Steady compressible Nevier-Stokes flow in a square

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by  Tomasz Piasecki

Steady compressible Navier-Stokes flow in a square
9 Tomasz Piasecki
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MathematicalInstitute,Polish AcademyofSciences
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J ul. Sniadeckich 8, 00-956Warszawa, Poland

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e-mail: [email protected]

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- Abstract
h
t
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m We investigate a steady flow of compressible fluid with inflow boundary condition on the
[ density and slip boundary conditions on the velocity in a square domain Q R2. We show
∈
existence if a solution (v,ρ) W2(Q) W1(Q) that is a small perturbation of a constant flow
1 ∈ p × p
v (v¯ [1,0],ρ¯ 1). We also show that this solution is unique in a class of small perturbations
6 ≡ ≡
of the constant flow (v¯,ρ¯). In order show the existence of the solution we adapt the techniques
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9 know from the theory of weak solutions. We apply the method of elliptic regularization and a
3 fixed pointargument.
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MSC:35Q30;76N10
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v Keywords: Navier-Stokes equations, steady compressibleflow, inflow boundary condition, slip
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X boundaryconditions,strongsolutions
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1 Introduction and main results
The problems of steady compressible flows described by the Navier-Stokes equations are usu-
allyconsidered withthehomogeneousDirichletboundaryconditionsonthevelocity. Itis worth
fromthemathematicalpointofview,aswellasintheeyeofapplications,toinvestigatedifferent
types of boundary conditions. A significant feature of the compressible Navier-Stokes system
is its mixed character: the continuity equation is elliptic in the velocity whereas the continuity
equation is hyperbolic in the density. If we assume that the flow enters the domain, then the
hyperbolicityofthecontinuityequationmakesitnecessarytoprescribethedensityontheinflow
part of the boundary. A time-dependent compressible flow with inflow boundary condition has
been considered by Valli and Zajaczkowski in [21]. The authors showed existence of a global
in time solutions under some smallness assumptions on the data. They also obtained a stability
1
result and existence of a stationary solution. Plotnikov and Sokolovski investigated shape opti-
mization problems with inflow boundary condition in 2D [16] and 3D [15], working with weak
solutions. Regularsolutionstoproblemswithinflowboundaryconditionshavebeeninvestigated
mainly by Kweon in a joint work with Kellogg [7] and with Song [9]. The results obtained by
these authors require some assumptions on the geometry of the boundary in the neighbourhood
ofthepointswheretheinflowandoutflowpartsoftheboundarymeet. In[8]KweonandKellogg
investigatedthecasewhen theinflowand outflowparts oftheboundaryare separated, obtaining
regular solutions. What seems to be interesting is to investigatean inflow condition on the den-
sitycombined with slipboundary conditionson thevelocity,that allow to describeprecisely the
action between the fluid and the boundary. The slip boundary conditionshavebeen investigated
byMucha[10]forincompressibleflows,andalsobyFujita[4]and MuchaandPokorny[11]for
compressibleflows.
Here we investigate a steady flow of a viscous, barotropic, compressible fluid in a square
domaininR2 satisfyinginhomogeneousslipboundaryconditionsonthevelocitycombinedwith
an inflow condition on the density. We impose that there is no flux across the bottom and the
topofthesquare,sothatitcanbeconsideredafinite,twodimensionalpipe. From theanalytical
point of view our domain prevents the singularity that appears in a general domain where the
inflowand outflowpartsoftheboundarycoincide.
Weshowexistenceofasolutionthatcanbeconsideredasaperturbationofaconstantsolution
(v¯ (1,0),ρ¯ 0). Under some smallness assumptions we can show an a priori estimate in
≡ ≡
a space W2(Q) W1(Q) that is crucial in the proof of existence of the solution. Now let us
p × p
formulatetheproblemunderconsiderationmoreprecisely.
The stationary compressible Navier-Stokes system describing the motion of the fluid, sup-
pliedwiththeslipboundaryconditions,reads
ρv v µ∆v (µ+ν) div v + p(ρ) = 0 in Q,
·∇ − − ∇ ∇
div (ρv) = 0 in Q,
n T(v,ργ) τ +fv τ = b on Γ, (1.1)
· · ·
n v = d on Γ,
·
ρ = ρ on Γ ,
in in
where Q = [0,1] [0,1] is a square domain in R2 with the boundary Γ and Γ = x Γ :
in
× { ∈
v¯ n(x) < 0 . We will also denote Γ = x Γ : v¯ n(x) > 0 and Γ = x Γ :
out 0
· } { ∈ · } { ∈
v¯ n(x) = 0 . Next,b W1−1/p(Γ),d W2−1/p(Γ)andρ W1−1/p(Γ )aregivenfunctions.
p p in p in
· } ∈ ∈ ∈
v = (v(1),v(2)) is the velocity field of the fluid and ρ is the density of the fluid. We assume that
thepressureisafunctionofthedensityoftheformp(ρ) = ργ forsomeγ > 1. Theoutwardunit
normal and tangent vectors are denoted respectively by n and τ. We assume d = 0 on Γ , what
0
meansthat thereisno flowacross thesepartsoftheboundary. Moreover,
T(v,p) = 2µD(v)+νdivvI pI
−
isthestresstensorand
1
D(v) = vi +vj
2{ xj xi}i,j=1,2
2
is the deformation tensor. µ and ν are viscosity constants satisfying µ > 0 and ν + 2µ > 0
and f > 0 is a friction coefficient. The slip boundary conditions (1.1) are supplied with the
3,4
condition (1.1) prescribing the values of the density on the inflow part of the boundary. Under
5
theassumptionsonµandν themomentumequation(1.1) isellipticinu,whereasthecontinuity
1
equation(1.1) is hyperbolicin ρ.
2
Our method would also work with no modification if we considered a perturbation of the
constant flow (v¯,ρ¯) satisfying(1.1) with a term ρF on the r.h.s providedthat F was small
1 || ||Lp
enough.
SinceT(v¯,ρ¯γ) = 0,theconstantflow(v¯,ρ¯)fulfillsequations(1.1)withboundaryconditions
fv¯ τ = b fτ(1) and n v¯ = d fτ(1).
· − · −
Ourmainresult is
Theorem 1. Assumethat b fτ(1) , d n(1) and ρ 1 are
|| − ||Wp1−1/p(Γ) || − ||Wp2−1/p(Γ) || in− ||Wp1−1/p(Γin)
small enough and f is large enough. Then there exists a solution (v,ρ) W2(Q) W1(Q) to
∈ p × p
thesystem(1.1)and
v v¯ + ρ ρ¯ E, (1.2)
W2 W1
|| − || p || − || p ≤
where E is a constant depending on the data, i.e. on d, ρ , b, the constants in the equation and
in
thedomain,thatcan bearbitrarilysmallprovidedthatthedataissmallenough.
Moreover, if (v ,ρ ) and (v ,ρ ) are two solutions to (1.1) satisfying the estimate (1.2) then
1 1 2 2
(v ,ρ ) =(v ,ρ ).
1 1 2 2
ThereareseveraldifficultiesintheproofofTheorem1thatresult,roughlyspeaking,fromthe
mixedcharacteroftheproblem. Inageneraldomainasingularityappearsinthepointswherethe
inflowandoutflowpartsoftheboundarymeetandwecannotapplythemethodusedinthispaper
to obtain an a priori estimate. However, there is another difficulty in the analysis of the steady
compressible Navier-Stokes system, independent on the domain. This difficulty lies in the term
u w. Namely,ifwewanttoapplysomefixedpointmethodthenthistermmakesitimpossible
·∇
to show the compactness of the solution operator. We overcome this difficulty applying the
method of elliptic regularization. We solve a sequence of approximate elliptic problems and
showthat thissequenceconvergestothesolutionof(1.1). Thisisawell-knownmethodthathas
been usuallyappliedtotheissueofweaksolutions([14],[11]), anddiffersfromtheapproach of
Kweon andKelloggusedto deriveregularsolutionsin[7], [8].
Letusnowoutlinethestrategyoftheproof,andthusthestructureofthepaper. Insection2we
start with removinginhomogeneityfrom the boundaryconditions(1.1) . It leads to thesystem
3,4
(2.3),andwecanfocusonthissysteminsteadof(1.1). Inthesamesectionwedefineanǫ-elliptic
regularizationtothesystem(2.3)andintroduceitslinearization(2.4). Insection3wederiveanǫ-
independentestimateonasolutionofthelinearizedellipticsystem(Theorem2). Althoughlinear,
the system (2.4) has variablecoefficients and thus its solution is not straightforward. In order to
solve (2.4) we apply the Leray-Schauder fixed point theorem in section 4, using a modification
of the estimatefrom Theorem 2. In section 5 we use the a priori estimateto apply the Schauder
fixed point theorem to solve the approximate elliptic systems. In section 6 we prove our main
result, Theorem 1. The proof is divided into two steps. First we show that the sequence of
approximate solutions converges to the solution of (2.3) and thus prove the existence of the
3
solutionto(1.1)satisfyingtheestimate(1.2). Nextweshowthatthissolutionisuniqueinaclass
ofsmallperturbationsof theconstant flow (v¯,ρ¯). We seethat theestimatefrom Theorem 3 isin
fact used at threestagesoftheproof,therefore weshowit inadetailed way insection3.
2 Preliminaries
In this section we remove the inhomogeneity from the boundary conditions (1.1) . Then we
4,5
defineanǫ-ellipticregularizationtothesystem(1). Wealsomakesomeremarksconcerningthe
notation. Let us constructu W2(Q) andw W1(Q) such that
0 ∈ p 0 ∈ p
n u = d n(1) and w = ρ 1. (2.1)
· 0|Γ − 0|Γin in −
Duetotheassumptionofsmallnessofd n(1) andρ 1 wecan assumethat
− |Γ in − |Γin
u , w << 1. (2.2)
0 W2 0 W2
|| || p || || p
Nowweconsider
u = v v¯ u and w = ρ ρ¯ w .
0 0
− − − −
Onecan easilyverify that(u,w)satisfies thefollowingsystem:
∂ u µ∆u (ν +µ) divu+γ(w+w +1)γ−1 w = F(u,w) in Q,
x1 − − ∇ 0 ∇
(w+w +1)divu+∂ w+(u+u ) w = G(u,w) in Q,
0 x1 0 ·∇
n 2µD(u) τ +f u τ = B on Γ, (2.3)
· · ·
n u = 0 on Γ,
·
w = 0 on Γ ,
in
where
F(u,w) = (w +w +1)(u u+u u ) w(u u )
0 0 0 0 0
− ·∇ ·∇ − ·∇
(w+w +1)u u γ(w+w +1)γ−1 w +µ∆u +(ν+µ) divu (w +1)u u ,
0 0 0 0 0 0 0 0
− ·∇ − ∇ ∇ − ·∇
G(u,w) = (w +w +1)divu (u+u ) w ∂ w
− 0 0− 0 ·∇ 0 − x1 0
and
B = b 2µn D(u ) τ fτ(1).
0
− · · −
InordertoproveTheorem1itisenoughtoprovetheexistenceofasolution(u,w)tothesystem
(2.3) provided that u , w and B are small enough. As we already men-
|| 0||Wp2 || 0||Wp1 || ||Wp1−1/p(Γ)
tioned,thepresenceofthetermu winthecontinuityequationmakesitimpossibletoshowthe
·∇
compactness of a solution operator if we try to apply fixed point methods directly to the system
(2.3). We overcome this difficulty applying the method of elliptic regularization. The method
consists of adding an elliptic term ǫ∆w to the r.h.s of (2.3) and introducing an additional
2
−
Neumann boundary condition. Since the density is already prescribed on the inflow part of the
boundaryby(2.3) ,weimposetheNeumannconditiononlyontheremainingpartofthebound-
5
ary. WhilewearepassingtothelimitwiththedensityinW1 -norm,theNeumannconditionwill
p
4
disappear. Similarapproachhasbeenappliedtotheissueofinviscidlimitfortheincompressible
Eulersystemin [6]. Considera followinglinearsystemwithvariablecoefficients:
∂ u µ∆u (ν +µ) divu +γ(w¯ +w +1)γ−1 w = F (u¯,w¯) in Q,
x1 ǫ − ǫ − ∇ ǫ 0 ∇ ǫ ǫ
(w¯ +w +1)divu +∂ w +(u¯+u ) w ǫ∆w = G (u¯,w¯) in Q,
0 ǫ x1 ǫ 0 ·∇ ǫ − ǫ ǫ
n 2µD(u ) τ +f u τ = B on Γ,
· ǫ · ǫ · (2.4)
n u = 0 on Γ,
ǫ
·
w = 0 on Γ ,
ǫ in
∂wǫ = 0 on Γ Γ .
∂n \ in
where (u¯,w¯) W2(Q) W1(Q) are given functions and F (u¯,w¯) and G (u¯,w¯) are regular-
∈ p × p ǫ ǫ
izations to F(u¯,w¯) and G(u¯,w¯) obtained by replacing the functions u and w by their regular
0 0
approximationsuǫ andwǫ.
0 0
Let us definean operatorT : W2(Q) W1(Q) W2(Q) W1(Q) :
ǫ D ⊂ p × p → p × p
(u ,w ) = T (u¯,w¯) (u ,w ) isa solutionto(2.4), (2.5)
ǫ ǫ ǫ ǫ ǫ
⇐⇒
where is a subset of W2(Q) W1(Q) that we will define later. Using the operator T we
D p × p ǫ
definean ǫ -ellipticregularizationto thesystem(2.3).
Definition 1. Byanǫ -ellipticregularizationtothesystem(2.3)we meana system
(u ,w ) = T (u ,w ). (2.6)
ǫ ǫ ǫ ǫ ǫ
Wewanttoshowtheexistenceofasolutiontotheǫ-ellipticregularizationtothesystem(2.3)
applyingtheShauderfixed pointtheorem. Thestrategyhasbeen outlinedin theintroduction. In
section4weshowthatT iswelldefined,whichmeansthatforgiven(u¯,w¯)thereexistsaunique
ǫ
solution to (2.4) (Theorem 3). In fact we show that T is well defined for ǫ small enough, but it
ǫ
suffices sinceweare interestedin smallvaluesofǫ.
In section 5 we show that T satisfies the assumptions of the Schauder fixed point theorem
ǫ
and thuswesolvethesystem(2.6)forǫsmallenough.
Aswealreadysaid,thekeypointistoderiveanǫ-independentestimateforthesystem(2.4),
whichisusedatdifferentstagesoftheproof. Wederivesuchestimateinthenextsection. Before
weproceed, wewillfinish thisintroductorypart withafew remarksconcerning notation.
Forsimplicitywewilldenote
γ(w¯ +w +1)γ
0
a (w¯) = ,
0
ν +2µ
(2.7)
a (w¯) = γ(w¯ +w +1)γ−1,
1 0
a (w¯) = γ(w¯ +w +1)γ−2.
2 0
ByC wewilldenoteaconstantthatdependonthedataandthuscanbecontrolled,notnecesarily
arbitrarily small. If the constant depend not only on the data, but also on ǫ, we will denote it by
C . Finally, by E we will denote a constant dependent on the data that can be arbitrarily small
ǫ
providedthatthedatais smallenough.
5
SincewewillusuallyusethespacesoffunctionsdefinedonQ,wewillomitQinthenotation
of a space, for example we will denote the space L (Q) by L . The spaces of functions defined
2 2
on theboundarywillbedenoted byL (Γ)etc.
2
We do not distinguish between the spaces of vector-valued and scalar-valued functions, for
examplewewillwriteu W2 insteadofu (W2)2.
∈ p ∈ p
3 A priori estimate for the linearized elliptic system
In this section we show an ǫ - independent estimate on u + w , where (u ,w ) is a
ǫ W2 ǫ W1 ǫ ǫ
|| || p || || p
solution to (2.4). The first step is an estimate in H1 L . Next we eliminate the term divu
2
×
from the second equation applying the Helmholtz decomposition and the properties of the slip
boundaryconditions. Then wederivethehigherestimateusinginterpolation.
3.1 Estimate in H1 L
2
×
InordertoproveaprioriestimatesonH1 -normofthevelocityandL2 -normofthedensityfor
thesystem(2.4)let usdefinea space
V = v H1(Q;R2) : v n = 0 . (3.1)
Γ
{ ∈ · | }
Theestimateisstatedin thefollowinglemma.
Lemma1. Assumethatǫ, u¯ and w¯ aresmallenoughandf islargeenough. Thenfor
W2 W1
|| || p || || p
sufficientlysmoothsolutionsto system(2.4)thefollowingestimateisvalid
||u||W21 +||w||L2 ≤ C ||F(u¯,w¯)||V∗ +||G(u¯,w¯)||L2 +||B||L2(Γ) +E||w||Wp1 . (3.2)
(cid:2) (cid:3)
where V∗ isthedualspaceof V.
Before westarttheproof, weshallmakearemark concerningtheterm w , thatisrather
W1
|| || p
unexpected in an energy estimate. Its presence is due to the functions a (w¯) and (w¯ +w +1)
1 0
on the r.h.s. of (2.4). However, this term does not cause any problems when we apply (3.2) to
interpolateintheproofofTheorem2, sinceit ismultipliedbyasmallconstant.
Proof. The proofis dividedinto threesteps. First wemultiply(2.4) by u and integrateover
1
Q. We obtain an estimate on u in terms of the data and w . Then we apply the second
|| ||H1 || ||L2
equationtoestimate w and finallycombinetheseestimatestoobtain(3.2).
|| ||L2
Step1. Wemultiply(2.4) byuandintegrateoverQ. Usingtheboundaryconditions(2.4)
1 3,4
weget
2µD2(u)+νdiv2udx+ (f + n(1)) u 2dσ + [a (w¯)] wudx =
Q Γ 2 | | Q 1 ∇ (3.3)
R + FuRdx+ B(u τ)dσ.R
Q Γ ·
R R
Theboundarytermonthel.h.swillbepositiveprovidedthatf islargeenough. Nextweintegrate
by partsthelastterm ofthel.h.sof(3.3). Using(2.4) weobtain:
2
[a (w¯)] wudx = [a (w¯)]div uwdx uw [a (w¯)]dx =
1 1 1
Z ∇ −Z −Z ∇
Q Q Q
6
[a (w¯)] 1 1
= 2 w2n(1)dσ w2[∂ a (w¯)+(u¯+u ) a (w¯)]dx [a (w¯)]div(u¯+u )w2dx
Z 2 −2 Z x1 2 0 ∇ 2 −2 Z 2 0
Γ Q Q
[a (w¯)]G(u¯,w¯)wdx ǫ [a (w¯)]w∆wdx uw [a (w¯)]dx.
2 2 1
−Z − Z −Z ∇
Q Q Q
Sincen(1) 1,using(3.3)and theKorninequality((7.1), Appendix)weget:
|Γout ≡
C u 2 + [a (w¯)]w2dσ
Q|| ||W21 Γout 2 ≤
R
a (w¯)div(u¯+u )w2dx+ [a (w¯)]Gwdx+ Fudx+ B(u τ)dσ
2 0 2
≤ Z Z Z Z ·
Q Q Q Γ
I1 I2
+ǫ |[a (w¯)]w∆w{dzx+ uw }[a|(w¯)]dx+ w2[∂ a{(zw¯)+(u¯+u ) a (w¯)}]dx.
Z 2 Z ∇ 1 Z x1 2 0 ∇ 2
Q Q Q
I3 I4 I5
(3.4)
| {z } | {z } | {z }
Obviouslywe haveI E w . Nowwe haveto deal withthe term with∆w. Dueto the
1 ≤ || ||L2
boundaryconditions(2.4) wehave
5,6
I = ǫ [a (w¯)]w∆wdx = ǫ [a (w¯)] w 2dx ǫ w [a (w¯)] wdx. (3.5)
3 2 2 2
Z − Z |∇ | − Z ∇ ∇
Q Q Q
UsingHölderinequalityweget
w [a (w¯)] wdx [a (w¯)] w w
|Z ∇ 2 ∇ | ≤ ||∇ 2 ||Lp|| ∇ ||Lp∗ ≤
Q
[a (w¯)] w w C w 2 ,
≤ ||∇ 2 ||Lp||∇ ||L2|| ||Lq ≤ ||∇ ||L2
where q = 2p < + and p∗ = p . Thus the term with ǫ on the r.h.s of (3.4) will be negative
p−2 ∞ p−1
providedthat w¯ willbesmallenough. Next,
W1
|| || p
I uw [a (w¯)]dx C [a (w¯)] u w E u 2 + w 2 .
4 ≤ |Z ∇ 1 | ≤ ||∇ 1 ||Lp|| ||W21|| ||L2 ≤ || ||W21 || ||L2
Q (cid:0) (cid:1)
The last term of the r.h.s. is the most inconvenient and it must be estimated by W1 - norm of
p
w, and this is the reason why this term appears in (3.2). Fortunately it is multiplied by a small
constantwhat willturn outvery importantintheproofofTheorem2. Wehave
I C a (w¯) w 2 E w 2 .
5 ≤ || 2 ||Wp1|| ||Wp1 ≤ || ||Wp1
Provided that the data is small enough, using the trace theorem to estimate the boundary term
and theHölderinequalityweget
u 2 +C w2dσ
|| ||W21 Γout ≤ (3.6)
≤ C ||F(u¯,w¯)||V∗ +||G(u¯,w¯)||L2 +||BR||L2(Γ) (||u||W21 +||w||L2)+E||w||2Wp1.
(cid:2) (cid:3)
7
Step 2. In order to derive (3.2) from (3.6) we need to find a bound on w . From (2.4)
|| ||L2 2
wehave
∂ w = G (u¯+u ) w (w¯ +w +1)div u+ǫ∆w,
x1 − 0 ·∇ − 0
thus
x1
w2(x ,x ) = w2(0,x )+ 2ww (s,x )ds =
1 2 2 s 2
Z
0
x1 x1 x1
2w[G (w¯ +w +1)divu]ds 2w (u¯+u ) w ds+2ǫ w ∆w dx.
0 2 0 2 2 2
Z − −Z ·∇ Z
0 0 0
S1 S2
S c|an beestimatedd{izrectly: }| {z }
1
S ( G +C u ) w . (3.7)
Z 1 ≤ || ||L2 || ||H1 || ||L2
Q
It isalittlemorecomplicatedto estimateS . Wehave
2
x1 x1 x1
S = (u¯+u )(1)∂ w2(s,x )ds (u¯+u )(2)∂ w2(s,x )ds+2ǫ w∆w.
2 −Z 0 s 2 −Z 0 x2 2 Z
0 0 0
Now we integrate both components by parts. In the second component we use the fact that the
integrationintervaldoesnot dependon x . We get
2
x1
S = (u¯+u )(1)w2(x ,x )+ (u¯+u )(1)w2(s,x )ds
2 − 0 1 2 Z 0 x1 2
0
∂ x1 x1 x1
(u¯+u )(2)w2(s,x )ds+ (u¯+u )(2)w2(s,x )ds+2ǫ w∆w =
−∂x Z 0 2 Z 0 x2 2 Z
2 0 0 0
x1 ∂ x1
= (u¯+u )(1)w2(x ,x )+ w2div (u¯+u )(s,x )ds (u¯+u )(2)w2(s,x )ds+
0 1 2 0 2 0 2
− Z − ∂x Z
0 2 0
x1
2ǫ w∆w =: S1 +S2 +S3 +S4.
2 2 2 2
Z
0
TheintegralsofS1 and S2 can beestimatedin adirectway:
2 2
S1 , S2 E w 2 . (3.8)
Z | 2| Z | 2| ≤ || ||L2
Q Q
Next,
∂ x1 x1
S3 = u(2)w2(s,x )ds dx = n(2) (u¯+u )(2)w2(s,x )ds dσ.
Z 2 Z ∂x Z 2 Z Z 0 2
Q Q 2(cid:2) 0 (cid:3) Γ (cid:2) 0 (cid:3)
Now we remind that w = 0 on Γ . Moreover, the boundary conditions yields (u¯ +u )(2) = 0
in 0
on Γ . Finally,on Γ wehaven(2) = 0. Thus
0 out
S3 = 0. (3.9)
2
Z
Q
8
Finally,
1 1 x1 1
S4dx = w∆w(s,x )dsdx dx = w∆w(x)dx dx ,
2 2 2 1 1
Z Z Z Z Z Z
Q 0 (cid:2) 0 0 (cid:3) 0 (cid:2) Px1 (cid:3)
whereP := [0,x ] [0,1]. We have
x1 1 ×
(2.4)5,6 1
w∆wdx = w 2dx+ w w ndσ ww (x ,x )dx ,
Z −Z |∇ | Z ∇ · ≤ Z x1 1 2 2
Px1 Px1 ∂Px1 0
thus
1 1
S4dx 2ǫ ww (x ,x )dx dx = ǫ ∂ w2dx = ǫ w2n(1)dσ. (3.10)
Z 2 ≤ Z Z x1 1 2 2 1 Z x1 Z
Q 0 0 Q Γout
Combining(3.8), (3.9)and (3.10)weget
S = S1 +S2 +S3 +S4 E w 2 +ǫ w2dσ.
Z 2 Z 2 2 2 2 ≤ || ||L2 Z
Q Q Γout
Combiningthisestimatewith(3.7)weget:
w 2 C G(u¯,w¯) + u 2 +E w +ǫ w2dσ,
|| ||L2 ≤ || ||L2 || ||W21 || ||L2 Z
(cid:0) (cid:1) Γout
and thus
w 2 C G(u¯,w¯) + u 2 +Cǫ w2dσ. (3.11)
|| ||L2 ≤ || ||L2 || ||W21 Z
(cid:0) (cid:1) Γout
Step 3. Substituting(3.11)to(3.6)weget:
u 2 + w2dσ CD( u + w )+CD2 +E w 2 , (3.12)
|| ||W21 Z ≤ || ||W21 || ||L2 || ||Wp1
Γout
whereD = F(u¯,w¯) V∗ + G(u¯,w¯) L2+ B L2(Γ). Combiningthisinequalitywith(3.11)we
|| || || || || ||
get
( u + w )2 +(C ǫ) w2dσ CD( u + w )+D2 +E w 2 ,
|| ||W21 || ||L2 − Z ≤ || ||W21 || ||L2 || ||Wp1
Γout
thusforǫsmallenoughweobtain(3.2). (cid:3)
3.2 Estimate for u + w
W2 W1
|| || p || || p
The following theorem gives an ǫ - independent estimate on u + w where (u ,w )
ǫ W2 ǫ W1 ǫ ǫ
|| || p || || p
isasolutionto (2.4).
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Theorem 2. Suppose that (u ,w ) is a solution to (2.4). Then the following estimate is valid
ǫ ǫ
providedthatthedata, u¯ and w¯ aresmallenoughandf islargeenough.
W2 W1
|| || p || || p
u + w C F (u¯,w¯) + G (u¯,w¯) + B , (3.13)
|| ǫ||Wp2 || ǫ||Wp1 ≤ || ǫ ||Lp || ǫ ||Wp1 || ||Wp1−1/p(Γ)
(cid:2) (cid:3)
where theconstantC dependson thedatabut doesnotdepend onǫ.
The proof will be divided into three lemmas. In the first lemma we eliminate the term divu
from (2.4) .
2
Lemma 2. Let usdefine
H¯ := (ν +2µ)divu +[a (w¯)]w . (3.14)
ǫ 1 ǫ
−
where (u ,w ) isa solutionto(2.4)anda (w¯) is definedin (2.7). Then
ǫ ǫ 1
H¯ C F (u¯,w¯) + B + u +E w (3.15)
||∇ ||Lp ≤ h|| ǫ ||Lp || ||Wp1−1/p(Γ) || ||Wp1−1/p(Γ)i || ||Wp1
andw satisfiesthefollowingequation
ǫ
[a (w¯)]w+w +(u¯+u ) w ǫ∆w = H˜, (3.16)
0 x1 0 ·∇ −
where
¯
H(w¯ +w +1)
H˜ = 0 +G(u¯,w¯). (3.17)
ν +2µ
Proof. Letus rewrite(2.4) as
1
∂ u µ∆u (ν +µ) divu +γ w = F (u¯,w¯) [a (w¯) γ] w .
x1 ǫ − ǫ − ∇ ǫ ∇ ǫ ǫ − 1 − ∇ ǫ
Takingthetwodimensionalvorticityof(2.4) weget
1
∂ α µ∆α = rot[F (u¯,v¯) (a (w¯) γ) w ] in Q,
x1 ǫ − ǫ ǫ − 1 − ∇ ǫ (3.18)
α = f(u τ)+ B on Γ,
ǫ −µ ǫ · µ
where α = rot u = u(2) u(1) . The boundary condition (3.18) has been shown in [10] in a
ǫ ǫ ǫ,x1 − ǫ,x2 2
moregeneralcase;asimplificationofthisproofyields(3.18) . Sinceourdomainisasquare,we
2
canusethesymmetrytodealwithcornersingularitesandapplythestandardLp theoryofelliptic
equations([5])to obtaintheestimate
f B
α C F (u¯,w¯) + (a (w¯) γ) w + (u τ)+ . (3.19)
|| ǫ||Wp1 ≤ || ǫ ||Lp(Q) || 1 − ∇ ǫ||Lp ||−µ ǫ· µ||Wp1−1/p(Γ)
(cid:2) (cid:3)
Fromthedefinitionofa (w¯)(2.7)weseethat (a (w¯) γ) canbearbitrarilysmallprovided
1 || 1 − ||L∞
that w¯ is small enough. Moreover, from the boundary condition (2.4) we have u =
W1 4 ǫ
|| || p
τ(u τ) onΓ, thus(3.19)can berewrittenas
ǫ
·
α C F (u¯,w¯) + B + u +E w . (3.20)
|| ǫ||Wp1 ≤ || ǫ ||Lp(Q) || ||Wp1−1/p(Γ) || ǫ||Wp1−1/p(Γ) || ǫ||Wp1
(cid:2) (cid:3)
10