On the Domain of Analyticity and Small Scales for the Solutions of the Damped-driven 2D Navier-Stokes Equations

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by  Alexei A. Ilyin

ON THE DOMAIN OF ANALYTICITY AND SMALL SCALES FOR
THE SOLUTIONS OF THE DAMPED-DRIVEN 2D NAVIER–STOKES
EQUATIONS
Alexei A. Ilyin1 and Edriss S. Titi2
7
0
0
2 Abstract. Weobtainalogarithmicallysharpestimateforthespace-analyticityradiusofthesolutions
of the damped-driven 2D Navier–Stokes equations with periodic boundary conditions and relate this
n
to the small scales in this system. This system is inspired by the Stommel–Charney barotropic ocean
a
circulationmodel.
J

9 Key words: Analyticity, Gevrey regularity, Navier–Stokes equations, dissipative length scales,
1 Stommel–Charneymodel.

AMS subject classification: 35B41,35Q30,37L30.
]
P
A 1. Introduction
.
h
It was shown in [15] (see also [3], [12]) that the solutions of the 2D Navier–Stokes equa-
t
a tions with periodic boundary conditions belong to the Gevrey class of analytic functions
m
(if the forcing term does). Using the Gevrey regularity approach the following estimate
[
for the spatial analyticity radius for the solutions that lie on the global attractor (or are

1
near it) was obtained
v
0 c|Ω|1/2
3 la ≥ G2logG, (1.1)
5
1 where G = kfk |Ω|/ν2 is the Grashof number and |Ω| = L2/γ is the area of the periodic
0 L2
domain Ω = [0,L/γ]×[0,L], γ ≤ 1.
7
0 Therefore, the Fourier coefficients uˆ are exponentially small for |k| ≫ L/l , and l
k a a
/
h naturally forms a lower bound for the small dissipative length scale for the system (see,
t
a for instance, [11]).
m There are other ways of estimating the dissipative small length scale for the Navier–
: Stokes system, for instance, in terms of the dimension of the global attractor [1], [6], [7],
v
i [12], [39]. TheHausdorff andfractaldimensions oftheglobalattractorsatisfythefollowing
X
estimate [8] (see also [6], [39]):
r
a
dim A ≤ c G2/3(log(1+G))1/3, c = c (γ)
F 1 1 1
which has been shown in [33] (following ideas of [1]) to be logarithmically sharp.
If we accept the point of view that the small length scale can be defined as follows (see
[7], [12], [36], [39])
1/2
|Ω|
l ∼ , (1.2)
f
dim A
(cid:18) F (cid:19)
DATE: JANUARY 18, 2007.
1
Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya Sq. 4, 125047
Moscow, Russia, E-mail: [email protected]
2
Department of Mathematics and Department of Mechanical and Aerospace Engineering, University
of California,Irvine, California 92697,USA, E-mail: [email protected]. Also: Department of Computer
Science and Applied Mathematics, Weizmann Institute of Science, P.O. Box 26, Rehovot, 76100, Israel,
E-mail: [email protected]
1
2 A.A. ILYINANDE.S. TITI
then up to logarithmic correction we have
|Ω|1/2
l ∼ . (1.3)
f G1/3
This heuristic estimate for the small length scale is probably the best one can hope for
sinceitmatches, uptologarithmicterm,thephysicallyassertedestimatesfortheenstrophy
dissipation length scale [30] . We also observe that the estimate (1.3) is extensive, that is,
independent of the size of the spatial domain provided that its shape is fixed.
Another rigorousdefinition of the small length scale can be given in terms of the number
ofdeterminingmodes, nodes, orvolumeelements (see[12],[14],[16],[28]andthereferences
therein). It was shown that if N is sufficiently large and N equal squares of size l tile the
dn
periodic spatial domain, then any collection of points (one in each square) are determining
for the long time dynamics of the 2D Navier–Stokes system. The best to date estimate
for N was obtained in [28]:
N ≤ c G,
2
where c = c (γ) depends only on the aspect ratio γ ≤ 1. (An explicit estimate for c was
2 2 2
obtained in [26]: c (γ) = (68/(γπ))1/2.)
2
Therefore the small length scale defined in terms of the lattice of determining nodes
satisfies
|Ω|1/2
l ≥ c (γ)−1/2 . (1.4)
dn 2
G1/2
We observe that this estimate is not extensive, that is, l scales like λ−1/2 if Ω is replaced
dn
by λΩ, λ > 0.
Wepoint out here that for the2DNavier–Stokes system withanalytic forcing the results
of [17], [18] provide the existence of a finite number N of instantaneously determining
nodes comparable with the fractal dimension of the attractor. These nodes, however, can
be chosen arbitrarily (up to a subset of ΩN with 2N-dimensional Lebesgue measure zero)
and therefore do not naturally define a regular lattice of determining nodes.
The best to date estimate for the analyticity radius of the solutions of the Navier–Stokes
equations with analytic forcing term f was obtained in [31]:
|Ω|1/2
l ≥ c (γ) . (1.5)
a 3
G1/2(1+logG)1/4
Relating the radius of analyticity to the dissipative small length scale (see also [23] in this
regard) we note that up to a logarithmic correction the estimate (1.5) coincides with (1.4),
but both are worse than (1.2), where the latter coincides, as we have already pointed out,
with the physically asserted estimate of [30].
In this paper we focus on the 2D space periodic Navier–Stokes system with damping
2
∂ u+ ui∂ u = −µu+ν∆u−∇p+f,
t i
(1.6)
i=1
X
divu = 0.
By adding the Coriolis forcing term to (1.6) one obtains the well-known Stommel–Charney
barotropic model of ocean circulation [4], [10], [35], [37]. Here the damping µu represents
the Rayleigh friction term and f is the wind stress. For an analytical study of this system
see, for instance [5], [22], [24], [41], and the references therein. In a follow up work we will
be studying the effect of adding rotation (Coriolis parameter) on the size of small scales
and the complexity of the dynamics of (1.6). Therefore, we will focus in this work on the
system (1.6). We also point out that in this geophysical context the viscosity plays a much
ANALYTICITY OF THE SOLUTIONS OF THE 2D DAMPED-DRIVEN NS SYSTEM 3
smaller role in the mechanism of dissipating energy than the Rayleigh friction. That is
why in this work the friction coefficient µ > 0 will be fixed and we consider the system at
the limit when ν → 0+.
Sharp estimates (as ν → 0) for the Hausdorff and the fractal dimensions of the global
attractor of the system (1.6) were first obtained in the case of the square-shaped domain
in [25] (γ = 1). Then the case of an elongated domain was studied in [27] (γ → 0), where
it was shown that
krotfk |Ω|
∞
dim A ≤ c D, D = , (1.7)
F 4
µν
where c is an absolute constant (c ≤ 12). This estimate is sharp as both ν → 0 and
4 4
γ → 0. Therefore the small length scale defined as in (1.2) is of the order of
|Ω| 1/2 µν 1/2 |Ω|1/2
l ∼ ∼ ∼ . (1.8)
f dim A krotfk D1/2
(cid:18) F (cid:19) (cid:18) ∞(cid:19)
This heuristic estimate is, in fact, a rigorous bound for the small length scale expressed in
terms of the number of determining modes and nodes [26]:
1/2 1/2
|Ω| µν
l = c = c , c = 681/4. (1.9)
dn 5 5 5
D krotfk
(cid:18) (cid:19) (cid:18) ∞(cid:19)
This means that any lattice of points in Ω at a typical distance l ≤ l is determining.
dn
The main result of this paper is in showing that the analyticity radius l of the solutions
a
of the damped-driven Navier–Stokes system (1.6) lying on the global attractor is bounded
from below and satisfies the estimate:
c|Ω|1/2
l ≥ , (1.10)
a D1/2(1+logD)1/2
which up to a logarithmic correction agrees both with the smallest scale estimate (1.8)
and the rigorously defined typical distance between the determining nodes (1.9).
It is worth mentioning that this point of view of relating the radius of analyticity of
solutions on the Navier–Stokes equations to small scales in turbulence was also presented
in [23].
This paper is organized as follows. In section 2 we employ the Gevrey–Hilbert space
technique of [15] to derive a lower bound for the radius of analyticity of the order
c|Ω|1/2
. (1.11)
D2logD
This bound considerably improves, for a fixed µ > 0, the lower bound (1.1) for the clas-
sical Navier–Stokes system as ν → 0+ (see also Remark 2.1). Let us remark that as an
alternative to the Gevrey regularity technique for estimating small scales one can apply
the ladder estimates approach presented in [9] to obtain estimates for the small scales
in (1.6) (see also [19]).
In section 3 the estimate (1.10) is proved for the system (1.6) following [31].
2. Gevrey regularity of the damped Navier–Stokes system
As usual (see, for instance, [1],[6],[32],[38]), we write (1.6) as an evolution equation in
the Hilbert space H which is the closed subspace of solenoidal vectors in (L (Ω))2 with
2
zero average over the torus Ω = [0,L/γ]×[0,L]:
∂ u+B(u,u)+νAu+µu = f, u(0) = u . (2.1)
t 0
4 A.A. ILYINANDE.S. TITI
Here A = −P∆ is the Stokes operator with eigenvalues 0 < λ ≤ λ ≤ ..., B(u,v) =
1 2
P 2 ui∂ v is the nonlinear term, f = Pf ∈ H, and P : (L (Ω))2 → H.
i=1 i 2
We restrict ourselves to the case γ = 1 and, in addition, assume that Ω = [0,2π]2 (this
(cid:0)P (cid:1)
simplifies the Fourier series below). The case of the square-shaped domain Ω = [0,L]2
reduces to this case by scaling. Furthermore, any domain with aspect ratio γ < 1 can be
treated in the similar way, the absolute dimensionless constants c ,c ,... below will then
1 2
depend on γ, however.
A vector field u ∈ H has the Fourier series expansion
u = u eij·x, u ∈ C2, u = u¯ , u ·j = 0, u = 0,
j j −j j j 0
j∈Z2
X
and
kuk2 = kuk2 = (2π)2 |u |2.
L2 j
j∈Z2
X
The eigenvalues of the Stokes operator A are the numbers |j|2, and the domain of its
powers is the set of vector functions u such that
(2π)2 |j|4α|u |2 = kAαuk2 < ∞.
j
j∈Z2
X
For τ,s > 0 we define the Gevrey space D(eτAs) of functions u satisfying
(2π)2 e2τ|j|2s|u |2 = keτAsuk2 < ∞. (2.2)
j
j∈Z2
X
We suppose that the forcing term f belongs to the Gevrey space of analytic functions
f ∈ D(eσ1A1/2A1/2), so that (2π)2 |j|2e2σ1|j||u |2 = keσ1A1/2A1/2fk < ∞ (2.3)
j
j∈Z2
X
for some σ > 0. We set
1
1/2
ϕ(t) = min(νλ t,σ ).
1 1
The norm and the scalar product in D(eϕ(t)A1/2) are denoted by k·k and (·,·) , respec-
ϕ ϕ
tively.
Weassume thatu ∈ D(A1/2)andtake thescalar product of(2.1)andAuinD(eϕ(t)A1/2)
0
1/2
for sufficiently small t ≤ σ /(νλ ). Since
1 1
1
eϕ(t)A1/2∂ u(t),eϕ(t)A1/2u(t) = ∂ kA1/2u(t)k2 −νλ1/2(Au(t),A1/2u(t)) ,
t 2 t ϕ 1 ϕ
(cid:0) (cid:1)
we obtain
1
∂ kA1/2uk2 +νkAuk2 +µkA1/2uk2 = (B(u,u)Au) +νλ1/2(Au,A1/2) +(A1/2f,A1/2u) .
2 t ϕ ϕ ϕ ϕ 1 ϕ ϕ
(2.4)
Next we use the key estimate (see [15], [12], [40]) for the nonlinear term in Gevrey spaces
kAuk2 1/2
(B(u,u),Au) ≤ c kA1/2uk2kAuk 1+log ϕ
ϕ 1 ϕ ϕ λ kA1/2uk2
(cid:18) 1 ϕ(cid:19)
ANALYTICITY OF THE SOLUTIONS OF THE 2D DAMPED-DRIVEN NS SYSTEM 5
and use Young’s inequality for this estimate and for the last two terms in (2.4):
∂ kA1/2uk2+νkAuk2 ≤
t ϕ ϕ
2c2 kAuk2 kA1/2fk2
≤ 1kA1/2uk4 1+log ϕ +2νλ kA1/2uk2 + ϕ ≤
ν ϕ λ kA1/2uk2 1 ϕ 2µ
(cid:18) 1 ϕ(cid:19)
c kAuk2 kA1/2fk2
≤ 2kA1/2uk4 1+log ϕ +ν3λ2 + ϕ,
ν ϕ λ kA1/2uk2 1 2µ
(cid:18) 1 ϕ(cid:19)
where c = 2c2 +1. Next, using the inequality −αz +β(1+logz) ≤ βlogβ/α (see [12],
2 1
[13]), we find
c kAuk2 c c kA1/2uk2
−νkAuk2 + 2kA1/2uk4 1+log ϕ ≤ 2kA1/2uk4 log 2 ϕ,
ϕ ν ϕ λ kA1/2uk2 ν ϕ λ ν2
(cid:18) 1 ϕ(cid:19) 1
and obtain the differential inequality
c c kA1/2uk2 kA1/2fk2
∂ kA1/2uk2 ≤ 2kA1/2uk4 log 2 ϕ +ν3λ2 + ϕ .
t ϕ ν ϕ λ ν2 1 2µ
1
Hence the function
c kA1/2uk2 kA1/2fk2
y(t) = 2 ϕ + σ1 +e,
λ ν2 λ ν3/2µ1/2
1 1
where lne = 1, satisfies
∂ y(t) ≤ νλ c y2logy, c = max(1,c /2).
t 1 3 3 2
Therefore y(t) ≤ 2y(0) for as long as
t ≤ (2νλ c y(0)log2y(0))−1.
1 3
In other words,
kA1/2uk2 ≤ 2kA1/2u k2 +c (ν/µ)1/2kA1/2fk +c λ ν2, c = e/c ,
ϕ 0 4 σ1 4 1 4 2
as long as 0 ≤ t ≤ T∗(kA1/2u k), where
0
1
T∗(kA1/2u k) = .
0 2c νλ c2kA1/2u0k2 + kA1/2fkσ1 +e log 2 c2kA1/2u0k2 + kA1/2fkσ1 +e
3 1 λ1ν2 λ1ν3/2µ1/2 λ1ν2 λ1ν3/2µ1/2
(cid:16) (cid:17) (cid:16) (cid:16) (cid:17)(cid:17)
We now observe (see Lemma 3.1) that on the global attractor or in the absorbing ball
we have, respectively,
kA1/2fk kA1/2fk
kA1/2u(t)k ≤ , t ∈ R, kA1/2u(t)k ≤ 2 , t ≥ T (kA1/2u k).
0 0
µ µ
Therefore we have the following lower bound for T∗:
kA1/2fk2 kA1/2fk kA1/2fk2 kA1/2fk −1
T∗ ≥ c νλ + σ1 +1 log + σ1 +1
5 1
λ ν2µ2 λ ν3/2µ1/2 λ ν2µ2 λ ν3/2µ1/2
(cid:20) (cid:18) 1 1 (cid:19) (cid:18) 1 1 (cid:19)(cid:21)
In the limit ν → 0+ we have
kA1/2fk2 kA1/2fk
≫ σ1,
λ ν2µ2 λ ν3/2µ1/2
1 1
and we can write the lower bound for T∗ as follows
T∗ ≥ c νλ D2logD −1,
6 1
(cid:2) (cid:3)
6 A.A. ILYINANDE.S. TITI
where
kA1/2fk krotfk|Ω|1/2 1 krotfk |Ω|
∞
= ≤ D, where D = .
λ1/2νµ 2πνµ 2π νµ
1
In terms of the analyticity radius l the lower bound for T∗ takes the form
a
c |Ω|1/2
7
l ≥ .
a D2logD
Thus, we have proved the following theorem.
Theorem 2.1. Suppose that f ∈ D(A1/2eσ1A1/2) for some σ > 0. Then a solution u lying
1
on the global attractor A is analytic with analyticity radius
c |Ω|1/2
7
l ≥ min , σ ,
a (D2 +D +1)log(D2 +D +1) 1
(cid:18) 1 1 (cid:19)
where
krotfk |Ω| kA1/2fk
D = ∞ , D = σ1.
νµ 1 λ ν3/2µ1/2
1
Moreover,
c |Ω|1/2
l ≥ 8 as ν → 0+. (2.5)
a D2logD
The constants c and c depend only on the aspect ratio of the periodic domain Ω.
7 8
Remark 2.1. We observe that the estimate (2.5) for the system (1.6) is of the order
ν−2log(1/ν) as far as the dependence on ν → 0+ is concerned, while the estimate (1.1)
for the classical Navier–Stokes system is, in this respect much larger; namely, is of the
order ν−4log(1/ν).
However, theestimate (2.5)isnotsharpandwillbeimproved inthenext section. Ashas
been demonstrated in [34] the Gevrey–Hilbert space technique does not always provide
sharp estimates for the radius of analyticity. The mechanism explaining this has been
reported in [34] by means of an explicitly solvable model equation.
3. Sharper bounds
In this section we obtain sharper lower bounds for the analyticity radius l . This is
a
achieved by combining the ν-independent estimate for the vorticity contained in the fol-
lowing lemma and the L -technique developed in [21], [31] for the uniform analyticity
p
radius of the solutions of the Navier–Stokes equations. We observe that similar technique
has been earlier established in [2] for studying the analyticity of the Euler equations.
Applying the operator rot to (1.6) we obtain the well-known scalar vorticity equation
∂ ω +u·∇ω = ν∆ω −µω +F, (3.1)
t
where ω = rotu, F = rotf, u = ∇⊥∆−1ω, so that u·∇ω = ∇⊥∆−1ω·∇ω = J(∆−1ω,ω),
and ∇⊥ = (−∂ ,∂ ), J(a,b) = ∇⊥a·∇b.
2 1
Lemma 3.1. (See [26].) The solutions u(t) lying on the global attractor A satisfy the
following bound:
krotfk
kω(t)k ≤ L2k , t ∈ R, (3.2)
L2k µ
where 1 ≤ k ≤ ∞.
ANALYTICITY OF THE SOLUTIONS OF THE 2D DAMPED-DRIVEN NS SYSTEM 7
Proof. We use the vorticity equation (3.1) and take the scalar product with ω2k−1, where
k ≥ 1 is integer, and use the identity
(J(ψ,ϕ),ϕ2k−1) = (2k)−1 J(ψ,ϕ2k)dx = (2k)−1 div(ϕ2k∇⊥ψ)dx = 0.
Z Z
We obtain
kωk2k−1∂ kωk +(2k −1)ν |∇ω|2ω2k−2dx+µkωk2k =
L2k t L2k L2k
Z
= (rotf,ω2k−1) ≤ krotfk kωk2k−1.
L2k L2k
Hence, by Gronwall’s inequality
kω(t)k ≤ kω(0)k e−µt +µ−1krotfk (1−e−µt),
L2k L2k L2k
and passing to the limit as k → ∞ we find
kω(t)k ≤ kω(0)k e−µt +µ−1krotfk (1−e−µt).
∞ ∞ ∞
Now, we let t → ∞ in the above inequalities and obtain
krotfk
limsupkω(t)k ≤ L2k , 1 ≤ k ≤ ∞,
L2k µ
t→∞
which gives (3.2) since the solutions lying on the attractor are bounded for t ∈ R. (cid:3)
As before we consider the square-shaped domain Ω = [0,L]2 and it is now convenient
to write (1.6) in dimensionless form. We introduce dimensionless variables x′, t′, u′ and p′
by setting
x = Lx′, t = (L2/ν)t′, u = (ν/L)u′, p = (ν2/L2)p′, µ = (ν/L2)µ′.
We obtain
2
∂t′u′ + u′i∂i′u′ = −µ′u+∆′u′ −∇′p′ +f′,
(3.3)
i=1
X
div′u′ = 0,
where x′ ∈ Ω′ = [0,1]2, f′ = (L3/ν2)f. Accordingly, the dimensionless form of (3.1) is as
follows (we omit the primes):
∂ ω +u·∇ω = ∆ω −µω +F. (3.4)
t
Remark 3.1. For dimensionless variables u′ and ω′ the estimate (3.2) with k = ∞ takes
the form
krot′f′k krotfk L2
kω′k = krot′u′k ≤ ∞ = ∞ = D. (3.5)
∞ ∞
µ′ νµ
The next lemma is similar to the main estimate for the space analyticity radius in [31].
Lemma 3.2. Suppose that F is a restriction to Ω (that is, y = 0) of a bounded x-periodic
analytic function F(x+iy)+iG(x+iy) in the region |y| ≤ δ and
F
M2 = sup (F(x+iy)2 +G(x+iy)2). (3.6)
F
x∈Ω, |y|≤δF
Let p ≥ 3/2 and let
M2
2p
t = .
0 CM2/µ
F
8 A.A. ILYINANDE.S. TITI
Here (and throughout) C is a sufficiently large universal constant and M ≥ kω k .
2p 0 L2p
Then the solution ω(t) is analytic for t > 0 and for 0 < t ≤ t the space analyticity radius
0
of ω(t) is greater than
t1/2 1 1 1
δ(t) = min , , , ,δ .
C Cpt(2p−3)/4pM2p Cpt(2p−3)/(4p+6)M2p/(2p+3) pt1/2M2p F!
2p
Proof. We solve (3.4) by a sequence of approximating solutions (see [29], [31]). We set
u(0) = 0 and ω(0) = 0. Then for ω(n), u(n) we have the equation
∂ ω(n) −∆ω(n) +u(n−1) ·∇ω(n) +µω(n) = F
t
(3.7)
ω(n)(0) = ω = rotu , u(n) = ∇⊥∆−1ω(n).
0 0
The solutions ω(n) and u(n) for t > 0 have analytic extensions ω(n)+iθ(n) and u(n)+iv(n)
and since the system (3.7) is linear, their analyticity radius is at least δ . They satisfy
F
the equation
∂ (ω(n)+iθ(n))−∆(ω(n)+iθ(n))+(u(n−1)+iv(n−1))·∇(ω(n)+iθ(n))+µ(ω(n)+iθ(n)) = F+iG,
t
or, equivalently, the system
∂ ω(n) −∆ω(n) +µω(n) +u(n−1) ·∇ω(n) −v(n−1) ·∇θ(n) = F,
t
(3.8)
∂ θ(n) −∆θ(n) +µθ(n) +u(n−1) ·∇θ(n) +v(n−1) ·∇ω(n) = G,
t
where, as before, u(n) = ∇⊥∆−1ω(n), v(n) = ∇⊥∆−1θ(n), and the differential operators are
taken with respect to x. In view of the analyticity of the solutions we have the Cauchy–
Riemann equations
∂ω(n) ∂θ(n)
= − ,
∂y ∂x
j j
(3.9)
∂ω(n) ∂θ(n)
= , j = 1,2,
∂x ∂y
j j
and the similar equations for u(n) and v(n).
Let ε > 0. We consider the functional
1
ψ (t) = ω(n)(x,αts,t)2 +θ(n)(x,αts,t)2 +ε pdxds. (3.10)
n
Z0 ZΩ
(cid:0) (cid:1)
We also set
Q (x,s,t) = ω(n)(x,αts,t)2 +θ(n)(x,αts,t)2 +ε.
n
Here t ∈ R+ and α ∈ R2. The combination αts will play the roleof the variable y; p ≥ 3/2,
and ε > 0 is arbitrary.
We differentiate ψ (t) taking into account (3.8) and use the Cauchy–Riemann equa-
n
tions (3.9) to handle the derivatives with respect to y. We obtain
1
∂ ψ (t)+I = I +I +I +I , (3.11)
t n 0 1 2 3 4
2p
where
1
I = Qp−1 |∇ω(n)|2 +|∇θ(n)|2 +µ(ω(n))2 +µ(θ(n))2 dxds+
0 n
Z0 ZΩ
(cid:0) 1 (cid:1)
+2(p−1) Qp−2 ω(n)∇ω(n) +θ(n)∇θ(n) 2dxds,
n
Z0 ZΩ
(cid:0) (cid:1)
ANALYTICITY OF THE SOLUTIONS OF THE 2D DAMPED-DRIVEN NS SYSTEM 9
and
1
I = Qp−1 −ω(n)∇θ(n) +θ(n)∇ω(n) ·αsdxds,
1 n
Z0 ZΩ
1 (cid:0) (cid:1)
I = Qp−1 ω(n)∇ω(n) +θ(n)∇θ(n) ·u(n−1)dxds,
2 n
Z0 ZΩ
1 (cid:0) (cid:1)
I = Qp−1 −ω(n)∇θ(n) +θ(n)∇ω(n) ·v(n−1)dxds,
3 n
Z0 ZΩ
1 (cid:0) (cid:1)
I = Qp−1 ω(n)F +θ(n)G)dxds.
4 n
Z0 ZΩ
The arguments of Q are x,s,t, an(cid:0)d the arguments of ω(n), θ(n), u(n), and v(n) are x, αts,
n
and t.
For an arbitrary η > 0 we have
1
I ≤ η Qp−1 |∇ω(n)|2 +|∇θ(n)|2 dxds+
1 n
Z0 ZΩ (3.12)
1 (cid:0) (cid:1)
C Qp−1 (ω(n))2 +(θ(n))2 |α|2s2dxds ≤ ηI +C |α|2ψ (t).
η n 0 η n
Z0 ZΩ
(cid:0) (cid:1)
Next,
1 1
I = ∇Qp ·u(n−1)dxds = 0. (3.13)
2 2p n
Z0 ZΩ
For I we have
3
1
I ≤ ηI +C Qp |v(n−1)|2dxds ≤ ηI +C I′I′′, (3.14)
3 0 η n 0 η 3 3
Z0 ZΩ
where
1 (p−1)/p 2 1 1/p
I′ = Q (x,s,t)p2/(p−1)dxds , I′′ = |v(n−1)(x,αts,t)|2pdxds .
3 n 3 j
(cid:18)Z0 ZΩ (cid:19) j=1 (cid:18)Z0 ZΩ (cid:19)
X
We write I′ as follows
3
I′ = kQp/2k2 , Ω = Ω×[0,1] ⊂ R3, β = 2p/(p−1), 2 ≤ β ≤ 6,
3 n Lβ(Ω0) 0
and use in Ω the Gagliardo–Nirenberg inequality
0
3/β−1/2 3/2−3/β
kAk ≤ CkAk k∇ Ak +CkAk
Lβ(Ω0) L2(Ω0) x,s L2(Ω0) L2(Ω0)
p/2
for A = A(x,s) = Q (x,s,t). We have
n
k∇ Ak2 = k∇ Qp/2k2 =
x,s L2(Ω0) x,s n L2(Ω0)
1
p2 Qp−2 (ω(n)∇ω(n) +θ(n)∇θ(n))2 +t2(θ(n)α·∇ω(n) −ω(n)α·∇θ(n))2 dxds ≤
n
Z0 ZΩ
≤ C(cid:0)p2(1+|α|2t2)I . (cid:1)
0
Hence,
k∇ Ak3/2−3/β = k∇ Qp/2k3/2p ≤ C(1+|α|2t2)3/4pI3/4p.
x,s L2(Ω0) x,s n L2(Ω0) 0
Next, kQp/2k2 = ψ (t),
n L2(Ω0) n
kAk3/β−1/2 = kQp/2k(2p−3)/2p = ψ (t)(2p−3)/4p
L2(Ω0) n L2(Ω0) n
10 A.A. ILYINANDE.S. TITI
and
I′ ≤ C(1+|α|2t2)3/2pI3/2pψ (t)(2p−3)/2p +Cψ (t). (3.15)
3 0 n n
We now consider I′′. Since v(n−1)(x,0,t) = 0 (the solution restricted to y = 0 is real-
3 j
valued), we have (using the Cauchy–Riemann equations for v )
j
2 1
(n−1) (n−1)
|v (x,αts,t)| = α ts ∂ v (x,αtsτ,t)dτ =
j k yk j
(cid:12)(cid:12)Xk=1 Z0 (cid:12)(cid:12)
(cid:12) (cid:12)
(cid:12) 2 1 (cid:12)
(cid:12) α ts ∂ u(n−1)(x,αtsτ,t)dτ (cid:12).
k k j
(cid:12)(cid:12)Xk=1 Z0 (cid:12)(cid:12)
(cid:12) (cid:12)
Then (cid:12) (cid:12)
(cid:12) (cid:12)
1/p
2p
2 1 2 1
I′′ = α ts ∂ u(n−1)(x,αtsτ,t)dτ dxds ≤
3  k k j 
Xj=1 Z0 ZΩ(cid:12)(cid:12)Xk=1 Z0 (cid:12)(cid:12)
(cid:12) (cid:12)
 (cid:12) 1 1 (cid:12) 1/p
C|α|2t2 (cid:12) ∇u(n−1)(x,αtsτ,t) 2ps2p(cid:12)dτdxds =
(cid:18)Z0 ZΩZ0 (cid:19)
C|α|2t2 1s2pds (cid:12)(cid:12) 1dτ ∇u(n−1)(x(cid:12)(cid:12),αtsτ,t) 2pdx 1/p.
(cid:18)Z0 Z0 ZΩ (cid:19)
(cid:12) (cid:12)
Since u = ∇⊥∆−1ω, we have (see [20], [42]) (cid:12) (cid:12)
1/2p
|∇u(x)|2pdx = k∇∇⊥∆−1ωk ≤ k∆−1ωk ≤ Cpkωk .
L2p W22p L2p
(cid:18)ZΩ (cid:19)
Therefore
1 1 1/p
I′′ ≤ Cp2|α|2t2 ω(n−1)(x,αtsτ,t) 2pdxs2pdsdτ ≤
3
(cid:18)Z0 Z0 ZΩ (cid:19)
1 (cid:12)(cid:12) 1/p (cid:12)(cid:12)
Cp2|α|2t2 Qp dxds ≤ Cp2|α|2t2ψ (t)1/p,
n−1 n−1
(cid:18)Z0 ZΩ (cid:19)
where we have used 1 1h(sτ)s2pdsdτ ≤ (2p)−1 1h(s)ds. Combining this with (3.14)
0 0 0
and (3.15) we obtain
R R R
I ≤
3
η′I0 +Cη′p2|α|2t2(1+|α|2t2)3/2pI03/2pψn(t)(2p−3)/2pψn−1(t)1/p +Cp2|α|2t2ψn−1(t)1/pψn(t) ≤
ηI +C p2(|α|t)4p/(2p−3)(1+|α|2t2)3/(2p−3)ψ (t)2/(2p−3)ψ (t)+Cp2|α|2t2ψ (t)1/pψ (t).
0 η n−1 n n−1 n
(3.16)
Finally, we estimate I :
4
1
I ≤ Qp−1 (ω(n))2 +θ(n))2)ηµ+(F2 +G2)/(4ηµ) dxds ≤
4 n
Z0 ZΩ (3.17)
(cid:0) 1 (cid:1)
ηI +C (M2/µ) Qp−1dxds ≤ ηI +C (M2/µ)ψ (t)(p−1)/p,
0 η F n 0 η F n
Z0 ZΩ
where M is defined in (3.6).
F