- Email: [email protected]
The rate of decay for solutions in nonlinear dissipative partial differential equations
22 December
1997
PHYSICS
Physics
Letters
A 236 (1997)
LETTERS
A
415-424
The rate of decay for solutions in nonlinear dissipative partial differential equations C.J. Woolcock l, M.V. Bartuccelli 2 Department Received
of Mathematical
21 November
1996; revised
and Computing
Sciences,
University
of Surrey,
Guildford
manuscriptreceived16 October 1997;acceptedfor Communicatedby A.P. Fordy
GU2 SXH, UK
publication
3 November
1997
Abstract
We show,applyingthe methodusedby Dyer and Edmunds[Proc. LondonMath. Sot. 18 (1968) 1691.for the NavierStokesequations,that the rate of decay for certaindissipativepartial differential equationscan be boundedbelow by an exponential.In particularwe showthat this holdstrue for a generaliseddiffusionequationclosely linked to the complex Ginzburg-Landauequation[M.V. Battuccelli et al., Phys. Ser.,in press]and that a similarmethodcan be appliedto the Kuramoto-Sivashinsky equationandthe Burgersequation.Wealsodiscussthe implicationsof theseresultsfor the estimates of the dissipativelength scalederivedfor dissipativepartial differentialequationsusingladdermethods.@ 1997Elsevier ScienceB.V. Keywords;
Decay
rates; Nonlinear;
Dissipative;
Partial differential
equations
1. Introduction Dyer and Edmunds in their papers [ 1,2], based on the work done by Ogawa [ 31, showed that the rate of decay of solutions of the Navier-Stokes equation is bounded below by an exponential. In these papers they showed that under the assumptionthat the m-norm of Vu was bounded and finite then a lower bound could be found for (d log lu( t) 12)/dt and hence through integration that lu( t) I2 is bounded below by an exponential. In this Letter we will show that by adapting this method to other equations, the rate of decay of the solutions of the chosen equation is bounded below by an exponential. These results are very important in a number of cases, (see, for example, Refs. [4,5] ) . Moreover, they are also necessary to obtain the time average of the dissipative length scale which arises naturally from ladder methods [ 6-91, which is our main reason for studying this area. The dissipative length scale is a measureof the smallest spatial structure of the solutions below which the dynamical behaviour of the solutions is smoothed out by dissipation. Deriving a bound for this quantity is necessaryfor understandingthe spatio-temporalpatterns I E-mail: 2 E-mail:
[email protected]. [email protected].
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Science B.V. All rights reserved.
C.J. Woolcock, M.V Bartuccelli/Physics Letters A 236 (1997) 415-424
416
that arise in the solutions of dissipative partial differential equations. It is also important if we are to create an accurate numerical schemewhich can capture all the dynamics of the solutions. Ladder methods give regularity and global existence for the solutions of the equation and a Co3 attractor. Moreover, bounds on the time-averaged dissipative length scale present in the dynamical flow can be obtained using the time average of the ladder. In Refs. [6-91 a ladder is created from a set of functionals which involve the norms of the solutions of the dissipative nonlinear equation. The rungs of the ladder are of the form + F (lower powers of HN) .
(1)
For clarity we shall define H,, = so ]V”U~*~.X, a,, = d( H,)/dt and HO= /, ]u]* dx = ]u];, which are the functionals used to create the ladder for the Kuramoto-Sivashinsky equation. Variations on these quantities have been used to create ladders for other dissipative PDEs; however, the resulting ladders are similar in structure. We can see that if an upper bound can be found for the bottom rung of the ladder (HO) then we can bound the behaviour of the higher norms. These functionals (H,,) can also be related to the dissipative length scale. From the theory of global attractors in dissipative PDEs we have that the number of degreesof freedom (or relevant modes) N is related to the minimum scale 1 by the formula N = (L/l)d, d being the spatial dimension and Ld the volume. The number of relevant modes can be obtained through a Fourier expansion of the solutions giving Iu~‘,/~u[; < ~1~~. By applying the Gagliardo and Nirenberg inequality to the term on the left-hand side of the inequality we can connect the functionals in the laddersto the dissipative length scale, so that Zmdz ( H,/Ho)~/*~. So we have an estimate for the length scaleswhich can be related to the ladders created for H,,. However, this estimate is time dependentand it is very hard to follow pointwise through time. Therefore, we take its time average obtaining (I-‘) z (( Hn/Ho) ‘/2n), w here the time average (f) is defined as 1 r sup f(T) f-cc2 u&n f s
(f) = lim
dr.
0
If we now look back at the rungs of the ladder ( 1) we can see that by taking time averages, the first term on the right-hand side can be made equivalent to our definition of the time average of the dissipative length scales. However, to find a bound for this estimate of the time average we have to prove that (fin/H,,) is bounded. It can be seenthat if the solutions do not decay faster than exponentially then from the definition of time average, (fin/H,) is bounded. Therefore, to have a well-defined dissipative length scale, some way of estimating lower bounds for the rate of decay of solutions is of crucial importance. The material below is organised in the following way. Section 2 outlines the proof of bounded decay for solutions of the generaliseddiffusion equation. This equation was originally studied (for the case q = 1) as a model for a population with a more general diffusion mechanismthan Fickian diffusion. Sections 3 and 4 contain the proof of lower bounded decay for solutionsof the Kuramoto-Sivashinsky equation and the Burgers equation. In Section 5 we outline the conclusions that can be drawn from the application of this method, and its implications for dissipative length scalesderived using ladder methods. 2. Generalised diffusion model We begin by considering the equation ur = -c&l
- pv*u + yVWf
+ Au - 6l2,
where u = U(X, t) , r > 0 and x E [0, Lid, d is the number of spatial dimensions.
(2)
C.J. Woolcock,
h4.V Bartuccelli/Physics
L&ten
A 236 (1997)
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417
The parameters a, y, A,S, q are all real and positive. p can be of either sign, but for spatially structured solutions to occur we require p > 0. As these are the solutions which are of the most interest to us we take ,6 > 0, but the method can equally be applied to p < 0. From Ref. [ 1 l] we have regularity and global existence of the solutions of this equation. In the case q = 1, this model has been studied in the context of population dynamics by Cohen and Murray [ lo]. In structure it closely resembles the Ginzburg-Laudau systems and it has been shown [ 111 that the resulting system closely parallels those systems. For convenience we adopt the following notation: (f(t) , g(t)) = s f( x, t)g( x, t) dx, If(t) I2 = (f(t) , f(t) ) , , xd), where d is the spatial dimension, and where the integrals are calculated on the set [ 0, L] d. x=(x1,x*,... Theorem 1. For u E 0 suppose we have that B(t)
= IUI,
< co,
If(t)
= IVul,
< co,
lu(t)
= p*lt,
< co,
where u is a solution of the generalised diffusion model and we take our equation on periodic boundary conditions in d spatial dimensions,i.e. u( 0, t) = u( L, t) , Vt 2 0 , where L is the cell length. (Through ladder methodswe can show that under the conditions for which the ladder is bounded then the quantities O(t), ZT(t) and P(t) are also bounded and finite; however, for clarity we shall simply assumethat these quantities are bounded.) Then there is a positive constant M such that for all t > 0 Iul* 2 lu(O)]*exp (--2art provided cr -pi*
- M( 1 + t)t’-“P),
2 0, T= IV*U(O)]~]U(O)]-*,
(3)
i* = (L/~T)~
and p > 1.
Proof Assuming that ]u( t) I # 0, let us define Nu = yV2u2qi’ - 6u2 = u, + av4u + pv*u - Au. We can clearly seethat dlog (ul* = 12(u, dt lu12
24)
= -2alu]-*(V*U,
v*u) + 2p]u]-*(Vu,Vu)
+ 2A + 214-*(u,
Nu).
The second and third terms on the right-hand side are positive definite and are therefore > 0. We also know that yV(
u2q+’) = (2q+ 1)2qyzP-‘(vu)*
+ (2q+ l)yu*qv*u.
Substituting this back into (u, Nu), we have (u,Nu) =2q(2q+l)y = -(2q + 1)~
s
J
u*q(Vu)*dx+(2q+l)y
u2q(Vu)2dx - S
I
u*q+‘V*udx-6
s
u3dx
s
u3dx.
Using the Cauchy-Schwarz and Young inequalities on the first term of the r.h.s., we have
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Bartuccelli/Physics
Letters
A 236 (1997)
415-424
Hence 2(u7Nu) lu12
z -(2q+
l)yJu1z-4
- (2q + l>~jVc4l4, - Sluloo.
Therefore, dlog (u(* > -2a(L4-2)V*u~2 dt
- (2q+
l>@+4(t>
- (2q+
l)rI14(t)
-SO(t).
(4)
Looking at the first term, -2r~~u~-~~V*u~*, IUi4( - 2(V4 u,u,)(u~2+2~v2u~2(u,u,)) = j--q
lu14% ($y) = 2a(
v4u,
)ul* - 2/?(
V4u)
V3u) lu12 - 2h(V2u,
v3u,
- 2a(V2U~*V2U~* + 2pIv2u121vu(*
+ 2h(u(*lV*u[2
v%)Ju)*
- 2(
3 2(a - p&*)p4u~*)u~*
- 2pI!3v*u~9772u~*
+ 2/3(V%~97u~2
Nu) 11112
+ 2lV%\2( u, Nu).
The A terms cancel out and by Poincare we have -IV3u12 > -E2)V4u12, lu14-$ (P)
v4u,
- 2(V4u,Nu)~u~2
where I!,* = (L/27r)2,
hence
- 2(a - p&*)~v*u~*~v*u~*
+ 2(u, V4u) (u, Nu).
Using Poincare again the two p terms cancel out and by rearranging the variables, we have 11414; (pq
3 2(CY - pI3\u12
-2(a-pF)
v4u -
*
N” 2((Y - PC)
2 + 2(a- pL2) N” 2( CX- pL*> > 4((u - pI3*
u,v4u(
2((Y-@) - 4((u - pL+*
lN42l42
(u,Nu)2
*
Provided q >, 1 and LY- PI!,~ > 0, we have
(5)
pg3+2(alpt2)~. Now from definition, INu12 = 1 (2q(2q + l)y~*~-’ = 4qy2q + 1)2y2 - 2S2q(2q + l>r
J
(0~)~
+ (2q+
l)y~*~V*u
~~~-*(Vu)~dx+4q(2q+
J
u2q+‘(Vu)2dx
1)“~”
-2S(2q+
1)~ J
+(2q+1)*~/u4~(V2u)*dx+~*/u4dx. But through integration by parts
J
~~q-‘(Vu)~V*udx
= -3
4q-1 J
u(4q-2)(
Vu)4 dx,
- SU*)~ dx.
J
u4q-’ ( Vu)*V2u
u2q+2V2udx
dx
C.J.
Woolcock,
ML!
Bartuccelli/Physics
Letters
A 236
(1997)
415-424
419
and s
U(2q+2)V2,dx
= -(24+
2)
s
U2q+1(VU)2dx,
hence we have -/Nu/~
> -32q+
- (2q+ Applying $
l)2~2/U4q-2(VU)4dx-46(2q+
1)2y2
u4q( V2u)* dx - S*
I
the Cauchy-Schwarz (T)
u4 dx.
s
and Young inequalities to -INu12 and substituting
3 -3(a4pL2)
- 4;z+&
l)y/~2q+‘(Vu)2dx
y-qo8q-4(t)
(2q+
1)2y2(084-s(t) +ns(t))
+ %@‘4(t))-
this into (5) gives us
- ~;24;&3(@4q-2(r)
+P(t))
it52 02(t). 2( Ly- p.Lq
Integrating this gives
> -Iv2u(o)~2(u(o)I-2-
-Iv2U(t)l*lU(t)l-*
3(a ypL2)
@I+
l)2rij(@8q-8W
+ IT8(s>) ds
0
6(2q+ 1) (a-pL2)~
t s 0
-
a2 qa
_
@,2)
J
(2q+ 1)2 4(a -pEz)y2
(~q-2W+f14WW~-
’
(O8q-4(s) +9’(s))
ds
0
t
J02(s)ds. 0
So if we substitute this back into (4) we have d10;jui2 2 --2&T-
(2q+ l)@+(t)
- (2q+ 1)7174(t) -6@(t) t
2%7
-
3(cY- #m)
(2q+ l>‘y*
s
(@8q-8(s) +Z?(s))ds-
0
1)2cr 2 t (O8q-4(s) +ly4(s))ds2(a - pL*) IY s (2q+
0
26(2q+ 1)cyy (@4q-2(s) +n”(s))ds (a - PL2) s 0
t (a _as* pL2)
I
02(s> ds,
0
where T = IV*U(O)~~]U(O)]-*. Following Ref. [ 11, we assumethat O(t) E Lp(0, 00) for somep, where n < p < cc, and hence
420
CJ. Woolcock,
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Bartuccelli/Physics
Letters
for some constant M. A similar result holds true for n(t) dlog 1uj2 2 -2ar dt -
- (2q + 1) yo4+
Cl> - (2q+
A 236 (1997)
415-424
and !P( t), hence, we have
l)rzP(r)
-6@(t)
Integrating again gives us log 11112 2 log /u(O) 12- 2cYrt - M4(q)yt’-‘lP
(a _olpi2)(Mdq)y2+ M:!(q)$+
- M&-“P M3(s)s2)
t2-llp.
Hence we have (u12 2 lu(0) 12exp (-20~3 provided LY- j3e2 2 0 Now let us consider exist a ta such that lu( it must also hold for t assumption. Therefore,
- M( 1 + t)t’-‘lP).
(6)
M is a constant, and p > 1 as required. our original assumption that ]u(t) 1 # 0. Following Ref. [ 11, we assume that there to) I = 0 then it is clearly true that the theorem holds for 0 < t < ta and by continuity = to. However, from our conclusions (6), we have lu( to) ] # 0, which contradicts our we have that lu( t) I # 0 for all t > 0, and our proof is concluded.
3. Kuramoto-Sivashinsky
equation
We shall now apply the method to the Kuramoto-Sivashinsky
equation. The equation is defined as
ut + YV4U + v2u + u.vu = 0,
(7)
where v is the dissipative parameter. Unlike the Navier-Stokes equation the Kuramoto-Sivashinsky (K-S) equation contains a fourth-order term, but the two equations are similar, as the nonlinear term is of the same structure. Hence the method used by Dyer and Edmunds [l] can be applied to the K-S equation to get similar results. Through ladder methods we can show that the solutions of this equation have regularity and existence. Theorem 2. For u E 0 suppose we have that II(t)
= sup [Vu] < co, XEJJ
where u is a solution of the K-S equation, u is periodic with period L > 0 in x = (XI, x2,. . . xd), and d is the number of spatial dimensions. Then provided Y > e2, e2 = ( L/~T)~, and we assume that ]V*u(O) I < cc then there is a positive constant p depending only on n such that for all t > 0
where A = (v2u(o),v2u(o))l~(o)l-2
and p is a constant such that p > 2.
Proot We initially assume that lu( t) ) # 0, but we can apply the proof outlined in the previous section to show that lu( t) 1 # 0 for all r 2 0.
C.J. Woolcock,
M.V
Bariuccelli/Physics
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A 236 (1997)
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421
We now define, Nu = u.Vu = -ut - vV4u - V2u. It can easily be seen that dWuW12 dt
_
1 2(u u ) = -2b(t)I-2((u, 14t)12 ’ t
By applying integration (U,V2U) = -(Vu,VU),
d log k(t) I2 dt
Nu) + v(u,V4u)
by parts and the periodic boundary conditions therefore -v(V2U,V2U)
=2/u(~)l-2(-WW
+ (U,V2U)). we have (u, V~U) = (vzu, v?-~), and
+ (VU,VU)).
However, the final term on the r.h.s. (2/u(t) lp2( VU, VU)) is positive definite and -(u, Nu) Iu( t) l-2 > -@(t). Therefore, d10p2
> -2v~u~-2(v%,v%)
- 2IP(t).
Considering the r.h.s. term, -~vIu/-~(V~U, IuI”$
V2u). If we consider the quantity
( -(v;;i2v2u)))
we have l”14$ (-‘“t:;“‘“’
) = -2(V4U,Ut)
IU12 +
2(U, Ut)
(V2U,
V2U)
= ~v~V~U~~~U~~ - 2~V3U~2~u~2 + 2(V4u, NU)lU12- ~v~V~U~~~V~U~~ + 2~VU~2~V2U~2 - 2(U, Nu) (u, V4u). By applying PoincarC’sinequality in a similar way to the method used in the generaliseddiffusion model, we have 2
lM‘$ (
> 2( v - &2) IL42 v4u +
- 2(v!i2> l~121Nu12+2cv t
N” 2( v - P)
- 2(v - L2)
u, v4u + K
Nu 2( v - L2)
2
>I
L2J[(wW12.
Hence provided v - E2 > 0,
1~1”;( -“;;ip2’))
> -2(v ! L2)~uI~INu~~.
But jNu12= (Nu,NU) = Ju2(VU)2dx. d z
1
-(V2u,V2u)
Id2
Therefore,
> 3 -2(v
- P)
L12(t).
If we now integrate this we get
- (V2U(t),v2U(t))lU(t)l-2 2 -(v2u(o),v2u(o))lu(o)~-2-
l
2( v - L2)
J
Z12(s) ds.
0
(9)
422
C.J. Woolcock,
M.V
Bartuccelli/Physics
If we now substitute (9) into our formula for (d/dt)
d’“glu(f)l > -zv~ dt
_
’
’
v - L2
Letters
A 236 (1997)
415-424
log lu( t) ( we get
J’ n2(s) & - 2n2(t), 0
where A = lV*u(O) 12/~(O> le2. As in the previous theorem, by applying Holder’s with 2 < p < 00, we have
J
n2(s)
ds 6
inequality and assuming that 17 E LP(0, 00) for some p
Mt1-2’p,
0
for some M, p > 2, and so we have
d’wlu(t)l
> -2vA-
dt
’
Mv
+-2/p -2n2(t).
(v - I?)
Integrating this gives $-2/P
log JU( t) I > log lU(0) I - 2vAt -
-
2M2t’
-2/P
hence we have
as required. For exactly the same reasons as in Section 2, lu( t) ) # 0 for t > 0. 4. Burgers
equation
Similar results can be obtained for the Burgers equation. For the Burgers equation, we have
(‘0)
ut - vv2u + u.vu = 0, where tr is the dissipative parameter. Theorem 3. As before, if we set n(t)
= sup IVUl < cQ, en
where u is a solution of the Burgers equation and is periodic. Then we have lu(t)l
> Ju(0)lexp(-2vht-~(l+t)t2-2’p)a
pro05 Initially
we assume that lu( t) ( # 0 and define Nu = LOU = -ut + vV2u. So we have
d’oglu(t)12 = dt
-21u(t)l-2((u,Nu)
+v(Vu,Vu))
= -2n*(t)
-2~+(t)(-~(vu,vu).
C.J. Woolcock,
We have (d/df)
(VU, VU) = -2(u,,
M.V
BartuccelliIPhysics
V’u).
Letters
A 236 (1997)
415-424
423
If we use this in the calculation of
we can use a method similar to the method used for the Kuromoto-Sivashinsky to Poincare’s inequality, to obtain
But from the definition we know that -INI
equation, but without
resorting
> --IuI~I~~( t), therefore, we have
Thus for any t 2 0 f* - 274~(t>~~~~u(t)l-~
3 -2vll~(O)~~~~u(O)I-~
- J Lr2(s)ds.
(11)
0
If we use ( 11) in (d/dt)
dlog lu(t)l dt
log lu( r) I we get f
> -2vA
- 2&t)
-
s
H2( s) ds,
0
where A = ~~u(O)~~~~U(O)I-*. By following the same method as used in the previous theorem we have /u(f)1 2 lu(O)lexp(-2vAt
- p(l+
t)t2-2/P),
as required. For exactly the same reasons as in Section 2, lu( t) I f 0 for t > 0.
5. Conclusions We have shown that it is possible to prove that the rate of decay for solutions of several nonlinear partial differential equations is bounded below by an exponential. Furthermore through a simple application of Poincare inequality we can see that if [u(f)1 is bounded below by an exponential then so is IV”ul. This is an important result for resolving the assumptions necessary for obtaining the time average of the dissipative length scale, using ladder methods [ 6-9 I. As we showed in the introduction the estimate for the length scales which can be related to the ladders hinges on whether we can show that (fi”/Hn) is bounded. Since
from the definition of the time average, we can see that this depended on the assumption that the rate of decay by the solutions is not faster than exponential. Since we have shown that the rate of the decay is not faster than the exponential this is no longer a problem and bounds can be found for the time average of
424
C.J. Woolcock,
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A 236 (1997)
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the dissipative length scale. Furthermore, the method used throughout this Letter, based on the work done by Dyer and Edmunds in their paper [l] and by Ogawa [ 31, can be extended to cover many dissipative partial differential equations for bounding the solutions of these equations from below.
Acknowledgement We would like to thank the unknown
referee who has helped us to much improve this Letter.
References [I] [2] [3] [ 41 IS] [6] [7] [ 81 191 [lo] [ 11]
R.H. Dyer, D.E. Edmunds, Proc. London Math. Sot. 18 (1968) 78. D.E. Edmunds, Arch. Rat. Mech. Anal. 22 (1966) 15. H. Ogawa, Proc. Am. Math. Sot 16 (1965) 1241. C. Foias, J.C. Saut, Ind. Math. J. 33, No. 3 (1984). C. Foias, J.C. Saut, Ind. Math. J. Vol 33, No. 6 (1984) M.V. Bartuccelli, C.R. Doering, J.D. Gibbon, S.J.A. Malham, Nonlinearity 6 (1993) 549. M.V. Bartuccelli, J.D. Gibbon, M. Oliver, Physica D 89 (1996) 267. S.A. Gourley, M.V. Bartuccelli, Length scales in solutions of a scalar reaction-diffusion equation with delay. CR. Doering, J.D. Gibbon, Applied Analysis of the Navier-Stokes Equations (Cambridge Univ. Press, Cambridge, 1995). D.S. Cohen, J.D. Murray, J. Math. Biol. 12 (1981) 237. M.V. Bartuccelli, S.A. Gourley, C.J. Woolcock Length scales in Solutions of a Generalized Diffusion Model, Physica Scripta,
in press.
1997
PHYSICS
Physics
Letters
A 236 (1997)
LETTERS
A
415-424
The rate of decay for solutions in nonlinear dissipative partial differential equations C.J. Woolcock l, M.V. Bartuccelli 2 Department Received
of Mathematical
21 November
1996; revised
and Computing
Sciences,
University
of Surrey,
Guildford
manuscriptreceived16 October 1997;acceptedfor Communicatedby A.P. Fordy
GU2 SXH, UK
publication
3 November
1997
Abstract
We show,applyingthe methodusedby Dyer and Edmunds[Proc. LondonMath. Sot. 18 (1968) 1691.for the NavierStokesequations,that the rate of decay for certaindissipativepartial differential equationscan be boundedbelow by an exponential.In particularwe showthat this holdstrue for a generaliseddiffusionequationclosely linked to the complex Ginzburg-Landauequation[M.V. Battuccelli et al., Phys. Ser.,in press]and that a similarmethodcan be appliedto the Kuramoto-Sivashinsky equationandthe Burgersequation.Wealsodiscussthe implicationsof theseresultsfor the estimates of the dissipativelength scalederivedfor dissipativepartial differentialequationsusingladdermethods.@ 1997Elsevier ScienceB.V. Keywords;
Decay
rates; Nonlinear;
Dissipative;
Partial differential
equations
1. Introduction Dyer and Edmunds in their papers [ 1,2], based on the work done by Ogawa [ 31, showed that the rate of decay of solutions of the Navier-Stokes equation is bounded below by an exponential. In these papers they showed that under the assumptionthat the m-norm of Vu was bounded and finite then a lower bound could be found for (d log lu( t) 12)/dt and hence through integration that lu( t) I2 is bounded below by an exponential. In this Letter we will show that by adapting this method to other equations, the rate of decay of the solutions of the chosen equation is bounded below by an exponential. These results are very important in a number of cases, (see, for example, Refs. [4,5] ) . Moreover, they are also necessary to obtain the time average of the dissipative length scale which arises naturally from ladder methods [ 6-91, which is our main reason for studying this area. The dissipative length scale is a measureof the smallest spatial structure of the solutions below which the dynamical behaviour of the solutions is smoothed out by dissipation. Deriving a bound for this quantity is necessaryfor understandingthe spatio-temporalpatterns I E-mail: 2 E-mail:
[email protected]. [email protected].
0375-9601/97/$17.00 @ 1997 Elsevier PI1 SO375-9601(97)00859-l
Science B.V. All rights reserved.
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416
that arise in the solutions of dissipative partial differential equations. It is also important if we are to create an accurate numerical schemewhich can capture all the dynamics of the solutions. Ladder methods give regularity and global existence for the solutions of the equation and a Co3 attractor. Moreover, bounds on the time-averaged dissipative length scale present in the dynamical flow can be obtained using the time average of the ladder. In Refs. [6-91 a ladder is created from a set of functionals which involve the norms of the solutions of the dissipative nonlinear equation. The rungs of the ladder are of the form + F (lower powers of HN) .
(1)
For clarity we shall define H,, = so ]V”U~*~.X, a,, = d( H,)/dt and HO= /, ]u]* dx = ]u];, which are the functionals used to create the ladder for the Kuramoto-Sivashinsky equation. Variations on these quantities have been used to create ladders for other dissipative PDEs; however, the resulting ladders are similar in structure. We can see that if an upper bound can be found for the bottom rung of the ladder (HO) then we can bound the behaviour of the higher norms. These functionals (H,,) can also be related to the dissipative length scale. From the theory of global attractors in dissipative PDEs we have that the number of degreesof freedom (or relevant modes) N is related to the minimum scale 1 by the formula N = (L/l)d, d being the spatial dimension and Ld the volume. The number of relevant modes can be obtained through a Fourier expansion of the solutions giving Iu~‘,/~u[; < ~1~~. By applying the Gagliardo and Nirenberg inequality to the term on the left-hand side of the inequality we can connect the functionals in the laddersto the dissipative length scale, so that Zmdz ( H,/Ho)~/*~. So we have an estimate for the length scaleswhich can be related to the ladders created for H,,. However, this estimate is time dependentand it is very hard to follow pointwise through time. Therefore, we take its time average obtaining (I-‘) z (( Hn/Ho) ‘/2n), w here the time average (f) is defined as 1 r sup f(T) f-cc2 u&n f s
(f) = lim
dr.
0
If we now look back at the rungs of the ladder ( 1) we can see that by taking time averages, the first term on the right-hand side can be made equivalent to our definition of the time average of the dissipative length scales. However, to find a bound for this estimate of the time average we have to prove that (fin/H,,) is bounded. It can be seenthat if the solutions do not decay faster than exponentially then from the definition of time average, (fin/H,) is bounded. Therefore, to have a well-defined dissipative length scale, some way of estimating lower bounds for the rate of decay of solutions is of crucial importance. The material below is organised in the following way. Section 2 outlines the proof of bounded decay for solutions of the generaliseddiffusion equation. This equation was originally studied (for the case q = 1) as a model for a population with a more general diffusion mechanismthan Fickian diffusion. Sections 3 and 4 contain the proof of lower bounded decay for solutionsof the Kuramoto-Sivashinsky equation and the Burgers equation. In Section 5 we outline the conclusions that can be drawn from the application of this method, and its implications for dissipative length scalesderived using ladder methods. 2. Generalised diffusion model We begin by considering the equation ur = -c&l
- pv*u + yVWf
+ Au - 6l2,
where u = U(X, t) , r > 0 and x E [0, Lid, d is the number of spatial dimensions.
(2)
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The parameters a, y, A,S, q are all real and positive. p can be of either sign, but for spatially structured solutions to occur we require p > 0. As these are the solutions which are of the most interest to us we take ,6 > 0, but the method can equally be applied to p < 0. From Ref. [ 1 l] we have regularity and global existence of the solutions of this equation. In the case q = 1, this model has been studied in the context of population dynamics by Cohen and Murray [ lo]. In structure it closely resembles the Ginzburg-Laudau systems and it has been shown [ 111 that the resulting system closely parallels those systems. For convenience we adopt the following notation: (f(t) , g(t)) = s f( x, t)g( x, t) dx, If(t) I2 = (f(t) , f(t) ) , , xd), where d is the spatial dimension, and where the integrals are calculated on the set [ 0, L] d. x=(x1,x*,... Theorem 1. For u E 0 suppose we have that B(t)
= IUI,
< co,
If(t)
= IVul,
< co,
lu(t)
= p*lt,
< co,
where u is a solution of the generalised diffusion model and we take our equation on periodic boundary conditions in d spatial dimensions,i.e. u( 0, t) = u( L, t) , Vt 2 0 , where L is the cell length. (Through ladder methodswe can show that under the conditions for which the ladder is bounded then the quantities O(t), ZT(t) and P(t) are also bounded and finite; however, for clarity we shall simply assumethat these quantities are bounded.) Then there is a positive constant M such that for all t > 0 Iul* 2 lu(O)]*exp (--2art provided cr -pi*
- M( 1 + t)t’-“P),
2 0, T= IV*U(O)]~]U(O)]-*,
(3)
i* = (L/~T)~
and p > 1.
Proof Assuming that ]u( t) I # 0, let us define Nu = yV2u2qi’ - 6u2 = u, + av4u + pv*u - Au. We can clearly seethat dlog (ul* = 12(u, dt lu12
24)
= -2alu]-*(V*U,
v*u) + 2p]u]-*(Vu,Vu)
+ 2A + 214-*(u,
Nu).
The second and third terms on the right-hand side are positive definite and are therefore > 0. We also know that yV(
u2q+’) = (2q+ 1)2qyzP-‘(vu)*
+ (2q+ l)yu*qv*u.
Substituting this back into (u, Nu), we have (u,Nu) =2q(2q+l)y = -(2q + 1)~
s
J
u*q(Vu)*dx+(2q+l)y
u2q(Vu)2dx - S
I
u*q+‘V*udx-6
s
u3dx
s
u3dx.
Using the Cauchy-Schwarz and Young inequalities on the first term of the r.h.s., we have
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Hence 2(u7Nu) lu12
z -(2q+
l)yJu1z-4
- (2q + l>~jVc4l4, - Sluloo.
Therefore, dlog (u(* > -2a(L4-2)V*u~2 dt
- (2q+
l>@+4(t>
- (2q+
l)rI14(t)
-SO(t).
(4)
Looking at the first term, -2r~~u~-~~V*u~*, IUi4( - 2(V4 u,u,)(u~2+2~v2u~2(u,u,)) = j--q
lu14% ($y) = 2a(
v4u,
)ul* - 2/?(
V4u)
V3u) lu12 - 2h(V2u,
v3u,
- 2a(V2U~*V2U~* + 2pIv2u121vu(*
+ 2h(u(*lV*u[2
v%)Ju)*
- 2(
3 2(a - p&*)p4u~*)u~*
- 2pI!3v*u~9772u~*
+ 2/3(V%~97u~2
Nu) 11112
+ 2lV%\2( u, Nu).
The A terms cancel out and by Poincare we have -IV3u12 > -E2)V4u12, lu14-$ (P)
v4u,
- 2(V4u,Nu)~u~2
where I!,* = (L/27r)2,
hence
- 2(a - p&*)~v*u~*~v*u~*
+ 2(u, V4u) (u, Nu).
Using Poincare again the two p terms cancel out and by rearranging the variables, we have 11414; (pq
3 2(CY - pI3\u12
-2(a-pF)
v4u -
*
N” 2((Y - PC)
2 + 2(a- pL2) N” 2( CX- pL*> > 4((u - pI3*
u,v4u(
2((Y-@) - 4((u - pL+*
lN42l42
(u,Nu)2
*
Provided q >, 1 and LY- PI!,~ > 0, we have
(5)
pg3+2(alpt2)~. Now from definition, INu12 = 1 (2q(2q + l)y~*~-’ = 4qy2q + 1)2y2 - 2S2q(2q + l>r
J
(0~)~
+ (2q+
l)y~*~V*u
~~~-*(Vu)~dx+4q(2q+
J
u2q+‘(Vu)2dx
1)“~”
-2S(2q+
1)~ J
+(2q+1)*~/u4~(V2u)*dx+~*/u4dx. But through integration by parts
J
~~q-‘(Vu)~V*udx
= -3
4q-1 J
u(4q-2)(
Vu)4 dx,
- SU*)~ dx.
J
u4q-’ ( Vu)*V2u
u2q+2V2udx
dx
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Woolcock,
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(1997)
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419
and s
U(2q+2)V2,dx
= -(24+
2)
s
U2q+1(VU)2dx,
hence we have -/Nu/~
> -32q+
- (2q+ Applying $
l)2~2/U4q-2(VU)4dx-46(2q+
1)2y2
u4q( V2u)* dx - S*
I
the Cauchy-Schwarz (T)
u4 dx.
s
and Young inequalities to -INu12 and substituting
3 -3(a4pL2)
- 4;z+&
l)y/~2q+‘(Vu)2dx
y-qo8q-4(t)
(2q+
1)2y2(084-s(t) +ns(t))
+ %@‘4(t))-
this into (5) gives us
- ~;24;&3(@4q-2(r)
+P(t))
it52 02(t). 2( Ly- p.Lq
Integrating this gives
> -Iv2u(o)~2(u(o)I-2-
-Iv2U(t)l*lU(t)l-*
3(a ypL2)
@I+
l)2rij(@8q-8W
+ IT8(s>) ds
0
6(2q+ 1) (a-pL2)~
t s 0
-
a2 qa
_
@,2)
J
(2q+ 1)2 4(a -pEz)y2
(~q-2W+f14WW~-
’
(O8q-4(s) +9’(s))
ds
0
t
J02(s)ds. 0
So if we substitute this back into (4) we have d10;jui2 2 --2&T-
(2q+ l)@+(t)
- (2q+ 1)7174(t) -6@(t) t
2%7
-
3(cY- #m)
(2q+ l>‘y*
s
(@8q-8(s) +Z?(s))ds-
0
1)2cr 2 t (O8q-4(s) +ly4(s))ds2(a - pL*) IY s (2q+
0
26(2q+ 1)cyy (@4q-2(s) +n”(s))ds (a - PL2) s 0
t (a _as* pL2)
I
02(s> ds,
0
where T = IV*U(O)~~]U(O)]-*. Following Ref. [ 11, we assumethat O(t) E Lp(0, 00) for somep, where n < p < cc, and hence
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for some constant M. A similar result holds true for n(t) dlog 1uj2 2 -2ar dt -
- (2q + 1) yo4+
Cl> - (2q+
A 236 (1997)
415-424
and !P( t), hence, we have
l)rzP(r)
-6@(t)
Integrating again gives us log 11112 2 log /u(O) 12- 2cYrt - M4(q)yt’-‘lP
(a _olpi2)(Mdq)y2+ M:!(q)$+
- M&-“P M3(s)s2)
t2-llp.
Hence we have (u12 2 lu(0) 12exp (-20~3 provided LY- j3e2 2 0 Now let us consider exist a ta such that lu( it must also hold for t assumption. Therefore,
- M( 1 + t)t’-‘lP).
(6)
M is a constant, and p > 1 as required. our original assumption that ]u(t) 1 # 0. Following Ref. [ 11, we assume that there to) I = 0 then it is clearly true that the theorem holds for 0 < t < ta and by continuity = to. However, from our conclusions (6), we have lu( to) ] # 0, which contradicts our we have that lu( t) I # 0 for all t > 0, and our proof is concluded.
3. Kuramoto-Sivashinsky
equation
We shall now apply the method to the Kuramoto-Sivashinsky
equation. The equation is defined as
ut + YV4U + v2u + u.vu = 0,
(7)
where v is the dissipative parameter. Unlike the Navier-Stokes equation the Kuramoto-Sivashinsky (K-S) equation contains a fourth-order term, but the two equations are similar, as the nonlinear term is of the same structure. Hence the method used by Dyer and Edmunds [l] can be applied to the K-S equation to get similar results. Through ladder methods we can show that the solutions of this equation have regularity and existence. Theorem 2. For u E 0 suppose we have that II(t)
= sup [Vu] < co, XEJJ
where u is a solution of the K-S equation, u is periodic with period L > 0 in x = (XI, x2,. . . xd), and d is the number of spatial dimensions. Then provided Y > e2, e2 = ( L/~T)~, and we assume that ]V*u(O) I < cc then there is a positive constant p depending only on n such that for all t > 0
where A = (v2u(o),v2u(o))l~(o)l-2
and p is a constant such that p > 2.
Proot We initially assume that lu( t) ) # 0, but we can apply the proof outlined in the previous section to show that lu( t) 1 # 0 for all r 2 0.
C.J. Woolcock,
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421
We now define, Nu = u.Vu = -ut - vV4u - V2u. It can easily be seen that dWuW12 dt
_
1 2(u u ) = -2b(t)I-2((u, 14t)12 ’ t
By applying integration (U,V2U) = -(Vu,VU),
d log k(t) I2 dt
Nu) + v(u,V4u)
by parts and the periodic boundary conditions therefore -v(V2U,V2U)
=2/u(~)l-2(-WW
+ (U,V2U)). we have (u, V~U) = (vzu, v?-~), and
+ (VU,VU)).
However, the final term on the r.h.s. (2/u(t) lp2( VU, VU)) is positive definite and -(u, Nu) Iu( t) l-2 > -@(t). Therefore, d10p2
> -2v~u~-2(v%,v%)
- 2IP(t).
Considering the r.h.s. term, -~vIu/-~(V~U, IuI”$
V2u). If we consider the quantity
( -(v;;i2v2u)))
we have l”14$ (-‘“t:;“‘“’
) = -2(V4U,Ut)
IU12 +
2(U, Ut)
(V2U,
V2U)
= ~v~V~U~~~U~~ - 2~V3U~2~u~2 + 2(V4u, NU)lU12- ~v~V~U~~~V~U~~ + 2~VU~2~V2U~2 - 2(U, Nu) (u, V4u). By applying PoincarC’sinequality in a similar way to the method used in the generaliseddiffusion model, we have 2
lM‘$ (
> 2( v - &2) IL42 v4u +
- 2(v!i2> l~121Nu12+2cv t
N” 2( v - P)
- 2(v - L2)
u, v4u + K
Nu 2( v - L2)
2
>I
L2J[(wW12.
Hence provided v - E2 > 0,
1~1”;( -“;;ip2’))
> -2(v ! L2)~uI~INu~~.
But jNu12= (Nu,NU) = Ju2(VU)2dx. d z
1
-(V2u,V2u)
Id2
Therefore,
> 3 -2(v
- P)
L12(t).
If we now integrate this we get
- (V2U(t),v2U(t))lU(t)l-2 2 -(v2u(o),v2u(o))lu(o)~-2-
l
2( v - L2)
J
Z12(s) ds.
0
(9)
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If we now substitute (9) into our formula for (d/dt)
d’“glu(f)l > -zv~ dt
_
’
’
v - L2
Letters
A 236 (1997)
415-424
log lu( t) ( we get
J’ n2(s) & - 2n2(t), 0
where A = lV*u(O) 12/~(O> le2. As in the previous theorem, by applying Holder’s with 2 < p < 00, we have
J
n2(s)
ds 6
inequality and assuming that 17 E LP(0, 00) for some p
Mt1-2’p,
0
for some M, p > 2, and so we have
d’wlu(t)l
> -2vA-
dt
’
Mv
+-2/p -2n2(t).
(v - I?)
Integrating this gives $-2/P
log JU( t) I > log lU(0) I - 2vAt -
-
2M2t’
-2/P
hence we have
as required. For exactly the same reasons as in Section 2, lu( t) ) # 0 for t > 0. 4. Burgers
equation
Similar results can be obtained for the Burgers equation. For the Burgers equation, we have
(‘0)
ut - vv2u + u.vu = 0, where tr is the dissipative parameter. Theorem 3. As before, if we set n(t)
= sup IVUl < cQ, en
where u is a solution of the Burgers equation and is periodic. Then we have lu(t)l
> Ju(0)lexp(-2vht-~(l+t)t2-2’p)a
pro05 Initially
we assume that lu( t) ( # 0 and define Nu = LOU = -ut + vV2u. So we have
d’oglu(t)12 = dt
-21u(t)l-2((u,Nu)
+v(Vu,Vu))
= -2n*(t)
-2~+(t)(-~(vu,vu).
C.J. Woolcock,
We have (d/df)
(VU, VU) = -2(u,,
M.V
BartuccelliIPhysics
V’u).
Letters
A 236 (1997)
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423
If we use this in the calculation of
we can use a method similar to the method used for the Kuromoto-Sivashinsky to Poincare’s inequality, to obtain
But from the definition we know that -INI
equation, but without
resorting
> --IuI~I~~( t), therefore, we have
Thus for any t 2 0 f* - 274~(t>~~~~u(t)l-~
3 -2vll~(O)~~~~u(O)I-~
- J Lr2(s)ds.
(11)
0
If we use ( 11) in (d/dt)
dlog lu(t)l dt
log lu( r) I we get f
> -2vA
- 2&t)
-
s
H2( s) ds,
0
where A = ~~u(O)~~~~U(O)I-*. By following the same method as used in the previous theorem we have /u(f)1 2 lu(O)lexp(-2vAt
- p(l+
t)t2-2/P),
as required. For exactly the same reasons as in Section 2, lu( t) I f 0 for t > 0.
5. Conclusions We have shown that it is possible to prove that the rate of decay for solutions of several nonlinear partial differential equations is bounded below by an exponential. Furthermore through a simple application of Poincare inequality we can see that if [u(f)1 is bounded below by an exponential then so is IV”ul. This is an important result for resolving the assumptions necessary for obtaining the time average of the dissipative length scale, using ladder methods [ 6-9 I. As we showed in the introduction the estimate for the length scales which can be related to the ladders hinges on whether we can show that (fi”/Hn) is bounded. Since
from the definition of the time average, we can see that this depended on the assumption that the rate of decay by the solutions is not faster than exponential. Since we have shown that the rate of the decay is not faster than the exponential this is no longer a problem and bounds can be found for the time average of
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the dissipative length scale. Furthermore, the method used throughout this Letter, based on the work done by Dyer and Edmunds in their paper [l] and by Ogawa [ 31, can be extended to cover many dissipative partial differential equations for bounding the solutions of these equations from below.
Acknowledgement We would like to thank the unknown
referee who has helped us to much improve this Letter.
References [I] [2] [3] [ 41 IS] [6] [7] [ 81 191 [lo] [ 11]
R.H. Dyer, D.E. Edmunds, Proc. London Math. Sot. 18 (1968) 78. D.E. Edmunds, Arch. Rat. Mech. Anal. 22 (1966) 15. H. Ogawa, Proc. Am. Math. Sot 16 (1965) 1241. C. Foias, J.C. Saut, Ind. Math. J. 33, No. 3 (1984). C. Foias, J.C. Saut, Ind. Math. J. Vol 33, No. 6 (1984) M.V. Bartuccelli, C.R. Doering, J.D. Gibbon, S.J.A. Malham, Nonlinearity 6 (1993) 549. M.V. Bartuccelli, J.D. Gibbon, M. Oliver, Physica D 89 (1996) 267. S.A. Gourley, M.V. Bartuccelli, Length scales in solutions of a scalar reaction-diffusion equation with delay. CR. Doering, J.D. Gibbon, Applied Analysis of the Navier-Stokes Equations (Cambridge Univ. Press, Cambridge, 1995). D.S. Cohen, J.D. Murray, J. Math. Biol. 12 (1981) 237. M.V. Bartuccelli, S.A. Gourley, C.J. Woolcock Length scales in Solutions of a Generalized Diffusion Model, Physica Scripta,
in press.