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.
 [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
(85g:58058)
 [2]
J.
E. Billotti and J.
P. LaSalle, Dissipative periodic
processes, Bull. Amer. Math. Soc. 77 (1971), 1082–1088. MR 0284682
(44 #1906), http://dx.doi.org/10.1090/S000299041971128793
 [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
(56 #7231)
 [4]
Philip
Brenner, Michel
Crouzeix, and Vidar
Thomée, Singlestep methods for inhomogeneous linear
differential equations in Banach space, RAIRO Anal. Numér.
16 (1982), no. 1, 5–26 (English, with French
summary). MR
648742 (83d:65268)
 [5]
Philippe
G. Ciarlet, The finite element method for elliptic problems,
NorthHolland Publishing Co., AmsterdamNew YorkOxford, 1978. Studies in
Mathematics and its Applications, Vol. 4. MR 0520174
(58 #25001)
 [6]
P.
Constantin, C.
Foias, and R.
Temam, On the large time Galerkin approximation of the
NavierStokes equations, SIAM J. Numer. Anal. 21
(1984), no. 4, 615–634. MR 749361
(85i:65145), http://dx.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
(89c:65102), http://dx.doi.org/10.1090/S00255718198709061763
 [9]
M.
Crouzeix and V.
Thomée, The stability in 𝐿_{𝑝}
and 𝑊¹_{𝑝} of the 𝐿₂projection onto
finite element function spaces, Math. Comp.
48 (1987), no. 178, 521–532. MR 878688
(88f:41016), http://dx.doi.org/10.1090/S00255718198708786882
 [10]
Hiroshi
Fujita and Akira
Mizutani, On the finite element method for parabolic equations. I.
Approximation of holomorphic semigroups, J. Math. Soc. Japan
28 (1976), no. 4, 749–771. MR 0428733
(55 #1753)
 [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
(82b:34059)
 [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
(89b:58115)
 [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
(47 #2581)
 [14]
Jack
K. Hale and Orlando
Lopes, Fixed point theorems and dissipative processes, J.
Differential Equations 13 (1973), 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 #8529, Brown University, Providence, RI, October 1985.
 [16]
HansPeter
Helfrich, Fehlerabschätzungen für das Galerkinverfahren
zur Lösung von Evolutionsgleichungen, Manuscripta Math.
13 (1974), 219–235 (German, with English summary).
MR
0356513 (50 #8983)
 [17]
Daniel
Henry, Geometric theory of semilinear parabolic equations,
Lecture Notes in Mathematics, vol. 840, SpringerVerlag, BerlinNew
York, 1981. MR
610244 (83j:35084)
 [18]
John
G. Heywood and Rolf
Rannacher, Finite element approximation of the nonstationary
NavierStokes problem. II. Stability of solutions and error estimates
uniform in time, SIAM J. Numer. Anal. 23 (1986),
no. 4, 750–777. MR 849281
(88b:65132), http://dx.doi.org/10.1137/0723049
 [19]
John
G. Heywood and Rolf
Rannacher, Finite element approximation of the nonstationary
NavierStokes problem. III. Smoothing property and higher order error
estimates for spatial discretization, SIAM J. Numer. Anal.
25 (1988), no. 3, 489–512. MR 942204
(89k:65114), http://dx.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
(88k:65100), http://dx.doi.org/10.1090/S00255718198709061751
 [21]
Tosio
Kato, Frational powers of dissipative operators. II, J. Math.
Soc. Japan 14 (1962), 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. Ladyženskaja, The dynamical system that is generated by
the NavierStokes 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
(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
14 (1962), 233–241 (French). MR 0152878
(27 #2850)
 [27]
Paul
Massatt, Attractivity properties of 𝛼contractions, J.
Differential Equations 48 (1983), no. 3,
326–333. MR
702423 (85d:47062), http://dx.doi.org/10.1016/00220396(83)900979
 [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
(85f:34045), http://dx.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
(80i:65132), http://dx.doi.org/10.1016/0362546X(78)900019
 [32]
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)
 [33]
Vidar
Thomée, Galerkin finite element methods for parabolic
problems, Lecture Notes in Mathematics, vol. 1054,
SpringerVerlag, Berlin, 1984. MR 744045
(86k:65006)
 [34]
Vidar
Thomée and Lars
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
