Upper semicontinuity of attractors for approximations of semigroups and partial differential equations
Authors:
Jack K. Hale, XiaoBiao Lin and Geneviève Raugel
Journal:
Math. Comp. 50 (1988), 89123
MSC:
Primary 47H20; Secondary 35K55, 35K70, 65N30
MathSciNet review:
917820
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Abstract: Suppose a given evolutionary equation has a compact attractor and the evolutionary equation is approximated by a finitedimensional 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.
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G. Raugel, "Hilbert space estimates in the approximation of inhomogeneous parabolic problems by single step methods," submitted to Math. Comp.
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Rutkowski, Approximate solutions of eigenvalue problems with
reproducing nonlinearities, Z. Angew. Math. Phys. 34
(1983), no. 3, 310–321 (English, with German summary). MR 712275
(85f:34045), http://dx.doi.org/10.1007/BF00944852
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Schmitt, R.
C. Thompson, and W.
Walter, Existence of solutions of a nonlinear boundary value
problem via the method of lines, Nonlinear Anal. 2
(1978), no. 5, 519–535. MR 512149
(80i:65132), http://dx.doi.org/10.1016/0362546X(78)900019
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Roger
Temam, NavierStokes equations and nonlinear functional
analysis, CBMSNSF Regional Conference Series in Applied Mathematics,
vol. 41, Society for Industrial and Applied Mathematics (SIAM),
Philadelphia, PA, 1983. MR 764933
(86f:35152)
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problems, Lecture Notes in Mathematics, vol. 1054,
SpringerVerlag, Berlin, 1984. MR 744045
(86k:65006)
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Wahlbin, On Galerkin methods in semilinear parabolic problems,
SIAM J. Numer. Anal. 12 (1975), 378–389. MR 0395269
(52 #16066)
 [1]
 A. V. Babin & M. I. Vishik, "Regular attractors of semigroups and evolution equations," J. Math. Pures Appl., v. 62, 1983, pp. 442491. MR 735932 (85g:58058)
 [2]
 J. E. Billotti & J. P. LaSalle, "Periodic dissipative processes," Bull. Amer. Math. Soc., v. 77, 1971, pp. 10821088. 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. 218241. 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. 526. MR 648742 (83d:65268)
 [5]
 P. G. Ciarlet The Finite Element Method for Elliptic Problems, NorthHolland, Amsterdam, 1978. MR 0520174 (58:25001)
 [6]
 P. Constantin, C. Foias & R. Témam, "On the large time Galerkin approximation of the NavierStokes equations," SIAM J. Numer. Anal., v. 21, 1984, pp. 615634. 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. 359377. 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. 521532. 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. 749771. 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, SpringerVerlag, Berlin and New York, 1980, pp. 152172. 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. 142. 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. 177191. MR 0314029 (47:2581)
 [14]
 J. K. Hale & O. Lopes, "Fixed point theorems and dissipative processes," J. Differential Equations, v. 13, 1973, pp. 391402. 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 #8529, Brown University, Providence, RI, October 1985.
 [16]
 H. P. Helfrich, "Fehlerabschätzungen für das Galerkinverfahren zur Lösung von Evolutionsgleichungen," Manuscripta Math., v. 13, 1974, pp. 219235. MR 0356513 (50:8983)
 [17]
 D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Math., vol. 840, SpringerVerlag, Berlin and New York, 1981. MR 610244 (83j:35084)
 [18]
 J. G. Heywood & R. Rannacher, "Finite element approximation of the nonstationary NavierStokes problem, Part II: Stability of solutions and error estimates uniform in time," SIAM J. Numer. Anal., v. 23, 1986, pp. 750777. MR 849281 (88b:65132)
 [19]
 J. G. Heywood & R. Rannacher, "Finite element approximation of the nonstationary NavierStokes 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. 331357. MR 906175 (88k:65100)
 [21]
 T. Kato, "Fractional powers of dissipative operators II," J. Math. Soc. Japan, v. 14, 1962, pp. 242248. 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 NavierStokes equations," Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), v. 27, 1972, pp. 91115. MR 0328378 (48:6720)
 [24]
 O. A. Ladyzhenskaya, "Dynamical system generated by the NavierStokes equations," Soviet Phys. Dokl., v. 17, 1973, pp. 647649.
 [25]
 X. B. Lin & G. Raugel, "Approximation of attractors of MorseSmale 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. 234241. MR 0152878 (27:2850)
 [27]
 P. Massatt, "Attractivity properties of contractions," J. Differential Equations, v. 48, 1983, pp. 326333. 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. 310321. 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. 519535. MR 512149 (80i:65132)
 [32]
 R. Témam, NavierStokes Equations and Nonlinear Functional Analysis, CBMSNSF, 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, SpringerVerlag, 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. 378389. MR 0395269 (52:16066)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S0025571819880917820X
PII:
S 00255718(1988)0917820X
Article copyright:
© Copyright 1988 American Mathematical Society
