Comparison of decay of solutions to two compressible approximations to Navier-Stokes equations

0.17 MB·English

by  Cesar J. Niche

COMPARISON OF DECAY OF SOLUTIONS TO TWO
COMPRESSIBLE APPROXIMATIONS TO NAVIER-STOKES
EQUATIONS
5
1
0 CE´SARJ.NICHEANDMAR´IAE.SCHONBEK
2

n
Abstract. In this article,we usethe decay character of initialdata to com-
a
pare the energy decay rates of solutions to different compressible approxima-
J
tionstotheNavier-Stokesequations. Weshowthatthesystemhavinganonlin-
9
eardampingtermhasslowerdecaythanitscounterpartwithanadvection-like

term. Moreover, me characterize a set of initial data for which the decay of
]
P the first system is driven by the difference between the full solution and the
A solutiontothelinearpart,whileforthesecondsystemthelinearpartprovides
thedecayrate.
.
h
t
a
m
[ 1. Introduction

1 The Navier-Stokes equations for an incompressible homogeneous fluid in R3
v
5
0
∂ u+(u )u = ∆u p,
t
1 ·∇ −∇
2 divu = 0,
0
(1.1) u (x) = u(x,0)
. 0
1
0 have been extensively studied because of their importance in modelling a wide
5
rangeofphenomenainFluidMechanics. Takingdivergenceinthefirstlineof(1.1),
1
usingthedivergence-freeconditionandtheninvertingtheLaplacian,weobtainthe
:
v nonlocal relation p = ∆−1div(u )u, which poses very hard problems when
i − ·∇
X trying to solve these equations numerically.
r In order to avoid these problems, Temam [1] proposed a model approximating
a
(1.1) in which the pressure and the velocity are related through ǫp = divu, for
−
ǫ > 0, thus breaking the nonlocality. To “stabilize” this system, i.e. to have an
energyinequality,headdedthenonlinearterm 1(divuǫ)uǫ,whichthenleadstothe
2
compressible system
1 1
∂ uǫ+(uǫ )uǫ+ (divuǫ)uǫ = ∆uǫ+ divuǫ
t
·∇ 2 ǫ ∇·
(1.2) uǫ(x,0) = uǫ(x).
0
Date:January12,2015.
C.J. Niche acknowledges financial support from PRONEX E-26/110.560/2010-APQ1,
FAPERJ-CNPq and Ciˆencia sem Fronteiras - PVE 011/12. M. E. Schonbek was partially sup-
portedbyNSFGrantDMS-0900909.
1
2 CE´SARJ.NICHEANDMAR´IAE.SCHONBEK
This system has been used in many numerical experiments and has also been the
subjectofsomearticlesconcerningitsanalyticalproperties(seeFabrieandGalusin-
ski[2], Plecha´ˇcandSˇver´ak[3]). Recently Rusin [4] provedexistence ofglobalweak
solutions in R3 and their convergence in L3 (R3 R ), when ǫ goes to zero, to a
loc × +
suitable(inthesenseofCaffarelli-Kohn-Nirenberg[5])solutiontotheNavier-Stokes
equation.
Giventhe differencesinthe linearpartsandnonlineartermsof(1.1)and(1.2)it
is natural to ask whether these affect the decay rates of the L2 norm of solutions.
A useful tool to try to answer this question is the decay character of the initial
datum, introduced by Bjorland and M.E. Schonbek [6] and refined by Niche and
M.E.Schonbek[7]. Roughlyspeaking,thedecaycharacterisanumberassociatedto
everyu L2(R3)thatdescribesthebehaviourofu nearξ =0,whichcharacterizes
0 0
∈
the norm decay of solutions to linear systems u = u for a wide class of linear
t
c L
operators that includes =∆ and =∆+ 1 div. Using the sharp decay rate
L L ǫ ∇·
estimatesobtainedbyNiche andM.E.Schonbekandthe FourierSplitting method,
the following Theorem can be proved.
Theorem 1.1. (Niche and M.E. Schonbek [7]) Let u be a solution to either (1.1)
or (1.2), with u L2(R3) and decay character r∗ = r∗(u ), with 3 < r∗ < .
0 ∈ 0 −2 ∞
Then
ku(t)k2L2(R3) ≤C(1+t)−min{32+r∗,52}.
Remark 1.2. Bjorland and M.E. Schonbek [6] proved this result for the Navier-
Stokes equations (1.1).
The decay rate obtained in Theorem 1.1 provides plenty of information about
the similarities between (1.1) and (1.2). From Theorem 2.10 we have that
ketLu0k2L2(R3) ≤C(1+t)−(32+r∗), r∗ =r∗(u0),
so we see that for r∗ 1 the linear parts are the ones that have slower decay,
≤
while for r∗ > 1 the nonlinear terms are the ones driving the decay due to the
fast dissipation provided by etL. Also, the stabilizing term 1(divuǫ)uǫ, needed for
2
having an energy inequality in (1.2), does not change the relative strength of the
linearpartsandthe nonlinearterms withregardsto theirinfluence onthe decayof
energy. Wecanthenconcludethat,regardingenergydissipation,theNavier-Stokes
equations (1.1) and Temam’s approximation (1.2) have the same behaviour.
The main goal of this article is to study the decay of a different compressible
approximation to (1.1) and compare its decay rates to those from Theorem 1.1
using the decay character. More precisely, consider the system
1
∂ uǫ+(uǫ )uǫ+α uǫ 2uǫ = ∆uǫ+ divuǫ, α>0
t
·∇ | | ǫ ∇·
(1.3) uǫ(x,0) = uǫ(x),
0
DECAY OF SOLUTIONS FOR APPROXIMATIONS TO THE NSE 3
introduced by Leli`evre [8] and Lemari´e-Rieusset and Leli`evre [9] as a modifica-
tion of one used by Vishik and Fursikov [10] to construct statistical solutions to
Navier-Stokesequations. System(1.3) has the same scaling asNavier-Stokesequa-
tions, thus allowing for its analysis in spaces which contain homogeneous initial
data leading to selfsimilar solutions. Note that this system differs from Temam’s
approximation (1.2) in that, instead of having an advection-like term (divuǫ)uǫ,
it has a strongly nonlinear damping term uǫ 2uǫ. Leli`evre [8] proved that for
| |
uǫ L2(R3), there exists a global in time weak solution to (1.3) and that when α
0 ∈
andǫ goto zero,solutions to (1.3) converge,as distributions, to a suitable solution
to the Navier-Stokes equations.
We now state our main result which will allow us to compare the behaviour of
(1.1) and (1.2) to that of (1.3).
Theorem 1.3. Let uǫ L2(R3), with r∗ = r∗(uǫ), with 3 < r∗ < . Then for
0 ∈ 0 −2 ∞
any weak solution to (1.3) we have that for α > 4ǫ, C = C(α,ǫ,kuǫ0kL2) and any
δ >0
kuǫ(t)k2L2 ≤C(1+t)−min{23+r∗,32−δ}.
Thus, the strong nonlinear damping uǫ 2uǫ also leads to decay (see the energy
| |
inequality (3.8)). However, there are significant quantitative and qualitative differ-
ences regarding the decay of solutions. First, note that for initial data with r∗ 0,
≥
solutions to Navier-Stokes equations (1.1) and Temam’s approximation (1.2) have
faster decay rates than solutions to Lemari´e-Rieusset and Leli`evre’s system. This
is so precisely due to the presence of this nonlinear damping, which slows down
decay, as can be explicitly seen in the proof by comparing (3.13) to (3.14).
Moreover, for Navier-Stokes equations and Temam’s approximation, the linear
partdeterminesthedecayratesforr∗ <1,whileforLemari´e-RieussetandLeli`evre’s
system the linear part is the leading one only for r∗ < 0. Hence, for uǫ with 0 <
0
r∗ <1 not only the decay rates are different, but so is the dissipation mechanism,
given by the linear part in the first case, and by the nonlinear terms in the second
case. In Remark 2.8 we show that v = Λsuǫ, where uǫ is in L1(R3) Hs(R3)
0 0 0 ∩
with0<s<1,hasdecaycharacterr∗(v )=s,thus providingexplicitexamples of
0
initial data that lead to such behaviour.
Remark 1.4. For any fixed α>0, when ǫ goes to zero we obtainthe Navier-Stokes
equations with an extra damping term. The decay of solutions to this system has
been recently addressed, see Cai and Lei [11], Jia, Zhang and Dong [12], Jiang and
Zhu [13], Jiang [14]. Theorem 1.3 improves and generalizes the results obtained in
the case of β =3 in [14].
This article is organizedas follows. In Section2 we recallsome existence results
and properties of solutions to (1.3) and, following Niche and M.E. Schonbek [7],
we provide the definitions and results we need concerning the decay character and
the characterization of decay of linear systems. In Section 3, we prove our main
Theorem 1.3.
4 CE´SARJ.NICHEANDMAR´IAE.SCHONBEK
2. Settings
2.1. Solutions to (1.3). For the sake of completeness, we first recall results con-
cerning existence of solutions to (1.3) and their convergence to solutions to (1.1).
Theorem 2.1. (Theorem 3.3, Leli`evre [8]) Let u L2(R3). Then there exists
0
∈
a distributional solution uǫ to (1.3) such that uǫ L∞(R ,L2) L2(R ,H˙1)
+ +
∈ ∩ ∩
L4(R ,L4).
+
Theorem 2.2. (Theorem 4.3, Leli`evre [8]) Let u L2(R3), with u =0. Then
0 0
∈ ∇·
solutions from Theorem 2.1 converge when α and ǫ go to zero, as distributions, to
solutions to (1.1).
2.2. Decaycharacter. Asthelongtimebehaviourofsolutionstomanydissipative
systems is determined by the low frequencies of the solution, Bjorland and M.E.
Schonbek [6] introduced the idea of decay character of a function u in L2(Rn) in
0
order to characterize the decay of solutions to Navier-Stokes equations with that
initial datum. Recently, Niche and M.E. Schonbek [7] generalized this notion in
order to use data in Hs(Rn),s > 0 and to obtain results for other equations. We
recall now these definitions and results.
Definition 2.3. Letu0 L2(Rn)andΛ=( ∆)12. Fors 0the s-decay indicator
∈ − ≥
Ps(u ) corresponding to Λsu is
r 0 0
Ps(u )= limρ−2r−n ξ 2s u (ξ)2dξ
r 0 ρ→0 ZB(ρ)| | | 0 |
for r n +s, , where B(ρ) is the ball at the orbigin with radius ρ.
∈ −2 ∞
(cid:0) (cid:1)
[
Remark 2.4. Settingr =q+s,weseethatthes-decayindicatorcompares Λsu (ξ)2
0
| |
to f(ξ) = ξ 2(q+s) near ξ = 0. When s = 0 we recover the definition of decay
| |
indicator given by Bjorland and M.E. Schonbek [6].
Definition2.5. Thedecaycharacter ofΛsu ,denotedbyr∗ =r∗(u )istheunique
0 s s 0
r n +s, such that 0 < Ps(u ) < , provided that this number exists.
∈ −2 ∞ r 0 ∞
If su(cid:0)ch Ps(u ) d(cid:1)oes not exist, we set r∗ = n + s, when Ps(u ) = for all
r 0 s −2 r 0 ∞
r n +s, or r∗ = , if Ps(u )=0 for all r n +s, .
∈ −2 ∞ s ∞ r 0 ∈ −2 ∞
(cid:0) (cid:1) (cid:0) (cid:1)
Remark 2.6. (Examples 2.5 and 2.6, Niche and M.E. Schonbek [7]). Let u
0
∈
L2(Rn) such that u (ξ)=0, for ξ <δ, for some δ >0. Then, Ps(u )=0, for any
0 | | r 0
r n +s, and r∗(u )= . If u Lp(Rn) L2(Rn), with 1 p 2, then
∈ −2 ∞ b s 0 ∞ 0 ∈ ∩ ≤ ≤
r∗(u(cid:0)) = n 1(cid:1) 1 , so if u L1(Rn) L2(Rn) we have that r∗(u ) = 0 and if
0 − (cid:16) − p(cid:17) 0 ∈ ∩ 0
u L2(Rn) but u / Lp(Rn), for any 1 p<2, we have that r∗(u )= n.
0 ∈ 0 ∈ ≤ 0 −2
Let u Hs(Rn), with s > 0. As Λ\s(u )(ξ) = ξ su (ξ), the heuristics for the
0 0 0
∈ | |
decay character given in Remark 2.4 lead us to expect that r∗(u ) = r∗(Λsu ) =
s 0 0
c
s+r∗(u ). This is the content of the following Theorem.
0
Theorem2.7. (Theorem2.11,NicheandM.E.Schonbek[7])Letu Hs(Rn),s>
0
∈
0.
DECAY OF SOLUTIONS FOR APPROXIMATIONS TO THE NSE 5
(1) If n <r∗(u )< then n +s<r∗(u )< and r∗(u )=s+r∗(u ).
−2 0 ∞ −2 s 0 ∞ s 0 0
(2) r∗(u )= if and only if r∗(u )= .
s 0 ∞ 0 ∞
(3) r∗(u )= n if and only if r∗(u )=r∗(u )+s= n +s.
0 −2 s 0 0 −2
Remark 2.8. We can now justify the assertion made after the statement of Theo-
rem 1.3 about initial data for which the decay of solutions to (1.2) and (1.3) are
quantitatively and qualitatively different. Let u L1(Rn) L2(Rn) such that
0
∈ ∩
Λsu L2(Rn), with 0<s <1. From Remark 2.6 we have that r∗(u )=0, while
0 0
∈
from Theorem 2.7 we have that r∗(Λsu ) = r∗(u ) = s+r∗(u ) = s. This proves
0 s 0 0
that Λsu has decay character 0<r∗ <1.
0
2.3. Linear operators and characterization of decay. We describe now the
linear operators for which we characterize the decay of the L2 norm of solutions
in terms of the decay character. For a Hilbert space X on Rn, we consider a
pseudodifferential operator :Xn L2(Rn) n, with symbol (ξ) such that
L → M
(cid:0) (cid:1)
(2.4) (ξ)=P−1(ξ)D(ξ)P(ξ), ξ a.e.
M −
where P(ξ) O(n) and D(ξ) = c ξ 2αδ , for c > c >0 and 0 <α 1. Taking
i ij i
∈ − | | ≤
the Fourier Transform of the linear equation
(2.5) v = v,
t
L
multiplying by v, integrating in space and then using (2.4) we obtain
b 1 d
v(t) 2 C ξ 2α v 2dξ
2dtk kL2 ≤− ZRn| | | |
which is the key inequality for using the Fourier Splitbting method in the proofs.
Remark 2.9. The fractional Laplacian on vector fields in Rn
( u) =( ∆)αu , i=1, n,
L i − i ···
providesanexampleof(2.4),asitssymbolis( (ξ)) = C ξ 2αδ . Theoperator
M ij − | | ij
1
u=∆u+ divu, ǫ>0,
L ǫ ∇
i.e. thelinearpartof(1.3),providesasecondexample,as(2.4)holdswith( (ξ)) =
M ij
ξ 2δ 1ξ ξ , D(ξ)=diag( ξ 2, ξ 2, 1+ 1 ξ 2) and
−| | ij − ǫ i j −| | −| | − ǫ | |
(cid:0) (cid:1)
−ξ2 −ξ1ξ3 ξ
 √ξξ121+ξ22 √−1ξ−2ξξ332 ξ1 
P(ξ)= √ξ120+ξ22 √1−1−ξ32ξ32 ξ2 ,
 √1−ξ32 3 
where v =(ξ ,ξ ,ξ ) has norm one. As a result of this,
1 2 3
(2.6) (cid:16)etM(ξ)(cid:17)ij =e−t|ξ|2δij − ξξiξ2j (cid:16)e−t|ξ|2 −e−(1+1ǫ)t|ξ|2(cid:17),
| |
6 CE´SARJ.NICHEANDMAR´IAE.SCHONBEK
see Rusin [4].
We now state the Theoremthat describes decay in terms of the decay character
for linear operators as in (2.4).
Theorem 2.10. (Theorem 2.10, Niche and M.E. Schonbek [7]) Let v L2(Rn)
0
∈
have decay character r∗(v ) = r∗. Let v(t) be a solution to (2.5) with data v .
0 0
Then:
(1) if n <r∗ < , there exist constants C ,C >0 such that
−2 ∞ 1 2
C1(1+t)−α1(n2+r∗) ≤kv(t)k2L2 ≤C2(1+t)−α1(n2+r∗);
(2) if r∗ = n, there exists C =C(ǫ)>0 such that
−2
v(t) 2 C(1+t)−ǫ, ǫ>0,
k kL2 ≥ ∀
i.e. the decay of v(t) 2 is slower than any uniform algebraic rate;
k kL2
(3) if r∗ = , there exists C >0 such that
∞
v(t) 2 C(1+t)−m, m>0,
k kL2 ≤ ∀
i.e. the decay of v(t) L2 is faster than any algebraic rate.
k k
3. Proof of Theorem 1.3
Proof: The proof is based on the Fourier Splitting method, introduced by M.E.
Schonbek to study decay of parabolic conservation laws [15] and of Navier-Stokes
equations [16], [17]. As is usual in this context, we prove the estimate assuming
the solutions are regular enough (for the existence of regular approximations, see
Leli`evre [8]). The limiting argument that proves the estimate for weak solutions
follows that for the Navier-Stokes equations in pages 267–269 in Lemari´e-Rieusset
[18] and the Appendix in Wiegner [19], which we refer to for full details.
We first show that solutions obey an energy inequality. As
1
uǫ(uǫ )uǫdx= uǫ 2divuǫdx,
ZR3 ·∇ −2ZR3| |
it follows that
1 1
(cid:12)(cid:12)ZR3uǫ(uǫ·∇)uǫdx(cid:12)(cid:12) ≤ 2(cid:12)(cid:12)ZR3|uǫ|2divuǫdx(cid:12)(cid:12)= 2k|uǫ|2divuǫkL1
(cid:12) (cid:12) 1(cid:12) (cid:12) ǫ 1
(3.7(cid:12)) (cid:12) ≤ 2k(cid:12)|uǫ|2kL2kdivuǫkL(cid:12)2 ≤ 4kuǫk4L4 + 4ǫkdivuǫk2L2.
Multipyling (1.3) by uǫ, integrating in space and using (3.7) we obtain
1 d 3 ǫ
(3.8) uǫ(t) 2 uǫ(t) 2 divuǫ 2 α uǫ(t) 4 ,
2dtk kL2 ≤−k∇ kL2 − 4ǫk kL2 −(cid:16) − 4(cid:17)k kL4
which, given the hypotheses on ǫ and α, provides an energy inequality. Then
1 d
(3.9) uǫ(t) 2 uǫ(t) 2 C divuǫ 2 .
2dtk kL2 ≤−k∇ kL2 − k kL2
DECAY OF SOLUTIONS FOR APPROXIMATIONS TO THE NSE 7
Now let
r′(t)
B(t)= ξ R3 : ξ 2
{ ∈ | | ≤ 2Cr(t)}
where r is a positive increasing function with r(0) = 1 and C is an appropiate
constant. From (3.9) we have that
1 d r′(t)
uǫ(t) 2 C ξ 2 uǫ(ξ,t)2dξ uǫ(ξ,t)2dξ.
2dtk kL2 ≤− ZR3| | | | ≤−2r(t)ZB(t)c| |
b b
Addingandsubstractingatermsimilartotheoneontherightsideofthisinequality,
only that with B(t) as the domain of integration, and then mutiplying by r(t) we
obtain
d
(3.10) (r(t) uǫ(t) L2) r′(t) uǫ(ξ,t)2dξ.
dt k k ≤ Z | |
B(t)
b
We now prove a pointwise estimate for
t
uǫ(ξ,t)=etM(ξ)uǫ(ξ) e(t−s)M(ξ)G(ξ,s)ds
0 −Z
0
where etM(ξ) is asbin (2.6) and c
G(ξ,s)= (uǫ )uǫ+ uǫ 2uǫ ,
F(cid:16) ·∇ | | (cid:17)
where is also the Fourier transform. As
F
(uǫ )uǫ = (uǫ uǫ) (divuǫ)uǫ
·∇ ∇· ⊗ −
and
\
(divuǫ)uǫ(ξ,t) divuǫ(t) L2 uǫ(t) L2,
(cid:12) (cid:12)≤k k k k
we obtain (cid:12) (cid:12)
(cid:12) (cid:12)
1
|F((uǫ·∇)uǫ(t))|≤|ξ|kuǫ(t)k2L2 + 2kdivuǫ(t)kL2kuǫ(t)kL2 ≤C|ξ|kuǫ(t)k2L2.
Also
(cid:12)F(cid:16)|uǫ|2uǫ(cid:17)(t)(cid:12)≤kF(cid:16)|uǫ|2uǫ(cid:17)(t)kL∞ ≤k|uǫ(t)|2uǫ(t)kL1 ≤kuǫ(t)k3L3.
(cid:12) (cid:12)
Now,(cid:12)we estimate th(cid:12)e nonlinear transport term. We have
t t
e(t−s)M(ξ) ((uǫ )uǫ)(ξ,s)ds C e−C(t−s)|ξ|2 ξ uǫ(s) 2 ds
(cid:12)Z F ·∇ (cid:12) ≤ Z | |k kL2
(cid:12) 0 (cid:12) 0
(cid:12) (cid:12) t
(3.1(cid:12)1) (cid:12) C ξ uǫ(s) 2 ds .
≤ | |(cid:18)Z k kL2 (cid:19)
0
Suppose now that
8 CE´SARJ.NICHEANDMAR´IAE.SCHONBEK
(3.12) uǫ(t) 2 C(1+t)−β,
k kL2 ≤
for some β 0 and β =1. We then have, after choosing r(t)=(t+1)3
≥ 6
t 2 r′(t) 25
e(t−s)M(ξ) ((uǫ )uǫ)(ξ,s)ds dξ C (1+t)2(1−β)
Z (cid:18)Z F ·∇ (cid:19) ≤ (cid:18)r(t)(cid:19)
B(t) 0
(3.13) C(1+t)−(12+2β),
≤
where we used (3.11) and (3.12). We now estimate the damping term. We have
t 2 t 2
e(t−s)M(ξ) uǫ 2uǫ (ξ,s)ds dξ C uǫ(s) 3 ds dξ
ZB(t)(cid:18)Z0 F(cid:16)| | (cid:17) (cid:19) ≤ ZB(t)(cid:18)Z0 k kL3 (cid:19)
t 2
≤CZ (cid:18)Z kuǫ(s)kL2kuǫ(s)k2L4ds(cid:19) dξ
B(t) 0
t t
C uǫ(s) 2 ds uǫ(s) 4 ds dξ
≤ Z (cid:18)Z k kL2 (cid:19)(cid:18)Z k kL4 (cid:19)
B(t) 0 0
(3.14) C(t+1)−(21+β)
≤
where we used the interpolation
1 2
kuǫ(t)kL3 ≤kuǫ(t)kL32kuǫ(t)kL34,
Ho¨lder’sinequality,(3.12)andthefactthatuǫ L∞L2 L4L4. Ofthesetwoterms,
∈ t x∩ t x
(3.14) has the slower decay.
The only apriori estimate we have is uǫ(t) L2 C, i.e. β = 0. So, in the ball
k k ≤
B(t) we have
uǫ(ξ,t)2dξ C etM(ξ)u 2dξ
0
Z | | ≤ Z | |
B(t) B(t)
b t c 2
+ C e(t−s)M(ξ)G(ξ,s)ds dξ
Z (cid:18)Z (cid:19)
B(t) 0
(3.15) C(t+1)−(32+r∗)+C(t+1)−21,
≤
which leads to
(3.16) kuǫ(t)k2L2 ≤C(1+t)−min{32+r∗,21}.
We now use this to boostrap our decay estimates by improving the value of β.
When r∗ < 1, we have that β = 3 +r∗ and (3.14) provides no improvement.
− 2
When r∗ 1, then β = 1 which leads to
≥− 2
DECAY OF SOLUTIONS FOR APPROXIMATIONS TO THE NSE 9
uǫ(ξ,t)2dξ C etM(ξ)u 2dξ
0
Z | | ≤ Z | |
B(t) B(t)
b t c 2
+ C e(t−s)M(ξ)G(ξ,s)ds dξ
Z (cid:18)Z (cid:19)
B(t) 0
(3.17) C(t+1)−(23+r∗)+C(t+1)−1.
≤
Hence
(3.18) kuǫ(t)k2L2 ≤C(1+t)−min{32+r∗,1}.
Again, r∗ < 1 does not produce an improvement on the decay while r∗ 1
−2 ≥ −2
leads to β =1, for which (3.13) and (3.14) do not hold, as
t
uǫ(s) 2 ds Cln(t+1).
Z k kL2 ≤
0
However,working along the lines of these estimates we obtain
t 2 r′(t) 25
(3.19) e(t−s)M(ξ) ((uǫ )uǫ)(ξ,s)ds dξ C ln2(t+1)
Z (cid:18)Z F ·∇ (cid:19) ≤ (cid:18)r(t)(cid:19)
B(t) 0
and
t 2 r′(t) 32
(3.20) e(t−s)M(ξ) uǫ 2uǫ (ξ,s)ds dξ C ln(t+1).
ZB(t)(cid:18)Z0 F(cid:16)| | (cid:17) (cid:19) ≤ (cid:18)r(t)(cid:19)
Again, the second term has slower decay and choosing r(t) = (t +1)γ for large
enough γ >0 we have that, as ln(t+1) (t+1)δ for any δ >0
≤
uǫ(ξ,t)2dξ C etM(ξ)u 2dξ
0
Z | | ≤ Z | |
B(t) B(t)
b t c 2
+ C e(t−s)M(ξ)G(ξ,s)ds dξ
Z (cid:18)Z (cid:19)
B(t) 0
(3.21) C(t+1)−(23+r∗)+C(t+1)−(32−δ).
≤
Then
kuǫ(t)k2L2 ≤C(1+t)−min{23+r∗,32−δ}, ∀δ >0.
This boostrapping process does not improve the decay further, as now for r∗ 0,
≥
the integral of uǫ(t) L2 is bounded by a constant. (cid:3)
k k
10 CE´SARJ.NICHEANDMAR´IAE.SCHONBEK
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(C.J. Niche) Departamento de Matema´tica Aplicada, Instituto de Matema´tica, Uni-
versidade Federal do Riode Janeiro,CEP 21941-909,Rio de Janeiro-RJ, Brasil
E-mail address: [email protected]
(M.E.Schonbek)DepartmentofMathematics,UCSantaCruz,SantaCruz,CA95064,
USA
E-mail address: [email protected]