Navier-Stokes-Voigt equations with memory in 3D lacking instantaneous kinematic viscosity

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by  Francesco Di Plinio

NAVIER-STOKES-VOIGT EQUATIONS WITH MEMORY IN 3D
LACKING INSTANTANEOUS KINEMATIC VISCOSITY
FRANCESCO DI PLINIO, ANDREA GIORGINI, VITTORINO PATA AND ROGER TEMAM
7
1
0 Abstract. We consider a Navier-Stokes-Voigt fluid model where the instantaneous
2 kinematicviscosityhasbeencompletelyreplacedbyamemorytermincorporatinghered-
n itary effects, in presence of Ekman damping. The dissipative character of our model is
a weakerthantheonewherehereditaryandinstantaneousviscositycoexist,studiedin[12]
J by Gal and Tachim-Medjo. Nevertheless, we prove the existence of a regular exponen-

6 tial attractor of finite fractal dimension under rather sharpassumptions on the memory
2
kernel.

]
P
A
. 1. Introduction
h
at Let Ω R3 be a bounded domain with smooth boundary ∂Ω, and let α > 0 and β 0 be
m given c⊂onstants. For t > 0, we consider the dimensionless form of the Navier-Stokes≥-Voigt
[ (NSV) equations with memory and Ekman damping in the unknown velocity u = u(x,t)

1 and pressure π = π(x,t)
v
5 ∞
∂ (u α∆u) g(s)∆u(t s)ds+βu+(u )u+ π = f,
4 (1.1) t − − − ·∇ ∇
8  Z0
divu = 0,
7 
0
where f is an assigned external forcing term, subject to the nonslip boundary condition
.
1 
0 (1.2) u = 0.
∂Ω
7 |
1 The function g : [0, ) R, usually called memory kernel, is supposed to be convex
v: nonnegative, smooth∞on R→+ = (0, ), vanishing at infinity and satisfying the relation
i ∞
X
∞
g(s)ds = 1.
r
a Z0
The system is supplemented with the initial conditions
(1.3) u(0) = u and u( s) = ϕ (s),
0 s>0 0
− |
where u and ϕ ( ) are prescribed data. The initial velocity u and the past history of the
0 0 0
·
velocity ϕ , which need only be defined for almost every s > 0, will play a different role in
0
the translation of (1.1) into an evolution system, as detailed in the subsequent Section 3.
The drag/friction term βu, widely used in geophysical hydrodynamics, is known as
Ekman damping. The coefficient β is the Ekman pumping/dissipation constant, induced
by the so-called Ekman layer, appearing at the bottom of a rotating fluid (see [24]). The
2010 Mathematics Subject Classification. 35B40,35Q30,37L30, 45K05.
Keywords andphrases. Navier-Stokes-Voigtequations,Ekmandamping,memorykernels,dissipation,
exponential attractors.
1
2 F. DI PLINIO,A. GIORGINI,V. PATAAND R.TEMAM
term βu is also referred to as the Rayleigh friction, employed in oceanic models such as
the viscous Charney-Stommel barotropic ocean circulation model of the gulf stream. In
the last decades, in connection with damped Euler and Navier-Stokes equations, several
works involving a dissipation term of such kind have been made, in order to study the
limit case of vanishing viscosity (see e.g. [5, 6, 16]).
Coming back to the memory kernel, the formal choice of g equal to the Dirac mass at
0+ in (1.1) corresponds to the damped NSV system
∂ (u α∆u) ∆u+βu+(u )u+ π = f,
t
(1.4) − − ·∇ ∇
divu = 0.
(
This system was introduced by Oskolkov in [19], without the friction term βu, as a
model for the motion of a Kelvin-Voigt linear viscoelastic incompressible fluid. Indeed,
global well-posedness of (1.4), along with the existence of a finite dimensional global
attractor, has been established in connection with numerical simulations of turbulent
flows in statistical equilibrium, see [17] and references therein. Letting the viscoelastic
length scale α go to zero we recover the three-dimensional Navier-Stokes system. The
recent contribution [9] improves on the results of [17] by proving the existence of a family
of exponential attractors parameterized by the length scale α which converges to the
(weak) exponential attractor of the Navier-Stokes system as α 0. We are led to the
→
study of (1.1) as a generalization of (1.4) where the instantaneous kinematic viscosity
of the Kelvin-Voigt fluid has been completely replaced by the physically more realistic
integrodifferential (memory) term, rendering the diffusive effects dependent on the past
historyofthevelocity. Arigorousderivationof (1.1)fromthebalanceoflinearmomentum
and the constitutive equations of a Kelvin-Voigt fluid can be found in [12], to which we
send for a more comprehensive description of the physical motivations of the class of fluid
models including (1.1).
Theauthorsof[12]thenspecializetothestudyofthewell-posedness andtheasymptotic
properties of the system
∞
∂ (u α∆u) ν∆u (1 ν) g(s)∆u(t s)ds+(u )u+ π = f,
t
− − − − − ·∇ ∇
 Z0
divu = 0,

for some parameter ν (0,1). This is a version of (1.1) with β = 0, where instantaneous

∈
and memory-type viscosities coexist. The main result of [12] is the existence of a finite-
dimensional exponential attractorfortherelateddynamical system. Inaddition, replacing
the memory kernel g with a family of suitable rescalings g converging to the Dirac mass
ε
(see point III of Section 9), the corresponding family of ε-exponential attractors is proved
to be robust in the limit ε 0, where the (undamped) system (1.4) is recovered. Such a
→
coexistenceofinstantaneousandmemory-typeviscoustermshasbeenpreviouslyexploited
in [13], where the asymptotic behavior of a two-dimensional Jeffreys model is shown to
converge to its Navier-Stokes (formal) limit.
Dealing with (1.1) is substantially more challenging from a mathematical viewpoint, as
the rather strong dissipative effects of the instantaneous diffusive term ∆u exploited in
−
[12]arenotavailable. Theupshotofthepresentworkisthattheasymptoticfeaturesofthe
NSV EQUATIONS WITH MEMORY IN 3D 3
Navier-Stokes-Voigt system with memory are preserved also in absence of instantaneous
kinematicviscosity, providedthatthemuchweaker, butstillphysicallyinteresting, Ekman
damping βu is taken into account.
We now move onto the task of describing our results, as well as the crucial points in
our analysis, in more detail. At the same time, we provide an overview of the structure of
our article. After recalling the relevant mathematical framework for Navier-Stokes type
systems with memory in Section 2, (1.1) is recast in Section 3 as an abstract evolution
system in the Dafermos history space setting [10]. Therein, we prove that such an evolu-
tionsystem generatesastronglycontinuous semigroupwithcontinuous dependence onthe
initial data. The more challenging part of our entire analysis is found in Section 4, where
we establish the dissipative character of the semigroup. This is done by combining ad-hoc
energy functionals with a Gronwall-type lemma with parameter borrowed from [21], and
substantially relying on the Ekman damping term βu. If no forcing term is present, the
dissipative estimate remains true even if β = 0. This is the content of Section 5. In the
presence of memory terms, which are intrinsically hyperbolic, finite time regularization
of the solution cannot occur. As a substitute, in Section 6 we show that the trajecto-
ries are exponentially attracted by more regular bounded sets. This property is relied
upon in Section 7, where the main result of the paper, namely, the existence of a regular
exponential attractor for the semigroup of finite fractal dimension in the phase space, is
established. The global attractor is thus recovered as a byproduct. Precise statements are
given in Theorem 7.2 and Corollary 7.3, and the proof of the main theorem is carried out
in Section 8. Finally, we send to Section 9 for several directions of future investigation.
2. Mathematical Setting and Notation
Throughout this work, L2(Ω), H1(Ω), Hr(Ω) will be the standard Lebesgue-Sobolev
0
spaces on Ω. In particular, the external force f will be assumed to be a (constant-in-
time) vector of [L2(Ω)]3.
We introduce the Hilbert space (H, , , ) given by
h· ·i k·k
H = u [L2(Ω)]3 : divu = 0, u n = 0 ,
∂Ω
{ ∈ · | }
n being the outward normal to ∂Ω, along with the Hilbert space
V = u [H1(Ω)]3 : divu = 0 ,
∈ 0
with inner product and norm (cid:8) (cid:9)
u,v = u, v and u = u .
1 1
h i h∇ ∇ i k k k∇ k
We denote by V ′ its dual space. We will also encounter the space
W = V [H2(Ω)]3.
∩
Calling P : [L2(Ω)]3 H the Leray orthogonal projection onto H, we consider the
Stokes operator on H→
A = P∆ with domain D(A) = W.
−
4 F. DI PLINIO,A. GIORGINI,V. PATAAND R.TEMAM
It is well known that A is a positive selfadjoint operator with compact inverse (see e.g.
[28]). This allows us to define the scale of compactly nested Hilbert spaces
Vr = D(Ar2), r R,
∈
endowed with the inner products and norms
r r r
u,v r = A2u,A2v and u r = A2u .
h i h i k k k k
In particular,
V 1 = V ′, V0 = H, V1 = V , V2 = W.
−
More generally (see [1]),
Vr = V [Hr(Ω)]3, 1 r 2,
∩ ≤ ≤
Vr V [Hr(Ω)]3, r > 2.
⊂ ∩
We also recall the Poincar´e inequality
(2.1) λ u u , u V ,
1 1
k k ≤ k k ∀ ∈
where λ > 0 is the first eigenvpalue of A. The symbol , will also stand for the duality
1
h· ·i
product between Vr and its dual space V r.
−
As customary, we write the trilinear form on V V V
× ×
3
∂v
j
b(u,v,w) = (u )v wdx = u w dx,
i j
·∇ · ∂x
ZΩ i,j=1ZΩ i
X
satisfying the relation
b(u,v,v) = 0.
The associated bilinear form B : V V V ′ is defined as
× →
B(u,v),w = b(u,v,w).
h i
Next, we turn our attention to the memory kernel. Defining
µ(s) = g (s),
′
−
the prime being the derivative, we suppose µ nonnegative, absolutely continuous, decreas-
ing (hence µ 0 almost everywhere), and summable on R+ with total mass
′
≤
∞
κ = µ(s) ds > 0.
Z0
The classical Dafermos condition will be also assumed (see [10]), namely,
(2.2) µ(s)+δµ(s) 0,
′
≤
for some δ > 0 and almost every s > 0. Then, we introduce the L2-weighted Hilbert space
on R+
= L2(R+;V ),
M µ
with inner product and norm
1
η,ξ = ∞µ(s) η(s),ξ(s) ds and η = ∞µ(s) η(s) 2ds 2,
h iM h i1 k kM k k1
Z0 (cid:18)Z0 (cid:19)
NSV EQUATIONS WITH MEMORY IN 3D 5
along with the infinitesimal generator of the right-translation semigroup on
M
Tη = ∂ η with domain D(T) = η : ∂ η , η(0) = 0 .
s s
− ∈ M ∈ M
Here, ∂sη is the distributional derivative of η(s(cid:8)) with respect to the interna(cid:9)l variable s.
Finally, we define the extended memory space
= V
H ×M
endowed with the product norm
(u,η) 2 = α u 2 + u 2 + η 2 .
k k k k1 k k k k
H M
Due to the Poincar´e inequality (2.1), this norm is equivalent to the natural one on .
H
In this work, we will also make use of higher order memory spaces. To this end, we
define
1 = L2(R+;W),
M µ
with inner product and norm analogous to those of , and the corresponding higher
M
order extended memory space
1 = W 1
H ×M
with norm
(u,η) 2 = α u 2 + u 2 + η 2 .
k k 1 k k2 k k1 k k 1
H M
General agreement. Throughout the paper, the symbols C > 0 and ( ) will denote
Q ·
a generic constant and a generic increasing positive function, respectively, depending
only on the structural parameters of the problem, but independent of f (unless otherwise
specified). Moreover, given a Banach space and R > 0, we denote by
X
B (R) = x : x R
X { ∈ X k kX ≤ }
the ball of of radius R about zero.
X
3. The Dynamical System
As anticipated in the Introduction, the original problem (1.1)-(1.2) is translated into an
evolution system in the so-called Dafermos past history framework [10]. First, we observe
that, withoutanylossofgenerality, wecanassumethatf H whichamountstochanging
∈
π (by adding Pf f to π). Hence we apply the projection P to the first equation (1.1)
− ∇
and we transform (1.1)-(1.2) into
∞
(3.1) ∂ (u+αAu)+ g(s)Au(t s)ds+βu+B(u,u) = f.
t
−
Z0
In more generality, we will replace the damping βu with a term of the form βA ϑu, for
−
some ϑ 0. Then, we introduce the past history variable
≥
s
ηt(s) = u(t σ)dσ,
−
Z0
that satisfies the differential identity
∂ ηt(s) = ∂ ηt(s)+u(t).
t s
−
6 F. DI PLINIO,A. GIORGINI,V. PATAAND R.TEMAM
At this point, recalling the definition of the operator T, a formal integration by parts
leads to the differential problem in the unknown variables u = u(t) and η = ηt( )
·
∂ (u+αAu)+ ∞µ(s)Aη(s)ds+βA ϑu+B(u,u) = f,
t −
(3.2)
 Z0
∂ η = Tη+u,
 t
for some ϑ 0, where µ = g . As we said, this is actually a generalization of equa-
≥ − ′
tion (3.1), which corresponds to the case ϑ = 0 (the strongest damping within this class).
In turn, the initial conditions (1.3) transform into
u(0) = u and η0 = η ,
0 0
having set
s
η (s) = ϕ (σ)dσ.
0 0
Z0
We begin with the definition of weak solution.
Definition 3.1. Given U = (u ,η ) , a function U = (u,η) ([0, ), ), with
0 0 0
∂ u L2(0,τ;V ) for every τ > 0, is a w∈eaHk solution to (3.2) with in∈itiaCl dat∞umH
t
∈
U(0) = (u(0),η0) = U
0
if for every test function v V and almost every t > 0
∈
∂ u,v +α ∂ u,v + ∞µ(s) η(s),v ds+β A ϑu,v +b(u,u,v) = f,v ,
t t 1 1 −
h i h i h i h i h i
Z0
where η fulfills the explicit representation
s
u(t σ)dσ, 0 < s t,
− ≤
(3.3) ηt(s) = Z0 t
η0(s t)+ u(t σ)dσ, s > t.

− −
Z0
Remark 3.2. Given u ([0, ),V), the function η = ηt( ) satisfies the representation

∈ C ∞ ·
formula above if and only if it is a mild solution (in the sense of [23]) to the nonhomoge-
neous linear equation
∂ η = Tη +u.
t
Theorem 3.3. For every initial datum U , system (3.2) admits a unique solution
0
∈ H
U(t) = (u(t),ηt).
Accordingly, the problem generates a dynamical system, otherwise called strongly con-
tinuous semigroup,
S(t) : , t 0,
H → H ≥
acting by the formula
S(t)U = U(t).
0
This is a one-parameter family of maps S(t) on satisfying the properties:
H
S(0) = Id ;
⋄ H
S(t+τ) = S(t)S(τ), for every t,τ 0;
⋄ ≥
t S(t)U ([0, ), ), for every U .
0 0
⋄ 7→ ∈ C ∞ H ∈ H
NSV EQUATIONS WITH MEMORY IN 3D 7
Given an initial datum U , the corresponding energy at time t 0 reads
0
∈ H ≥
1 1
(t) = S(t)U 2 = u(t) 2 +α u(t) 2+ ηt 2 .
E 2k 0kH 2 k k k k1 k kM
h i
The existence of a weak solution is carried out via a Galerkin approximation scheme,
by exploiting standard energy estimates (much more immediate than the uniform ones
of the next Section 4), and then passing to the limit in the usual way. Indeed (see the
energy equality (4.2)), we easily get that, for any given τ > 0 and R 0,
≥
(3.4) U(t) (R+τ),
k kH ≤ Q
for all t τ and all initial data U(0) B (R). In particular, the increasing function
≤ ∈ H Q
does not depend on β 0. We refer the interested reader to [7] for more details on the
≥
Galerkin scheme in connection with equations with memory.
Remark 3.4. It is worth noting that the argument does not require β to be strictly
positive, nor inequality (2.2) is needed at this level. Indeed, both β > 0 and (2.2) will
come into play in connection with the dissipative properties of the semigroup.
Uniqueness is a consequence of the following continuous dependence result.
Proposition 3.5. For every τ > 0 and R 0, any two solutions U (t) and U (t) to
1 2
≥
equation (3.2) fulfill the estimate
(3.5) U (t) U (t) (R+τ) U (0) U (0) ,
1 2 1 2
k − kH ≤ Q k − kH
for all t τ and all initial data U (0) B (R).
i
≤ ∈ H
Proof. Let τ > 0 be fixed, and let U = (u ,η ), U = (u ,η ) be two solutions to (3.2) on
1 1 1 2 2 2
the time-interval [0,τ] such that U (0) B (R). Then, the difference
i
∈ H
U¯ = (u¯,η¯) = U U
1 2
−
solves
∂ (u¯+αAu¯)+ ∞µ(s)Aη¯(s)ds+βA ϑu¯+B(u ,u¯)+B(u¯,u ) = 0.
t − 1 2
Z0
Testing the equation by the admissible test function u¯, we easily find
d
Γ+ η¯,u¯ b(u¯,u ,u¯),
2
dt h iM ≤ −
where
1 α
Γ(t) = u¯(t) 2 + u¯(t) 2.
2k k 2k k1
In light of (3.4),
b(u¯,u ,u¯) C u u¯ 2 (R+τ)Γ,
− 2 ≤ k 2k1k k1 ≤ Q
and an integration in time yields
t t
Γ(t)+ η¯y,u¯(y) dy Γ(0)+ (R+τ) Γ(y)dy, t τ.
h iM ≤ Q ∀ ≤
Z0 Z0
Observing that η¯ fulfills the representation (3.3), from [7, Theorem 5.1] we learn that
1 1 t
η¯t 2 η¯0 2 η¯y,u¯(y) dy.
2k kM − 2k kM ≤ Z0 h iM
8 F. DI PLINIO,A. GIORGINI,V. PATAAND R.TEMAM
In summary,
t
U¯(t) 2 U¯(0) 2 + (R+τ) U¯(y) 2 dy,
k kH ≤ k kH Q Z0 k kH
(cid:3)
and the claim follows from the integral Gronwall lemma.
In particular, we draw from Proposition 3.5 that S(t) fulfills the further continuity
property
S(t) ( , ), for every t 0.
⋄ ∈ C H H ≥
4. Dissipativity
The main result of this section provides a uniform-in-time a priori estimate on the solu-
tions U(t) = S(t)U . To this end, we introduce the function
0
1
φ(ϑ) = ϑ ,
− 2
and we strengthen the assumptions on the external force by requiring
(4.1) f V̺ for some ̺ > φ(ϑ).
∈
Since φ(ϑ) < ϑ, there is no harm in assuming also ̺ ϑ. When φ(ϑ) < 0, which is the
≤
same as saying ϑ < 1/2, we only require ̺ = 0, i.e.
f V0 = H,
∈
which does not add anything to our general assumptions made at the beginning. As
expected, φ is increasing, since the influence of the damping term A ϑ becomes weaker
−
when ϑ is larger.
Remark 4.1. We highlight the fact that f H in the physically relevant case ϑ = 0,
∈
corresponding to the Ekman damping.
Theorem 4.2. Let β > 0 be fixed, and let (4.1) hold. Then, for any initial datum U ,
0
∈ H
the corresponding energy fulfills the estimate
(t) ( (0))e ωt +C f 2,
E ≤ Q E − k k̺
where ω > 0 is a universal constant depending only on the structural parameters of the
problem.
Remark 4.3. It is worth mentioning that a dissipative estimate of this kind seems to be
out of reach when β = 0.
The main consequence of Theorem 4.2 is that, when β > 0, the dynamical system
S(t) is dissipative, namely the trajectories originating from any given bounded set belong
uniformly in time to an absorbing set. By definition, this is a bounded set with
0
B ⊂ H
the following property: for any bounded set of initial data there is an entering
B ⊂ H
time t = t ( ) 0 such that
e e
B ≥
S(t) , t t .
0 e
B ⊂ B ∀ ≥
It is then immediate to see that one can take
= B (R ),
0 0
B H
NSV EQUATIONS WITH MEMORY IN 3D 9
for any fixed R > √2C f , with the constant C of the previous statement.
0 ρ
k k
The remaining of the section is devoted to the proof of Theorem 4.2. This will require
a number of steps.
4.1. Technical lemmas. The main tool needed in the proof is a Gronwall-type lemma
from [21].
Lemma 4.4. Let Λ be a family of absolutely continuous nonnegative functions on [0, )
ε
satisfying for every ε > 0 small and some κ > 0 and M 0 the differential inequality∞
≥
d M
Λ +κεΛ CεpΛq + ,
dt ε ε ≤ ε εr
where the nonnegative parameters p,q,r fulfill
p 1 > (q 1)(1+r) 0.
− − ≥
Moreover, let E be a continuous nonnegative function on [0, ) such that
∞
1
E(t) Λ (t) mE(t)
ε
m ≤ ≤
for every ε > 0 small and some m 1. Then, there exists ω > 0 such that
≥
E(t) (E(0))e ωt +CM.
−
≤ Q
We also recall a basic interpolation result (see e.g. [18]).
Lemma 4.5. Let a < b < c. Then
u u ̟ u 1 ̟, u Vc,
k kb ≤ k kc k ka− ∀ ∈
with
b a
̟ = − .
c a
−
4.2. Energy functionals. Let now
U(t) = (u(t),ηt)
be the solution to (3.2) originating from a given U . In what follows, we will use
0
∈ H
several times without explicit mention the Young, H¨older and Poincar´e inequalities. We
will also perform several formal computations, justified within a suitable regularization
scheme.
We introduce the (nonnegative) functional
1
Π(t) = ∞µ(s) ηt(s) 2 ds,
′
−2 Z0 k kM
satisfying the equality (see e.g. [15])
Π = Tη,η .
−h iM
Hence, recalling that b(u,u,u) = 0, the basic multiplication of (3.2) by U in gives
H
d
(4.2) +β u 2 +Π = f,u .
dtE k k ϑ h i
−
10 F. DI PLINIO,A. GIORGINI,V. PATAAND R.TEMAM
Next, inordertohandlethepossiblesingularityofµatzero, borrowinganideafrom[20]
we fix s > 0 such that
∗ s∗ κ
µ(s)ds .
≤ 2
Z0
Setting
µ(s ), 0 < s s ,
µ (s) = ∗ ≤ ∗
∗ (µ(s), s > s ,
∗
we consider the further functional
4
Φ(t) = ∞µ (s) ηt(s),u(t) ds.
1
−κ ∗ h i
Z0
Lemma 4.6. For any ε > 0, we have the differential inequality
d 8µ(s ) 4
(4.3) dtΦ+kuk21 ≤ κ2∗ Π+ ακεkηk2 +αεk∂tuk21.
M
Proof. We start from the identity
d 4 4 4
Φ+ ∞µ (s) u 2ds = ∞µ (s) Tη(s),u ds ∞µ (s) η(s),∂ u ds.
dt κ ∗ k k1 −κ ∗ h i1 − κ ∗ h t i1
Z0 Z0 Z0
According to the assumptions on µ and the definition of µ , we get
∗
4 4
∞µ (s) u 2ds ∞µ (s) u 2ds 2 u 2.
κ ∗ k k1 ≥ κ ∗ k k1 ≥ k k1
Z0 Zs∗
Integrating by parts, and using (2.2), we find the controls
4 4
∞ ∞
µ (s) Tη(s),u ds = µ(s) η(s),u ds
1 ′ 1
−κ ∗ h i −κ h i
Z0 Zs∗
4 2µ(s )
1
∗ u 1Π2
≤ κ k k
p
8µ(s )
u 2 + ∗ Π,
≤ k k1 κ2
and
4 4
∞ ∞
µ (s) η(s),∂ u ds µ (s) η(s) ∂ u ds
t 1 1 t 1
−κ ∗ h i ≤ κ ∗ k k k k
Z0 Z0
4
∞
µ(s) η(s) ∂ u ds
1 t 1
≤ κ k k k k
Z0
4
αε ∂ u 2 + η 2 .
≤ k t k1 ακεk kM
(cid:3)
Collecting the estimates above, we meet the thesis.
Finally, we define the functional
Ψ(t) = 2β u(t) 2 .
k k ϑ
−