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

Authors:
Jack K. Hale, Xiao-Biao Lin and Geneviève 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

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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 and M. I. Vishik,*Regular attractors of semigroups and evolution equations*, J. Math. Pures Appl. (9)**62**(1983), no. 4, 441–491 (1984). MR**735932****[2]**J. E. Billotti and J. P. LaSalle,*Dissipative periodic processes*, Bull. Amer. Math. Soc.**77**(1971), 1082–1088. MR**0284682**, https://doi.org/10.1090/S0002-9904-1971-12879-3**[3]**J. H. Bramble, A. H. Schatz, V. Thomée, and L. B. Wahlbin,*Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations*, SIAM J. Numer. Anal.**14**(1977), no. 2, 218–241. MR**0448926**, https://doi.org/10.1137/0714015**[4]**Philip Brenner, Michel Crouzeix, and Vidar Thomée,*Single-step methods for inhomogeneous linear differential equations in Banach space*, RAIRO Anal. Numér.**16**(1982), no. 1, 5–26 (English, with French summary). MR**648742****[5]**Philippe G. Ciarlet,*The finite element method for elliptic problems*, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1978. Studies in Mathematics and its Applications, Vol. 4. MR**0520174****[6]**P. Constantin, C. Foias, and R. Temam,*On the large time Galerkin approximation of the Navier-Stokes equations*, SIAM J. Numer. Anal.**21**(1984), no. 4, 615–634. MR**749361**, https://doi.org/10.1137/0721043**[7]**G. Cooperman, -*Condensing Maps and Dissipative Systems*, Ph.D. Thesis, Brown University, Providence, RI, June 1978.**[8]**Michel Crouzeix and Vidar Thomée,*On the discretization in time of semilinear parabolic equations with nonsmooth initial data*, Math. Comp.**49**(1987), no. 180, 359–377. MR**906176**, https://doi.org/10.1090/S0025-5718-1987-0906176-3**[9]**M. Crouzeix and V. Thomée,*The stability in 𝐿_{𝑝} and 𝑊¹_{𝑝} of the 𝐿₂-projection onto finite element function spaces*, Math. Comp.**48**(1987), no. 178, 521–532. MR**878688**, https://doi.org/10.1090/S0025-5718-1987-0878688-2**[10]**Hiroshi Fujita and Akira Mizutani,*On the finite element method for parabolic equations. I. Approximation of holomorphic semi-groups*, J. Math. Soc. Japan**28**(1976), no. 4, 749–771. MR**0428733**, https://doi.org/10.2969/jmsj/02840749**[11]**Jack K. Hale,*Some recent results on dissipative processes*, Functional differential equations and bifurcation (Proc. Conf., Inst. Ciênc. Mat. São Carlos, Univ. São Paulo, São Carlos, 1979) Lecture Notes in Math., vol. 799, Springer, Berlin, 1980, pp. 152–172. MR**585487****[12]**J. K. Hale,*Asymptotic behaviour and dynamics in infinite dimensions*, Nonlinear differential equations (Granada, 1984) Res. Notes in Math., vol. 132, Pitman, Boston, MA, 1985, pp. 1–42. MR**908897****[13]**J. K. Hale, J. P. LaSalle, and Marshall Slemrod,*Theory of a general class of dissipative processes*, J. Math. Anal. Appl.**39**(1972), 177–191. MR**0314029**, https://doi.org/10.1016/0022-247X(72)90233-8**[14]**Jack K. Hale and Orlando Lopes,*Fixed point theorems and dissipative processes*, J. Differential Equations**13**(1973), 391–402. MR**0333851**, https://doi.org/10.1016/0022-0396(73)90025-9**[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]**Hans-Peter Helfrich,*Fehlerabschätzungen für das Galerkinverfahren zur Lösung von Evolutionsgleichungen*, Manuscripta Math.**13**(1974), 219–235 (German, with English summary). MR**0356513**, https://doi.org/10.1007/BF01168227**[17]**Daniel Henry,*Geometric theory of semilinear parabolic equations*, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin-New York, 1981. MR**610244****[18]**John G. Heywood and Rolf Rannacher,*Finite element approximation of the nonstationary Navier-Stokes problem. II. Stability of solutions and error estimates uniform in time*, SIAM J. Numer. Anal.**23**(1986), no. 4, 750–777. MR**849281**, https://doi.org/10.1137/0723049**[19]**John G. Heywood and Rolf Rannacher,*Finite element approximation of the nonstationary Navier-Stokes problem. III. Smoothing property and higher order error estimates for spatial discretization*, SIAM J. Numer. Anal.**25**(1988), no. 3, 489–512. MR**942204**, https://doi.org/10.1137/0725032**[20]**Claes Johnson, Stig Larsson, Vidar Thomée, and Lars B. Wahlbin,*Error estimates for spatially discrete approximations of semilinear parabolic equations with nonsmooth initial data*, Math. Comp.**49**(1987), no. 180, 331–357. MR**906175**, https://doi.org/10.1090/S0025-5718-1987-0906175-1**[21]**Tosio Kato,*Frational powers of dissipative operators. II*, J. Math. Soc. Japan**14**(1962), 242–248. MR**0151868**, https://doi.org/10.2969/jmsj/01420242**[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. Ladyženskaja,*The dynamical system that is generated by the Navier-Stokes equations*, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI)**27**(1972), 91–115 (Russian). Boundary value problems of mathematical physics and related questions in the theory of functions, 6. MR**0328378****[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**14**(1962), 233–241 (French). MR**0152878**, https://doi.org/10.2969/jmsj/01420233**[27]**Paul Massatt,*Attractivity properties of 𝛼-contractions*, J. Differential Equations**48**(1983), no. 3, 326–333. MR**702423**, https://doi.org/10.1016/0022-0396(83)90097-9**[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.**34**(1983), no. 3, 310–321 (English, with German summary). MR**712275**, https://doi.org/10.1007/BF00944852**[31]**K. 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**, https://doi.org/10.1016/0362-546X(78)90001-9**[32]**Roger Temam,*Navier-Stokes equations and nonlinear functional analysis*, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 41, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1983. MR**764933****[33]**Vidar Thomée,*Galerkin finite element methods for parabolic problems*, Lecture Notes in Mathematics, vol. 1054, Springer-Verlag, Berlin, 1984. MR**744045****[34]**Vidar Thomée and Lars Wahlbin,*On Galerkin methods in semilinear parabolic problems*, SIAM J. Numer. Anal.**12**(1975), 378–389. MR**0395269**, https://doi.org/10.1137/0712030

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Additional Information

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

Article copyright:
© Copyright 1988
American Mathematical Society