Existence and uniqueness of global solutions for the modified anisotropic 3D Navier-Stokes equations

0.14 MB·

by  Hakima Bessaih

EXISTENCE AND UNIQUENESS OF GLOBAL SOLUTIONS FOR
THE MODIFIED ANISOTROPIC 3D NAVIER-STOKES
EQUATIONS
6
1
0
2 HAKIMABESSAIH,SABERTRABELSI,ANDHAMDIZORGATI

n
a Abstract. Westudyamodifiedthree-dimensionalincompressibleanisotropic
J Navier-Stokesequations. Themodificationconsistsintheadditionofapower
2 term to the nonlinear convective one. This modification appears naturally
2 in porous media when a fluid obeys the Darcy-Forchheimer law instead of
the classicalDarcy law. We prove global intimeexistence and uniqueness of
] solutions without assuming the smallness condition on the initial data. This
P improves the result obtained for the classical 3D incompressible anisotropic
A Navier-Stokes equations.
.
h
t
a MSC Subject Classifications: 35Q30,35Q35,76D05, 76D03, 76S05.
m
Keywords: Navier-Stokes equations, Brinkman-Forchheimer-extended Darcy
[ model, anisotropic viscosity.

1
Introduction
v
8 ThepurposeofthispaperistostudythefollowingmodifiedNavier-Stokessystem
7
1 ∂tu−ν∆hu+(u·∇)u+a|u|2αu = −∇p for (t,x)∈R3×R3,
6
0 S :  ∇·u = 0 for (t,x)∈R3×R3,
1. a 
0 u = u ,
|t=0 0
16 where ∂ denotes the partial derivative with respect to time, α ∈ R,a > 0,∆ :=
t h
:
v ∂12+∂22 and ∂i denotes the partial derivative in the direction xi. Clearly, S0 corre-
i sponds to the classical anisotropic Navier-Stokes equations. When a Coriolis force
X
1u×e is added, where e denotes the unit vertical vector and ǫ > 0 is the so
r ǫ 3 3
a called Rossby number, system S0 models rotating flows (see e.g. [12]). We refer
the reader to e.g. [6] and [2] where the relevance of considering anisotropic vis-
cosities of the form ν ∆ u+ǫν ∂2u is explained through the Ekman’s law. For a
h h v 3
complete discussionleading to the anisotropic Navier-Stokessystems, the reader is
referred to the book [12] or the introduction of the book [5]. System S with ∆
a h
replacedbytheclassicalLaplacianisnothingbutthethree-dimensionalBrinkman-
Forchheimer-extended Darcy model. The Brinkman-Forchheimer-extended Darcy
equations
∂ u−ν∆u+(u·∇)u+∇p+a|u|2αu=f, ∇·u=0, (1)
t
have been extensively studied. The existence of weak solutions for α ≥ 0 and
existence (for α ≥ 5) and uniqueness (for 5 < α ≤ 2) of strong solutions of
4 4
(1) is shown in [3]. Also, in [10], existence and uniqueness of weak and strong
solutions is shown for a larger range of α. In [7], the authors show existence and
1
2 HAKIMABESSAIH,S.TRABELSI,ANDHAMDIZORGATI
uniquenessofsolutionsforallα>1,withDirichletboundaryconditionsandregular
enough initial data. Their argument relies on the maximal regularity estimate
for the corresponding semi-linear stationary Stokes problem proved using some
modification of the nonlinear localization technique. In [10], the authors showed
theexistenceanduniquenessofweakandstrongsolutions,inparticularwithinitial
datain H1 insteadof H2 in [7] withperiodic boundaryconditions. Letus mention
thatthespaceL2α+2 appearsnaturallyinthe mathematicalanalysisofsystem(1),
and it coincides obviously with L4 for α=1. Since H˙ 12 is known to be the critical
Sobolev space for the classical 3D Navier-Stokes equations and H˙ 12 ⊂ L4, then it
makes sense to assume α > 1. For a more detailed discussion about equation (1)
and the various values of α that lead to the well posedness, we refer the reader to
[7, 10] and the references therein.
Beforegoingfurther,letusprecisethe notationandthe functionalsettingthatwill
be used along the paper. Since the horizontal variable x δ(x ,x ) does not play
h 1 2
thesameroleastheverticalvariablex ,itisnaturaltointroducefunctionalspaces
3
takinginto accountthis feature. These spacesarethe socalledanisotropicSobolev
spaces Hs,s′ for all s,s′ ∈ R. More precisely, the space Hs,s′ is the Sobolev space
withregularityHs inxh andHs′ inx3. Lethxidenotethequantityhxiδ(1+|x|2)12,
thenforalls,s′ ∈R, Hs,s′ isthe spaceoftempereddistributionsψ ∈S′(R3)which
satisfy
||ψ|| δ hξ′i2shξ i2s′|Fψ(ξ)|2dξ,
s,s′ 3
ZΩ
where ξ′ := (ξ1,ξ2) and F denotes the Fourier transform. The space ||ψ||Hs,s′
endowed with the norm || · || is a Hilbert space. Obviously, the homogenous
s,s′
anisotropic Sobolev space H˙s,s′(R3) are obtained by replacing h·i by |·|. We will
denote Lp(Lq) the space Lp(R ×R ;Lq(R )) endowed with the norm
h v x1 x2 x3
pq p1
||ψ|| δ||||ψ|| || = |ψ(x ,x )|qdx dx .
Lph(Lqv) Lq(Rx3) Lp(Rx1×Rx2) ZRx1×Rx2 ZRx3 h 3 3! h
Equivalently,wedenoteLq(Lp)thespaceLq(R (Lq(R ×R )))withtheassoci-
v h  x3 x1 x2 
atednorm givenby ||ψ|| :=||||ψ(·,x )|| || . The Lp(R3) norms
Lqv(Lph) 3 Lp(Rx1×Rx2) Lq(Rx3)
will be denoted ||·|| .
p
ThemathematicalanalysisoftheanisotropicNavier-StokessystemS wasoriginally
0
investigatedin[4]and[8]whereitisprovedthatthesystemS islocallywell-posed
0
for initial data in H0,s(R3) for all s> 1. Moreover,it has also been proved that if
2
the initial data u is such that
0
s−1 3−s
||u || 2 ||u ||2 ≤c,
0 L2(R3) 0 H˙0,s(R3)
for a sufficiently small constant c, then system S is globally well-posed. The
0
aim of this short paper is to show how the damping term |u|2αu gives rise to
a smoothing effect in the vertical velocity. Therefore it allows to get rid of the
smallness assumption (above) used in S . Even though, this result is still valid in
0
the critical Sobolev and Besov spaces H0,21 and B0,12 (see [11]), in order to avoid
technicalities, in this paper we chose to focus on a less optimal space to show how
we take advantage of the damping term. A similar result in the spaces H0,21 and
ANISOTROPIC 3D BRINKMAN-FORCHHEIMER-EXTENDED DARCY MODEL 3
B0,12 will be shown in a forthcoming paper soon. Here, we specifically show the
following
Theorem 0.1. Let a,ν >0,α>1 and u ∈H0,1(R3) such that divu =0. Then,
0 0
system S has a unique global solution u(t) satisfying
a
u(t)∈L∞(R3;H0,1(R3))∩L2 (R3;H1,1(R3))∩L2α+2(R3;L2α+2(R3)).
loc loc loc
Moreover, thesolution is in C0(R3;L2(R3))and depends continuouslyon theinitial
data.
Proof of Theorem 0.1
The rest of the paper is dedicated to the proof of Theorem 0.1. In general, the
proof is structured in four steps. First, one defines a family of approximate sys-
tems (San)n∈N andshow that this family has localin time smoothenoughsolutions
(u (t),p (t)). ThiscanbeachievedforinstancebytheclassicalFriedrich’smethod.
n n
Second, one proves uniform bounds for (u (t),p (t)) on some fixed time interval
n n
[0,T]. Next,oneshowsthatthesequenceofsolutionsto(San)n∈N convergestowards
some solution of S with adequate properties. Eventually, one exhibits a stability
a
kind estimate leading to the continuous dependence of the solutions on the initial
data, in particular their uniqueness. We refer to any textbook of fluid mechanics
fortechnicaldetailsofthisprocedure(seee.g. [1,13]). Toshortenthepresentation,
wewillonlypresentthenecessaryuniformboundsbyperformingformalcalculation
using system Sa instead of (San)n∈N and we will briefly explain how to pass to the
limit.
A priori Estimates. We start by looking for an L2 uniform estimate for the
velocity. For this purpose, we multiply the first equation of system S by u and
a
integrate1 over R3 to get
1 d
||u(t)||2+ν||∇ u(t)||2+a||u(t)||2α+2 =0,
2 dt 2 h 2 2α+2
thanks to the fact that (u·∇)u·udx=0. Now, we integrate this equality with
R3
respect to time
R
t t
||u(t)||2+2ν ||∇ u(τ)||2dτ +2a ||u(t)||2α+2dτ =||u ||2.
2 h 2 2α+2 0 2
Z0 Z0
This shows that if u ∈L2(R3), then for all t∈[0,T], it holds
0
u(t)∈L∞(R3;L2(R3))∩L2 (R3;H1,0(R3))∩L2α+2(R3;L2α+2(R3)). (2)
loc loc
Next, we multiply the first equation of S by −∂2u and integrate1 over R3 to get
3
1 d
||∂ u(t)||2+ν||∇ ∂ u(t)||2− (u(t)·∇)u(t)·∂2u(t)dx
2 dt 3 2 h 3 2 R3 3
Z
−a |u(t)|2αu(t)·∂2u(t)dx=0. (3)
3
R3
Z
1Thisshouldbedoneonthesmoothapproximatesolutionsun.
4 HAKIMABESSAIH,S.TRABELSI,ANDHAMDIZORGATI
Now, we handle the nonlinear terms. On the one hand, an integration by parts
leads clearly to the fact that
− |u|2αu·∂2udx= |∂ u|2|u|2αdx+2α (u·∂ )2|u|2α−2dx
3 3 3
R3 R3 R3
Z Z Z
=(1+2α)|||u|α∂ u||2. (4)
3 2
On the other hand, using integration by parts, we can write
3
− (u·∇)u·∂2udx= ∂ u ∂ u ∂ u dx
3 3 k k l 3 l
R3 R3
Z k,l=1 Z
X
2 3
= ∂ u ∂ u ∂ u dx
3 k k l 3 l
R3
k=1l=1 Z
XX
3
+ ∂ u ∂ u ∂ u dx:=T +T .
3 3 3 l 3 l 1 2
R3
l=1 Z
X
Now, integrating again by parts, we obtain that
2 3
T =− (u ∂ u ∂ ∂ u +u ∂ u ∂ ∂ u ) dx.
1 l 3 l k 3 k l 3 k k 3 l
R3
k=1l=1Z
XX
Moreover,usingthefactthat∇·u=0,wehave−∂ u =div u whereu δ(u ,u ).
3 3 h h h 1 2
Thus
3
T =− div u ∂ u ∂ u dx.
2 h h 3 l 3 l
R3
l=1Z
X
Next, using Ho¨lder and Young inequality, it is rather easy to see that for all f,g
and h, we have for all α>1 and ǫ ,ǫ >0
0 1
ZΩ fghdx≤ZΩ |f||g|α1 |g|1−α1 |h|dx≤|||f||g|α1||2α|||g|1−α1|||α2−α1 ||h||2
≤ 1 ||fαg||α2 ||g||2(1−α1)+ǫ ||h||2
4ǫ 2 2 0 2
0
1
≤ ǫ1 ||fαg||2+ ǫ11−α ||g||2+ǫ ||h||2.
4ǫ 2 4ǫ 2 0 2
0 0
Eventually, applying this inequality with f =u ,g =∂ u ,h=∂ ∂ u for the first
l 3 l k 3 k
part of T , f =u ,g =∂ u ,h=∂ ∂ u for the second part of T and proceeding
1 l 3 k k 3 l 1
equivalently for T , we obtain the existence of a constant γ > 0 independent of α
2
such that
1
γǫ γǫ1−α
(u·∇)u·∂2udx≤ 1 |||u|α∂ u||2+ 1 ||∂ u||2+γǫ ||∇ ∂ u||2. (5)
ZR3 3 4ǫ0 3 2 4ǫ0 3 2 0 h 3 2
The idea then is to tune ǫ and ǫ to compensate the first and third terms of
0 1
the right hand side of (5) using (4) and the second term of the left hand side of
(3). More precisely, setting ǫ = ν and ǫ = aν(1+4α) and using (4) and (5), the
0 2γ 1 γ2
equality (3) implies the existence of some η >0 such that
d
||∂ u(t)||2+ν||∇ ∂ u(t)||2+a|||u(t)|α∂ u(t)||2 ≤η||∂ u(t)||2.
dt 3 2 h 3 2 3 2 3 2
ANISOTROPIC 3D BRINKMAN-FORCHHEIMER-EXTENDED DARCY MODEL 5
Thus, thanks to Gronwall’s inequality, we obtain the following bound
||∂ u(t)||2 ≤||∂ u ||2eηt, for all t∈[0,T].
3 2 3 0 2
In particular, we have
t t
||∂ u(t)||2+ν ||∇ ∂ u(τ)||2dτ +a |||u(τ)|α∂ u(τ)||2dτ ≤ 1+eηt ||∂ u ||2.
3 2 h 3 2 3 2 3 0 2
Z0 Z0
(cid:0) (cid:1)
Thus, we infer
u(t)∈L∞(R3;H0,1)∩L2(R3;H1,1). (6)
Rigorously, these bounds hold for the approximate solutions constructed via the
Friederich’s regularization procedure. So, at this level, it remains only to pass to
the limit inthe sequence ofsolutionsof(San)n∈N. For thatpurpose,the mainpoint
to show is that
∂tu∈L2loc(R3,H−1(R3))+Ll1o+c2α1+1(R3,L1+2α1+1(R3))
= L2 (R3,H1(R3))∩L2α+2(R3,L2α+2(R3)) ⋆, (7)
loc loc
where the star stands(cid:0)for the dual symbol. Indeed, let us recall(cid:1)the Ladyˇzhenskaya
inequality, which is a special case of the Gagliardo-Nirenberg-Sobolev inequality
(see e.g. [9])
||ψ|| ≤δ ||ψ||41 ||∇ψ||34, for all ψ ∈H1(R3). (8)
4 1 2 2 0
Therefore, using Ho¨lder and Ladyˇzhenskayainequalities, we have
||(u·∇)u|| ≤||u|| ||∇u|| ≤δ ||u||14 ||∇u||47 ≤δ8||u||2+||∇u||2. (9)
4 4 2 1 2 2 1 2 2
3
Therefore (u·∇)u∈L2 (R3,H−1(R3)). Also, we have clearly
loc
|||u|2αu||1+2α1+1 =||u||2α+2, and (2α+2)−1+(1+1/2α+1)−1 =1.
1+ 1 2α+2
2α+1
Thus,(7)holdsthanksto(2),(6),(8)and(9). Recallthat(7)isneededinorderto
getsomecompactnessintime. Thepassagetothelimitfollowsusingusingclassical
argument by combining Ascoli’s theorem and the Cantor diagonal process.
Remark 0.2. It is rather standard to show that u ∈ C0(R3;L2) by using (6) and
(7).
Uniqueness. Now, we show the continuous dependence of the solutions on the
initial data, in particular their uniqueness. Let u(t) and v(t) be two solutions of
system S in the class u(t)∈L∞(R3;H0,1)∩L2(R3;H1,1). Let w(t) =u(t)−v(t),
then w satisfies
∂ w−ν∆ w+(w·∇)u+(v·∇)w+a|u|2αu−a|v|2αv = −∇(p −p ),
t h u v

∇·w = 0,

u = u −v ,
|t=0 0 0
We proceed as for the obtention of a priori estimates. Thanks to (7), the action of
∂ w on w leads to
t
1 d
||w||2+ν||∇ w||2+ (w·∇)u·wdx+a |u|2αu− |v|2αv wdx=0.
2dt 2 h 2 R3 R3
Z Z
(cid:0) (cid:1)
6 HAKIMABESSAIH,S.TRABELSI,ANDHAMDIZORGATI
On the one hand
2 3 3
(w·∇)u·wdx= w ∂ u w dx+ w ∂ u w dxδI +I .
k k l l 3 3 l l 1 2
R3 R3 R3
Z k=1l=1 Z l=1 Z
XX X
Now, using Ho¨lder inequality, it holds
2 3
I ≤ ||∂ u || ||w || ||w || dx
1 k l L2 k L4 l L4 3
R h h h
k=1l=1 Z
XX
2 3
≤ ||∂ u || ||w || ||w || .
k l L∞(L2) k L2(L4) l L2(L4)
v h v h v h
k=1l=1
XX
Now, using the Sobolev embedding H˙ 21 ֒→ L4 and interpolating H˙ 21 between H˙1
h h h h
and L2, we obtain clearly for all ψ ∈L2∩H˙1
h v h
1 1
||ψ|| ≤C||∇ ψ||2 ||ψ||2. (10)
L2(L4) h 2 2
v h
Also, we have
x3 d
||ψ(·,x )||2 = ||ψ(·,z)||2 dz
3 L2h Z−∞x3dz (cid:16) L2h(cid:17)
=2 ψ(x ,z)∂ ψ(x ,z)dx dz ≤2||ψ|| ||∂ ψ|| . (11)
h z h h 2 3 2
Z−∞ZR2
Therefore, using (10) with ψ =w and ψ =w and (11) with ψ =∂ u , we obtain
k l k l
thanks to Young’s inequality
I ≤C||∂ ∇ u||21 ||∇ u||12 ||∇ w|| ||w|| ≤ ν ||∇ w||2+C ||∂ ∇ u||2+||∇ u||2 ||w||2.
1 3 h 2 h 2 h 2 2 4 h 2 3 h 2 h 2 2
(cid:16) (cid:17)
Next, proceeding in the same way, we get
I ≤C||w || ||∂ ∇ u||12 ||∂ u||21 ||∇ w||12 ||w||12.
2 3 L∞(L2) 3 h 2 3 2 h 2 2
v h
But, using the fact that ∇·w=0, thus div w =−∂ w , we get
h h 3 3
x3
||w ||2 =2 w (x ,z)∂ w (x ,z)dx dz
3 L2 3 h 3 3 h h
h Z−∞ZR2
x3
=−2 w (x ,z)div w (x ,z)dx dz
3 h h h h h
Z−∞ZR2
≤2||div w || ||w || .
h h 2 3 2
Hence
I ≤C||div w ||21 ||w ||21 ||∂ ∇ u||12 ||∂ u||12 ||∇ w||12 ||w||12
2 h h 2 3 2 3 h 2 3 2 h 2 2
ν
≤ ||∇ w||2+C ||∂ ∇ u||2+||∂ u||2 ||w||2
4 h 2 3 h 2 3 2 2
(cid:16) (cid:17)
On the other hand, it is well known that there exists a nonnegative constant κ =
κ(α) such that
0≤κ|u−v|2 (|u|+|v|)2α ≤ |u|2αu−|v|2αv ·(u−v).
Thus (cid:0) (cid:1)
a |u|2αu− |v|2αv wdx≥aκ (|u|+|v|)2α w2dx=aκ||(|u|+|v|)α w||2.
2
R3 R3
Z Z
(cid:0) (cid:1)
ANISOTROPIC 3D BRINKMAN-FORCHHEIMER-EXTENDED DARCY MODEL 7
All in all, we have
d
||w||2+ν||∇ w||2+2aκ||(|u|+|v|)α w||2
dt 2 h 2 2
≤C ||∂ ∇ u||2+||∂ u||2+||∇ u||2 ||w||2
3 h 2 3 2 h 2 2
(cid:16) (cid:17)
Setting L(t) := C ||∂ ∇ u||2+||∂ u||2+||∇ u||2 and integrating the above in-
3 h 2 3 2 h 2
equality with respec(cid:16)t to time, we obtain that for a(cid:17)ll t∈[0,T]
t t
||w(t)||2+ν ||∇ w(τ)||2dτ +2aκ ||(|u|+|v|)α w(τ)||2dτ
2 h 2 2
Z0 Z0
t
≤||w ||2+ L(τ)||w(τ)||2dτ (12)
0 2 2
Z0
Now,since∂ u∈L∞(R3,L2(R3)),∂ ∇ u∈L2 (R3,L2(R3))and∇ u∈L2 (R3,L2(R3)),
3 loc 3 h loc h loc
it is clear that L(t) ∈ L1 (R3). Therefore, Gronwall’s Lemma applied to the in-
loc
equality (12) leads to the uniqueness for all α>1 and the proof is complete.
Acknowledgment: The research of Hakima Bessaih was partially supported
by NSF grantDMS-1418838. The researchof Saber Trabelsiwas supportedby the
King Abdullah University of Science and Technology.
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8 HAKIMABESSAIH,S.TRABELSI,ANDHAMDIZORGATI
(H. Bessaih) University of Wyoming, Department of Mathematics, Dept. 3036, 1000
EastUniversity Avenue,Laramie WY 82071,UnitedStates.
E-mail address: [email protected]
(S.Trabelsi)DivisionofMathandComputerSci. andEng.,KingAbdullahUniversity
ofScience andTechnology,Thuwal23955-6900,SaudiArabia
E-mail address: [email protected]
(H. Zorgati) D´epartementde Math´ematiques, Campus Universitaire, Universit´e Tunis
ElManar2092,Tunisia.
E-mail address: [email protected]