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Arbitrarily accurate approximate inertial manifolds of fixed dimension
23 June 1997 PHYSICS
LETTERS
A
Physics Letters A 230 ( 1997) 301-304
ELSEVIER
Arbitrarily accurate approximate inertial manifolds of fixed dimension James C. Robinson Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge CB3 9EW. UK
Received 24 September 1996; accepted for publication Communicated by A.P. Fordy
21 March 1997
Abstract By employing an embedding result due to Mat%, and its recent strengthening due to Foias and Olson it is shbwn that a global attractor with finite fractal (box counting) dimension d lies within an arbitrarily small neighbourhood of a smooth graph over the space spanned by the first [ [ 2d + 1] ] Fourier-Galerkin modes. The proof is, however, nonconstructive. @ 1997 Elsevier Science B.V.
1. Introduction
For many dissipative partial differential equations, the long term dynamics is determined by the behaviour on a much smaller set, the global attractor A. This is the maximal compact invariant set that attracts all bounded sets, see Refs. [ 1] and [ 21 for many examples. In many cases it can also be shown that the attractor has finite fractal (box counting) dimension [ 31. Such systems are commonly analysed in the form of an abstract evolution equation on a Hilbert space H, du/dt + Au = f(u)
,
where A is a positive self-adjoint operator with compact inverse and f a nonlinear term with some Lipschitz smoothness properties. This general semilinear parabolic equation covers many examples and forms the main subject of the monograph [4]. In what follows, the norm in H will be denoted by / . 1 and the inner product by (., .). H has a natural basis consisting of Fourier modes, the eigenfunctions w,, 037%9601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved PII SO3759601 (97)00245-4
of the operator A ( Aw, = A,w, ) By orgering the eigenvalues in increasing order (A,+ I > A,,) , one can define the projection P, onto the first n Fourier modes,
j=l
and its orthogonal complement Qn = I - Pn. For some systems one can show that thelglobal attractor is contained in a manifold, given as t e graph of h a C’ function over the finite-dimensional s ace P,H, P which is positively invariant under the flow ahd attracts all the orbits exponentially. This is termed bn inertial manifold [ 51, and has been found to exists for many (a equations. Nevertheless, the spectral gap co,b dltlon . . sufficient condition for existence, but one r&quired by all current proofs for equations of a general ~form. see, for example, Ref. [ 61) prevents this result eing verified for some important examples, includi !)g the twodimensional Navier-Stokes equations. Eve+ when an inertial manifold exists, the abstract existe ce results do not give an explicit form for the manif “,Id, show-
302
J.C. Robi~.~o~/~hy.~;cs Letters A 230 (I9971 301-304
ing only that there exists a function I$ : PNH 4 QNH whose graph {P + W)
: P E f’,vH)
is an inertial manifold. To remedy this, the idea of an “approximate inertial manifold” was introduced in Refs. [ 7,8]. In this case, the global attractor is shown to lie in some small neighbourhood of an explicitly given manifolds the width of this neigh~urhood shrinking, preferably rapidly, as the dimension of the manifold increases. In Ref. [ 71 a nonconstructive proof was given that there exists a family of approximate inertial manifolds MN of exponential order, i.e. such that
where MN is an N-dimensional manifold (and C, k, LYare all positive constants). Each manifold is the graph of a Lipschitz function 4~ : PNH -+ QNH with Lipschitz constant at most 1. (An explicit scheme for constructing such a family with exponential order was given later in Ref. [9] .) In this paper it is shown that there exist arbitrarily accurate approximate inertial manifolds at any Jired dimension, provided the dimension is higher than (approximately) twice the dimension of the global attractor. To compensate for this improvement, an increasingly accurate sequence of manifolds Me, with dist(M,,d)
< E,
has diverging Lipschitz constant as E --+ 0. This result, independent of any conditions, explains two potentially puzzling results that arise in the theory of approximate inertial manifolds (see, for example, Ref. [ lo] ) . In systems for which no true inertial manifold can be shown to exist, it is often possible to show the existence of an approximate inertial manifold; and for systems which do have an exact inertial manifold, approximate inertial manifolds generally have much smaller dimension.
2. Emb~ding
finite-dimensio~l
projections of a set X with a finite fractal dimension dF( X) = d onto subspaces of dimension [ [ 2d + 1 ] ] are injective ( [ [k] ] denotes the smallest integer greater than or equal to k). The best version of the theorem is due to Foias and Olson [ 111, who prove Holder continuity for the inverse of the projection. Theorem I (MaTi&,Foias-Olsonj. If H is a Hilbert space and X is a compact subset of H with dF (X) < d then given an orthogonal projection PO of rank [ [Zd + 1] ] , for every 6 > 0 there exists an orthogonal projection P = P(S) of the same rank such that IIP - Poll G 6 and P is injective on X. Furthermore there exist constants (depending on 6) C > 0 and 0 < 0 < 1 such that IP-‘x - P-‘yl
< CJX - yf+
x,y E PX.
It is of course immediate from this theorem that one can write the attractor as a graph over a subspuce of PH (Pd, in fact), so that x={~-i-&7r)
:rE
Pd},
= ( P-‘T) - 7~ is Holder continuous where +(v) for rr E Pd. If one extends 4 to a function defined over the whole of PH, using the extension theorem in Ref. [ 121 which guarantees that the extension is also Holder and bounded by its m~imum value on PA, it is clear that the global attractor A lies within a Holder continuous manifold, XCG+{~~+~(T)
:TEPH}.
(1)
This is reminiscent of the results of Refs. [ 131 and [ 141, which show that the attractor for a reactiondiffusion equation is a smooth graph. Note, however, that the graph of 4 is not (Lipschitz) smooth nor is it positively invariant under the dynamics, unlike an inertial manifold.
sets in 4%” 3. Approximate
Although the global attractor is finite dimensional, it lies within the infinite-dimensional Hilbert space H. However, in Ref. [3] Mai% showed that “most”
inertial manifolds
Direct application of Theorem 1 to the global attractor leads to (1). Indeed, Theorem 1 implies that
such a statement is possible with P arbitrarily close to the distinguished projection PN. However, in every case the particular projection P is unknown. Turning this around, one can show that (1) holds “arbitrarily accurately” when P is replaced by the Fourier projection PN. This observation relates the embedding result of Foias and Olson to the theory of approximate inertial manifolds. It will be shown that there exists a function @ : PwH --+ QNH, such that the global attractor lies within an arbitrarily small neighbourhood of the graph of +%,
Tlzeoretrr 2. Let A be a compact set with dF(d) < d,and N 2 [ [ 2d + I ] I. Then given e > 0 there exists a Lipschitz function il, : PNH --+ QNH, with graph G,++ as above, such that
independent. They thus form a basis for PH, and so P is a bijection from PNH onto PH. Similarly, PN maps PH bijectively onto PNH. Since P : PNH --+ PH is onto, the set
is exactly the set ~~ defined above ( I). For an approximate inertial manifold in the standard form, take
and observe that fi is also Holder continuoujs,
Now,for
p E P,vH,
IPp+4(Pp) Pruo$ Since A is compact it is bounded, and so A c ,12, z (1~1 6 p} (which implies that 11~/;11~6 p). Now, given an E > 0, use theorem 1 to choose a projection P which is injective on A and which satisfies
l/P .- PNII6 c/6p.
(2)
Note that it follows that i/Q -QN// G e/6p also. Given such a P, there is a co~esponding (f, : PH --+ QH as in ( I ), such that A c $&. The aim is now to construct a function J/ : PNH + QNH whose graph is close to 64. Now, P : P,vH + PH is injective, for otherwise there exist p, J7 E PNH withp # p and P(p - ~7) = 0. i.e. a tt’ t PNH with w # 0 and Pw = 0. But then
lP,bv -- fivl = lP,“wl = IWI, contradicting (2). P also maps PNH onto PH, since {w, ]Ei, the basis for PNH, is mapped into {Pw~)$!,; and if
52
,t,Pwt
= 0,
i=i
then the linearity of P and its injectivity on PNH imply that A, = 0 for all j; and hence the {aVj} are linearly
-P-Ili(p)I
6 IPP -
PI + IQNCP-
+ l&PP,
- QNrp(PP)l
PP)I
G E(lPl + lpi + I4(PP)l)
G E/2.
for u E oP > d. It follows from the bijectiive properties of P and PNH that the Hausdorff distance between the two sets G& 0 fi, and $& fl&, is, bounded by e/2, and hence dist( A, C&,) < c/2. Thus A lies within an e/2 neighbourhood of a graph over PNH, for arbitrarily small E. The function @ can be smoothed to produce a Lipschitz (rather than only continuous) manifold - if I,/I~is a smoothing that lies within ~12 of $I then clearly
and this an approximate quired form. El
inertial
manifold
of the re-
Note that if the global attractor is knowar to be a bounded subset of L)( A’), then one can simply replace H in the above argument with D(A’) and obtain a Lipschitz function into QNH n D( A’).
J.C. Robinson/Physics
304
Letrem A 230 (1997) 30/-304
4. Conclusion In general, the results of Mafit! and Foias-Olson assert that a finite-dimensional global attractor will lie within the graph of a Holder function over some finitedimensional space. If one selects the distinguished space P,vH spanned by the first N Fourier modes, the global attractor lies within an arbitrarily small neighbourhood of a Lipschitz graph over PNH, where N is comparable to the dimension of the attractor. As discussed in the introduction, this explains two phenomena from the theory of approximate inertial manifolds.
Acknowledgement Many thanks to Trinity College, Cambridge, their continue ~nanci~ support and generosity.
for
References I I I J.K. Hale, Asymptotic
behaviour
of dissipative
systems,
in:
Math. Surveys and Monographs, Vol. 25. Am. Math. Sot. (Providence, 1988). I21 R. Temam, Infinite dimensional dynamical systems in mechanics and physics, AMS 68 (Springer, Berlin, 1988). [3] R. Matie, Springer lecture notes in mathematics. Vol. 898 (Springer, Berlin, 1981) p. 230. 141 D. Henry, Geometric theory of semilinear parabolic equations, in: Lecture notes in mathematics, Vol. 840 (Springer, Berlin, 198 I ) 15 1 C. Foias, G.R. Sell and R. Temam, C. R. Acad. Sci. 1 301 (1985) 139. 161 S.-N. Chow, K. Lu and G.R. Sell, J. Math. Anal. Appl. 169 (1992) 283. [ 7 I C. Foias. 0. Manley and R. Temam, Math. Mod. Num. Anal. 22 (1988) 93. f8] C. Foias, G.R. Sell and E.S. Titi, J. Dyn. Diff. Eq. I (1989) 199. [ 9 J A. Debussche and R. Temam, J. Math. Put-es Appl. 73 ( 1994) 489. [IOJ M. Marion, J. Dyn. Diff. Eq. I (1989) 245. f 1 I j C. Foias and E. Olson, Indiana Univ. Math. J. 45 ( 1996) 603. [ 121 E.M. Stein, Singular integrals and differentiability properties of functions (Princeton Univ. Press, Princeton, 1970). [ 13 ] M.S. Jolly. J. Diff. Eq. 78, 220. [ 141 J? Brunovsky, J. Dyn. Diff. Eq. 2 (1990) 293.
LETTERS
A
Physics Letters A 230 ( 1997) 301-304
ELSEVIER
Arbitrarily accurate approximate inertial manifolds of fixed dimension James C. Robinson Department of Applied Mathematics and Theoretical Physics, Silver Street, Cambridge CB3 9EW. UK
Received 24 September 1996; accepted for publication Communicated by A.P. Fordy
21 March 1997
Abstract By employing an embedding result due to Mat%, and its recent strengthening due to Foias and Olson it is shbwn that a global attractor with finite fractal (box counting) dimension d lies within an arbitrarily small neighbourhood of a smooth graph over the space spanned by the first [ [ 2d + 1] ] Fourier-Galerkin modes. The proof is, however, nonconstructive. @ 1997 Elsevier Science B.V.
1. Introduction
For many dissipative partial differential equations, the long term dynamics is determined by the behaviour on a much smaller set, the global attractor A. This is the maximal compact invariant set that attracts all bounded sets, see Refs. [ 1] and [ 21 for many examples. In many cases it can also be shown that the attractor has finite fractal (box counting) dimension [ 31. Such systems are commonly analysed in the form of an abstract evolution equation on a Hilbert space H, du/dt + Au = f(u)
,
where A is a positive self-adjoint operator with compact inverse and f a nonlinear term with some Lipschitz smoothness properties. This general semilinear parabolic equation covers many examples and forms the main subject of the monograph [4]. In what follows, the norm in H will be denoted by / . 1 and the inner product by (., .). H has a natural basis consisting of Fourier modes, the eigenfunctions w,, 037%9601/97/$17.00 @ 1997 Elsevier Science B.V. All rights reserved PII SO3759601 (97)00245-4
of the operator A ( Aw, = A,w, ) By orgering the eigenvalues in increasing order (A,+ I > A,,) , one can define the projection P, onto the first n Fourier modes,
j=l
and its orthogonal complement Qn = I - Pn. For some systems one can show that thelglobal attractor is contained in a manifold, given as t e graph of h a C’ function over the finite-dimensional s ace P,H, P which is positively invariant under the flow ahd attracts all the orbits exponentially. This is termed bn inertial manifold [ 51, and has been found to exists for many (a equations. Nevertheless, the spectral gap co,b dltlon . . sufficient condition for existence, but one r&quired by all current proofs for equations of a general ~form. see, for example, Ref. [ 61) prevents this result eing verified for some important examples, includi !)g the twodimensional Navier-Stokes equations. Eve+ when an inertial manifold exists, the abstract existe ce results do not give an explicit form for the manif “,Id, show-
302
J.C. Robi~.~o~/~hy.~;cs Letters A 230 (I9971 301-304
ing only that there exists a function I$ : PNH 4 QNH whose graph {P + W)
: P E f’,vH)
is an inertial manifold. To remedy this, the idea of an “approximate inertial manifold” was introduced in Refs. [ 7,8]. In this case, the global attractor is shown to lie in some small neighbourhood of an explicitly given manifolds the width of this neigh~urhood shrinking, preferably rapidly, as the dimension of the manifold increases. In Ref. [ 71 a nonconstructive proof was given that there exists a family of approximate inertial manifolds MN of exponential order, i.e. such that
where MN is an N-dimensional manifold (and C, k, LYare all positive constants). Each manifold is the graph of a Lipschitz function 4~ : PNH -+ QNH with Lipschitz constant at most 1. (An explicit scheme for constructing such a family with exponential order was given later in Ref. [9] .) In this paper it is shown that there exist arbitrarily accurate approximate inertial manifolds at any Jired dimension, provided the dimension is higher than (approximately) twice the dimension of the global attractor. To compensate for this improvement, an increasingly accurate sequence of manifolds Me, with dist(M,,d)
< E,
has diverging Lipschitz constant as E --+ 0. This result, independent of any conditions, explains two potentially puzzling results that arise in the theory of approximate inertial manifolds (see, for example, Ref. [ lo] ) . In systems for which no true inertial manifold can be shown to exist, it is often possible to show the existence of an approximate inertial manifold; and for systems which do have an exact inertial manifold, approximate inertial manifolds generally have much smaller dimension.
2. Emb~ding
finite-dimensio~l
projections of a set X with a finite fractal dimension dF( X) = d onto subspaces of dimension [ [ 2d + 1 ] ] are injective ( [ [k] ] denotes the smallest integer greater than or equal to k). The best version of the theorem is due to Foias and Olson [ 111, who prove Holder continuity for the inverse of the projection. Theorem I (MaTi&,Foias-Olsonj. If H is a Hilbert space and X is a compact subset of H with dF (X) < d then given an orthogonal projection PO of rank [ [Zd + 1] ] , for every 6 > 0 there exists an orthogonal projection P = P(S) of the same rank such that IIP - Poll G 6 and P is injective on X. Furthermore there exist constants (depending on 6) C > 0 and 0 < 0 < 1 such that IP-‘x - P-‘yl
< CJX - yf+
x,y E PX.
It is of course immediate from this theorem that one can write the attractor as a graph over a subspuce of PH (Pd, in fact), so that x={~-i-&7r)
:rE
Pd},
= ( P-‘T) - 7~ is Holder continuous where +(v) for rr E Pd. If one extends 4 to a function defined over the whole of PH, using the extension theorem in Ref. [ 121 which guarantees that the extension is also Holder and bounded by its m~imum value on PA, it is clear that the global attractor A lies within a Holder continuous manifold, XCG+{~~+~(T)
:TEPH}.
(1)
This is reminiscent of the results of Refs. [ 131 and [ 141, which show that the attractor for a reactiondiffusion equation is a smooth graph. Note, however, that the graph of 4 is not (Lipschitz) smooth nor is it positively invariant under the dynamics, unlike an inertial manifold.
sets in 4%” 3. Approximate
Although the global attractor is finite dimensional, it lies within the infinite-dimensional Hilbert space H. However, in Ref. [3] Mai% showed that “most”
inertial manifolds
Direct application of Theorem 1 to the global attractor leads to (1). Indeed, Theorem 1 implies that
such a statement is possible with P arbitrarily close to the distinguished projection PN. However, in every case the particular projection P is unknown. Turning this around, one can show that (1) holds “arbitrarily accurately” when P is replaced by the Fourier projection PN. This observation relates the embedding result of Foias and Olson to the theory of approximate inertial manifolds. It will be shown that there exists a function @ : PwH --+ QNH, such that the global attractor lies within an arbitrarily small neighbourhood of the graph of +%,
Tlzeoretrr 2. Let A be a compact set with dF(d) < d,and N 2 [ [ 2d + I ] I. Then given e > 0 there exists a Lipschitz function il, : PNH --+ QNH, with graph G,++ as above, such that
independent. They thus form a basis for PH, and so P is a bijection from PNH onto PH. Similarly, PN maps PH bijectively onto PNH. Since P : PNH --+ PH is onto, the set
is exactly the set ~~ defined above ( I). For an approximate inertial manifold in the standard form, take
and observe that fi is also Holder continuoujs,
Now,for
p E P,vH,
IPp+4(Pp) Pruo$ Since A is compact it is bounded, and so A c ,12, z (1~1 6 p} (which implies that 11~/;11~6 p). Now, given an E > 0, use theorem 1 to choose a projection P which is injective on A and which satisfies
l/P .- PNII6 c/6p.
(2)
Note that it follows that i/Q -QN// G e/6p also. Given such a P, there is a co~esponding (f, : PH --+ QH as in ( I ), such that A c $&. The aim is now to construct a function J/ : PNH + QNH whose graph is close to 64. Now, P : P,vH + PH is injective, for otherwise there exist p, J7 E PNH withp # p and P(p - ~7) = 0. i.e. a tt’ t PNH with w # 0 and Pw = 0. But then
lP,bv -- fivl = lP,“wl = IWI, contradicting (2). P also maps PNH onto PH, since {w, ]Ei, the basis for PNH, is mapped into {Pw~)$!,; and if
52
,t,Pwt
= 0,
i=i
then the linearity of P and its injectivity on PNH imply that A, = 0 for all j; and hence the {aVj} are linearly
-P-Ili(p)I
6 IPP -
PI + IQNCP-
+ l&PP,
- QNrp(PP)l
PP)I
G E(lPl + lpi + I4(PP)l)
G E/2.
for u E oP > d. It follows from the bijectiive properties of P and PNH that the Hausdorff distance between the two sets G& 0 fi, and $& fl&, is, bounded by e/2, and hence dist( A, C&,) < c/2. Thus A lies within an e/2 neighbourhood of a graph over PNH, for arbitrarily small E. The function @ can be smoothed to produce a Lipschitz (rather than only continuous) manifold - if I,/I~is a smoothing that lies within ~12 of $I then clearly
and this an approximate quired form. El
inertial
manifold
of the re-
Note that if the global attractor is knowar to be a bounded subset of L)( A’), then one can simply replace H in the above argument with D(A’) and obtain a Lipschitz function into QNH n D( A’).
J.C. Robinson/Physics
304
Letrem A 230 (1997) 30/-304
4. Conclusion In general, the results of Mafit! and Foias-Olson assert that a finite-dimensional global attractor will lie within the graph of a Holder function over some finitedimensional space. If one selects the distinguished space P,vH spanned by the first N Fourier modes, the global attractor lies within an arbitrarily small neighbourhood of a Lipschitz graph over PNH, where N is comparable to the dimension of the attractor. As discussed in the introduction, this explains two phenomena from the theory of approximate inertial manifolds.
Acknowledgement Many thanks to Trinity College, Cambridge, their continue ~nanci~ support and generosity.
for
References I I I J.K. Hale, Asymptotic
behaviour
of dissipative
systems,
in:
Math. Surveys and Monographs, Vol. 25. Am. Math. Sot. (Providence, 1988). I21 R. Temam, Infinite dimensional dynamical systems in mechanics and physics, AMS 68 (Springer, Berlin, 1988). [3] R. Matie, Springer lecture notes in mathematics. Vol. 898 (Springer, Berlin, 1981) p. 230. 141 D. Henry, Geometric theory of semilinear parabolic equations, in: Lecture notes in mathematics, Vol. 840 (Springer, Berlin, 198 I ) 15 1 C. Foias, G.R. Sell and R. Temam, C. R. Acad. Sci. 1 301 (1985) 139. 161 S.-N. Chow, K. Lu and G.R. Sell, J. Math. Anal. Appl. 169 (1992) 283. [ 7 I C. Foias. 0. Manley and R. Temam, Math. Mod. Num. Anal. 22 (1988) 93. f8] C. Foias, G.R. Sell and E.S. Titi, J. Dyn. Diff. Eq. I (1989) 199. [ 9 J A. Debussche and R. Temam, J. Math. Put-es Appl. 73 ( 1994) 489. [IOJ M. Marion, J. Dyn. Diff. Eq. I (1989) 245. f 1 I j C. Foias and E. Olson, Indiana Univ. Math. J. 45 ( 1996) 603. [ 121 E.M. Stein, Singular integrals and differentiability properties of functions (Princeton Univ. Press, Princeton, 1970). [ 13 ] M.S. Jolly. J. Diff. Eq. 78, 220. [ 141 J? Brunovsky, J. Dyn. Diff. Eq. 2 (1990) 293.