0314029 (47:2581)****[14]**J. K. Hale & O. Lopes, "Fixed point theorems and dissipative processes,"*J. Differential Equations*, v. 13, 1973, pp. 391-402. MR**0333851 (48:12173)****[15]**J. K. Hale, X. B. Lin & G. Raugel,*Upper Semicontinuity of Attractors for Approximations of Semigroups and Partial Differential Equations*, LCDS report #85-29, Brown University, Providence, RI, October 1985.**[16]**H. P. Helfrich, "Fehlerabschätzungen für das Galerkinverfahren zur Lösung von Evolutionsgleichungen,"*Manuscripta Math.*, v. 13, 1974, pp. 219-235. MR**0356513 (50:8983)****[17]**D. Henry,*Geometric Theory of Semilinear Parabolic Equations*, Lecture Notes in Math., vol. 840, Springer-Verlag, Berlin and New York, 1981. MR**610244 (83j:35084)****[18]**J. G. Heywood & R. Rannacher, "Finite element approximation of the nonstationary Navier-Stokes problem, Part II: Stability of solutions and error estimates uniform in time,"*SIAM J. Numer. Anal.*, v. 23, 1986, pp. 750-777. MR**849281 (88b:65132)****[19]**J. G. Heywood & R. Rannacher, "Finite element approximation of the nonstationary Navier-Stokes problem, Part III: Smoothing property and higher order error estimates for partial discretization." (Preprint.) MR**942204 (89k:65114)****[20]**C. Johnson, S. Larsson, V. Thomée & L. B. Wahlbin "Error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data,"*Math. Comp.*, v. 49, 1987, pp. 331-357. MR**906175 (88k:65100)****[21]**T. Kato, "Fractional powers of dissipative operators II,"*J. Math. Soc. Japan*, v. 14, 1962, pp. 242-248. MR**0151868 (27:1851)****[22]**S. N. S. Khalsa, "Finite element approximation of a reaction diffusion equation. Part I: Application of a topological technique to the analysis of asymptotic behavior of the semidiscrete approximation." (Preprint.)**[23]**O. A. Ladyzhenskaya, "A dynamical system generated by the Navier-Stokes equations,"*Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov.*(*LOMI*), v. 27, 1972, pp. 91-115. MR**0328378 (48:6720)****[24]**O. A. Ladyzhenskaya, "Dynamical system generated by the Navier-Stokes equations,"*Soviet Phys. Dokl.*, v. 17, 1973, pp. 647-649.**[25]**X. B. Lin & G. Raugel, "Approximation of attractors of Morse-Smale systems given by parabolic equations." (In preparation.)**[26]**J. L. Lions, "Espaces d'interpolation et domaines de puissances fractionnaires d'opérateurs,"*J. Math. Soc. Japan*, v. 14, 1962, pp. 234-241. MR**0152878 (27:2850)****[27]**P. Massatt, "Attractivity properties of -contractions,"*J. Differential Equations*, v. 48, 1983, pp. 326-333. MR**702423 (85d:47062)****[28]**X. Mora, "Comparing the phase portrait of a nonlinear parabolic equation with that of its Galerkin approximations." (Preprint).**[29]**G. Raugel, "Hilbert space estimates in the approximation of inhomogeneous parabolic problems by single step methods," submitted to*Math. Comp.***[30]**P. Rutkowski, "Approximate solutions of eigenvalue problems with reproducing nonlinearities,"*Z. Angew. Math. Phys.*, v. 34, 1983, pp. 310-321. MR**712275 (85f:34045)****[31]**K. Schmitt, R. C. Thompson & W. Walter, "Existence of solutions of a nonlinear boundary value problem via the method of lines,"*Nonlinear Anal.*, v. 2, 1978, pp. 519-535. MR**512149 (80i:65132)****[32]**R. Témam,*Navier-Stokes Equations and Nonlinear Functional Analysis*, CBMS-NSF, vol. 41, SIAM, Philadelphia, Pa., 1983. MR**764933 (86f:35152)****[33]**V. Thomée,*Galerkin Finite Element Methods for Parabolic Problems*, Lecture Notes in Math., vol. 1054, Springer-Verlag, Berlin and New York, 1984. MR**744045 (86k:65006)****[34]**V. Thomée & L. Wahlbin, "On Galerkin methods in semilinear parabolic problems,"*SIAM J. Numer. Anal.*, v. 12, 1975, pp. 378-389. MR**0395269 (52:16066)**

Retrieve articles in *Mathematics of Computation*
with MSC:
47H20,
35K55,
35K70,
65N30

Retrieve articles in all journals with MSC: 47H20, 35K55, 35K70, 65N30

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1988-0917820-X

Article copyright:
© Copyright 1988
American Mathematical Society