Leray-Hopf and Continuity Properties for All Weak Solutions for the 3D~Navier-Stokes Equations

0.09 MB·English

by  Nataliia V. Gorban

T
Leray-Hopf and Continuity Properties for All Weak Solutions for
F
the 3D Navier-Stokes Equations
5
1
0
October12,2015
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t
c A
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NataliiaV.Gorban1,PavloO.Kasyanov2,OlhaV.Khomenko3,andLuisaToscano4.

8

]
P Abstract
A
. Inthisnoteweprovethateachweaksolutionforthe3DNavier-StokessystemsatisfiesLeray-Hopf
h
property. Moreover,each weak solutionis rightlycontinuousin the standardphasespace H endowed
t
a
withthestrongconvergencetopology.
m
[

2 1 IntroductionRand Main Result
v
9
7 LetΩ ⊂ R3 beabounded domainwithrathersmoothboundary Γ = ∂Ω,and[τ,T]beafixedtimeinterval
6
with−∞ < τ < T < +∞. Weconsider3DNavier-Stokes system inΩ×[τ,T]
0
0
. ∂y
1 −ν△y+(y·∇)y = −∇p+f, divy = 0,
0  ∂t (1.1)
5  y = 0, y = y ,
1 (cid:12)Γ (cid:12)t=τ τ
:  (cid:12) (cid:12)
v where y(x,t) means the unknown velocity, p(x,t) the unknown pressure, f(x,t) the given exterior force,
i D
X andy (x)thegiveninitialvelocitywitht ∈[τ,T],x ∈ Ω,ν > 0meanstheviscosity constant.
τ
r ThroughoutthisnoteweconsidergeneralizedsettingofProblem(1.1). Forthispurposedefinetheusual
a
function spaces
V = {u ∈ (C0∞(Ω))3 :divu= 0}, Vσ = cl(H0σ(Ω))3V, σ ≥ 0,
wherecl denotestheclosureinthespaceX. SetH := V ,V := V . ItiswellknownthateachV ,σ > 0,
X 0 1 σ
∗ ∗
is a separable Hilbert space and identifying H and its dual H we have V ⊂ H ⊂ V with dense and
σ σ
1InstituteforAppliedSystemAnalysis,NationalTechnicalUniversityofUkraine“KyivPolytechnicInstitute”,Peremogyave.,
37,build,35,03056,Kyiv,Ukraine,nata [email protected]
2InstituteforAppliedSystemAnalysis,NationalTechnicalUniversityofUkraine“KyivPolytechnicInstitute”,Peremogyave.,
37,build,35,03056,Kyiv,Ukraine,[email protected].
3InstituteforAppliedSystemAnalysis,NationalTechnicalUniversityofUkraine“KyivPolytechnicInstitute”,Peremogyave.,
37,build,35,03056,Kyiv,Ukraine,[email protected]
4University of Naples “Federico II”, Dep. Math. and Appl. R.Caccioppoli, via Claudio 21, 80125 Naples, Italy,
[email protected]
1
T
compactembeddingforeachσ > 0. Wedenoteby(·,·), k·kand((·,·)), k·k theinnerproductandnorm
V
∗
in H and V, respectively; h·,·i will denote pairing between V and V that coincides on H ×V with the
innerproduct(·,·). LetH bethespaceH endowedwiththeweaktopology. Foru,v,w ∈ V weput
w
F3 ∂v
j
b(u,v,w) = u w dx.
i j
Z ∂x
ΩiX,j=1 i
It is known that b is a trilinear continuous form on V and b(u,v,v) = 0, if u,v ∈ V. Furthermore, there
existsapositiveconstantC suchthat
|b(u,v,w)| ≤ Ckuk kvk kwk , (1.2)
V V V
foreachu,v,w ∈ V;see,forexample,ASohr[17,LemmaV.1.2.1]andreferences therein.
Let f ∈ L2(τ,T;V∗) + L1(τ,T;H) and y ∈ H. Recall that the function y ∈ L2(τ,T;V) with
τ
dy
∈ L1(τ,T;V∗)isaweaksolution ofProblem(1.1)on[τ,T],ifforallv ∈ V
dt
d
(y,v)+ν((y,v))+b(y,y,v) = hf,vi (1.3)
dt
inthesenseofdistributions, and
y(τ)= y . (1.4)
τ
The weak solution y of Problem (1.1) on [τ,T] is called a Leray-Hopf solution of Problem (1.1)on [τ,T],
R
ify satisfiestheenergy inequality:
V (y(t)) ≤ V (y(s)) forallt ∈ [s,T], a.e. s > τ ands = τ, (1.5)
τ τ
where
ς ς
1
V (y(ς)) := ky(ς)k2 +ν ky(ξ)k2 dξ− hf(ξ),y(ξ)idξ, ς ∈[τ,T]. (1.6)
τ 2 Z V Z
τ τ
For each f ∈ L2(τ,T;V∗) + L1(τ,T;H) and y ∈ H there exists at least one Leray-Hopf so-
τ
lution ofDProblem (1.1); see, for example, Temam [18, Chapter III] and references therein. Moreover,
y ∈ C([τ,T],Hw) and dy ∈ L34(τ,T;V∗) + L1(τ,T;H). If f ∈ L2(τ,T;V∗), then, additionally,
dt
dy 4 ∗
∈ L3(τ,T;V ). Inparticular, theinitialcondition (1.4)makessense.
dt
The following Theorem 1.1 implies that each weak solution of the 3D Navier-Stokes system is Leray-
Hopf one and it is rightly strongly continuous in H at all the points t ∈ [τ,T). This theorem is the main
resultofthisnote.
Theorem 1.1. Let −∞ < τ < T < +∞, y ∈ H, f ∈ L2(τ,T;V∗)+ L1(τ,T;H), and y be a weak
τ
solution ofProblem(1.1)on[τ,T]. Thenthefollowing statementshold:
(a) y ∈ C([τ,T],H )andthefollowingenergyinequality holds:
w
V (y(t)) ≤ V (y(s)) forallt,s ∈ [τ,T], t ≥ s, (1.7)
τ τ
whereV isdefinedinformula(1.6);
τ
2
T
(b) foreacht ∈ [τ,T)thefollowing convergence holds:
y(s) → y(t)strongly inH ass → t+;
(c) thefunction t → ky(t)k2 isofbounded variation on[τ,T].
F
Remark 1.2. Since a real function of bounded variation has no more than countable set of discontinuity
points, then statement (a) of Theorem 1.1, weak continuity in Hilbert space H of each weak solution of
Problem (1.1) on [τ,T], yield that each weak solution of the 3D Navier-Stokes system has no more than
countable set of discontinuity points in the phase space H endowed with the strong convergence topology.
Theorem 1.1partially clarifiestheresults provided inBall[1];Balibrea etal. [2];Barbuetal. [3];Caoand
Titi [4]; Chepyzhov and Vishik [5]; Cheskidov and Shvydkoy [6]; Kapustyan et al. [9, 10]; Kloeden et al.
A
[13];Sohr[17]andreferences therein.
2 Topological Properties of Solutions for Auxiliary Control Problem
Let−∞ < τ < T < +∞. Weconsider thefollowingspaceofparameters:
U := (L2(τ,T;V))× L2(τ,T;V∗)+L1(τ,T;H) ×H.
τ,T
(cid:0) (cid:1)
Eachtriple(u,g,z ) ∈ U iscalledadmissible forthefollowingauxiliary controlproblem:
τ τ,T
dz
Problem (C) on [τ,T] with (u,g,z ) ∈ U : find z ∈ L2(τ,T;V) with ∈ L1(τ,T;V∗) such that
τ τ,T
R dt
z(τ) = z andforallv ∈ V
τ
d
(z,v)+ν((z,v))+b(u,z,v) = hg,vi (2.1)
dt
inthesenseofdistributions; cf. Kapustyanetal. [9,10];Kasyanovetal. [11,12];MelnikandToscano[14];
Zgurovskyetal. [19,Chapter6].
∗
As usual, let A : V → V be the linear operator associated with the bilinear form ((u,v)) = hAu,vi,
∗
u,v ∈ V. For u,v ∈ V we denote by B(u,v) the element of V defined by hB(u,v),wi = b(u,v,w),
forallw ∈ V. ThenProblem(C)on[τ,T]with(u,g,z ) ∈ U canberewrittenas: findz ∈ L2(τ,T;V)
τ τ,T
dz D
with ∈ L1(τ,T;V∗)suchthat
dt
dz
∗
+νAz+B(u,z) = g, inV , andz(τ) = z . (2.2)
τ
dt
Thefollowingtheorem establishes theuniqueness properties forsolutions ofProblem(C).
Theorem2.1. Let−∞ < τ < T < +∞andu∈ L2(τ,T;V). ThenProblem(C)on[τ,T]with(u,¯0,¯0)∈
U hastheuniquesolution z ≡ ¯0.
τ,T
We recall, that {w ,w ,...} ⊂ V is the special basis, if ((w ,v)) = λ (w ,v) for each v ∈ V and
1 2 j j j
j = 1,2,...,where 0 < λ ≤ λ ≤ ...is thesequence ofeigenvalues. LetP bethe projection operator
1 2 m
of H onto H := span{w ,...,w }, that is P v = m (v,w )w for each v ∈ H and m = 1,2,....
m 1 m m i=1 i i
Of course we may consider P as a projection operatoPr that acts from V onto H for each σ > 0 and,
m σ m
sincePm∗ = Pm,wededuce thatkPmkL(Vσ∗;Vσ∗) ≤ 1. Notethat(wj,v)Vσ = λσj(wj,v)foreachv ∈ Vσ and
j = 1,2,....
3
T
ProofofTheorem2.1. Let−∞ < τ < T < +∞,u ∈ L2(τ,T;V),and z beasolution ofProblem (C)on
[τ,T]with(u,¯0,¯0)∈ U . Provethatz ≡ ¯0.
τ,T
Letusfixanarbitrary m = 1,2,....According tothedefinitionofasolution forProblem(C)on[τ,T]
with(u,¯0,¯0) ∈ U ,thefollowingequality holds:
τ,T
F
1 d
kP z(t)k2 +νkP z(t)k2 = b(u(t),P z(t),z(t)), (2.3)
2dt m m V m
fora.e. t ∈ (τ,T).Sinceb(u(t),P z(t),P z(t)) = 0fora.e. t ∈ (τ,T),theninequality (1.2)yieldsthat
m m
b(u(t),P z(t),z(t)) ≤ Cku(t)k kP z(t)k kz(t)−P z(t)k ,
m V m V m V
fora.e. t ∈ (τ,T).Therefore, equality(2.3)implythefollowinginequality
A
1 d
kP z(t)k2 +kP z(t)k (νkP z(t)k −Cku(t)k kz(t)−P z(t)k ) ≤ 0, (2.4)
m m V m V V m V
2dt
fora.e. t ∈ (τ,T).
Letussetψ (t) := kP z(t)k (νkP z(t)k −Cku(t)k kz(t)−P z(t)k ),foreachm = 1,2,...
m m V m V V m V
anda.e. t ∈ (τ,T).Thefollowingstatements hold:
(i) ψ ∈ L1(τ,T)foreachm = 1,2,...;
m
(ii) ψ (t) ≤ ψ (t)foreachm = 1,2,... anda.e. t ∈ (τ,T);
m m+1
R
(iii) ψ (t) → νkz(t)k2 asm → ∞,fora.e. t ∈ (τ,T).
m V
Indeed, statement (i) holds, because u,z ∈ L2(τ,T;V) and P z ∈ L∞(τ,T;V) for each m = 1,2,....
m
Statement(ii)holds,becausekP z(t)k ≤ kP z(t)k and−kz(t)−P z(t)k ≤ −kz(t)−P z(t)k
m V m+1 V m V m+1 V
for each m = 1,2,... and a.e. t ∈ (τ,T). Statement (iii) holds, because P z(t) → z(t) strongly in V as
m
m → ∞,fora.e. t ∈ (τ,T).
Since kz(·)k2 ∈ L1(τ,T), then statements (i)–(iii) and Lebesgue’s monotone convergence theorem
V
yield
Dt t t
lim ψ (s)ds = lim ψ (s)ds = kz(s)k2 ds, (2.5)
m→∞Z m Z m→∞ m Z V
τ τ τ
foreacht ∈ [τ,T].Inequality (2.4)implies
1 t t 1 d t
kP z(t)k2 +ν ψ (s)ds = kP z(t)k2 +ν ψ (s)ds ≤ 0, (2.6)
m m m m
2 Z Z 2dt Z
τ τ τ
foreachm = 1,2,... andt ∈[τ,T]. Wenotethattheequalityin(2.6)holds,because z(τ) = ¯0.
Equality(2.5)andinequality (2.6)yieldthat
1 t
kz(t)k2 +ν kz(s)k2 ds ≤ 0,
2 Z V
τ
for a.e. t ∈ (τ,T), because P z(t) → z(t) strongly in H for a.e. t ∈ (τ,T). Thus, z(t) = ¯0 for a.e.
m
t ∈ (τ,T). Sincez ∈ C([τ,T];V∗),thenz ≡ ¯0,thatis,Problem(C)on[τ,T]with(u,¯0,¯0) ∈ U hasthe
τ,T
uniquesolution z ≡ ¯0.
4
T
ThefollowingtheoremestablishessufficientconditionsfortheexistenceofanuniquesolutionforProb-
lem(C).Thisisthemainresultofthissection.
Theorem 2.2. Let −∞ < τ < T < +∞, y ∈ H, f ∈ L2(τ,T;V∗)+ L1(τ,T;H), and y be a weak
τ
solutionofProblem(1.1)on[τ,T]. Then(y,f,y )F∈ U andProblem(C)on[τ,T]with(y,f,y ) ∈ U
τ τ,T τ τ,T
hastheuniquesolution z = y. Moreover,y satisfies inequality (1.5).
BeforetheproofofTheorem2.2weremarkthatAC([τ,T];H ),m = 1,2,...,willdenotethefamily
m
ofabsolutely continuous functions actingfrom[τ,T]intoH ,m = 1,2,....
m
ProofofTheorem2.2. Prove that z = y is the unique solution of Problem (C) on [τ,T] with (y,f,y ) ∈
τ
U . Indeed,yisthesolutionofProblem(C)on[τ,T]with(y,f,y ) ∈U ,becausey isaweaksolution
τ,T τ τ,T
A
of Problem (1.1) on [τ,T]. Uniqueness holds, because if z is a solution of Problem (C) on [τ,T] with
(y,f,y ) ∈ U ,thenz−y ≡ ¯0istheunique solution ofProblem (C)on[τ,T]with(y,¯0,¯0) ∈ U (see
τ τ,T τ,T
Theorem2.1).
The rest of the proof establishes that y satisfies inequality (1.5). We note that y can be obtained via
d
standard Galerkin arguments, that is, if y ∈ AC([τ,T];H ) with y ∈ L1(τ,T;H ), m = 1,2,...,
m m m m
dt
istheapproximate solution suchthat
dy
m
+νAy +P B(y,y ) = P f, inH , y (τ) = P y(τ), (2.7)
m m m m m m m
dt
R
thenthefollowingstatements hold:
(i) y satisfythefollowingenergyequality:
m
1 t1 t1
ky (t )k2 +ν ky (ξ)k2 dξ − hf(ξ),y (ξ)idξ
2 m 1 Z m V Z m
s s (2.8)
1 t2 t2
= ky (t )k2+ν ky (ξ)k2 dξ− hf(ξ),y (ξ)idξ,
2 m 2 Z m V Z m
s s
foreacht ,t ∈ [τ,T],foreachm = 1,2,...;
1 2
D
(ii) there exists a subsequence {y } ⊆ {y } such that the following convergence (as
mk k=1,2,... m m=1,2,...
m → ∞)hold:
(ii) y → y weaklyinL2(τ,T;V);
1 mk
∞
(ii) y → y weaklystarinL (τ,T;H);
2 mk
(ii) P B(u,y ) → B(u,y)weaklyinL2(τ,T;V∗);
3 mk mk 3
2
(ii) P f → f stronglyinL2(τ,T;V∗)+L1(τ,T;H);
4 mk
dy dy
(ii) mk → weaklyinL2(τ,T;V∗)+L1(τ,T;H).
5 dt dt 3
2
Indeed, convergence (ii) and (ii) follow from (2.8) (see also Temam [18, Remark III.3.1, pp. 264, 282])
1 2
1 1
and Banach-Alaoglu theorem. Since there exists C1 > 0 such that |b(u,v,w)| ≤ CkukVkwkVkvkV2kvk2,
for each u,v,w ∈ V (see, for example, Sohr [17, Lemma V.1.2.1]), then (ii) , (ii) and Banach-Alaoglu
1 2
5
T
theorem imply (ii) . Convergence (ii) holds, because of the basic properties of the projection operators
3 4
{P } . Convergence (ii) directly follows from (ii) , (ii) and (2.7). We note that we may not to
m m=1,2,... 5 3 4
pass to a subsequence in (ii) –(ii) , because z = y is the unique solution of Problem (C) on [τ,T] with
1 5
(y,f,y ) ∈U .
τ τ,T
F
Moreover, thereexistsasubsequence {y } ⊆ {y } suchthat
kj j=1,2,... mk k=1,2,...
y (t) → y(t)stronglyinH fora.e. t ∈(τ,T)andt = τ, j → ∞. (2.9)
kj
t
Indeed,accordingto(2.7),(2.8)and(ii) ,thesequence{y −F } ,whereF (t) := P f(s)ds,
3 mk mk k=1,2,... mk τ mk
m = 1,2,..., t ∈ [τ,T], is bounded in a reflexive Banach space W := {w ∈ L2(τ,T;VR) : dw ∈
τ,T dt
L2(τ,T;V∗)}. Compactness lemma yields that W ⊂ L2(τ,T;H) with compact embedding. There-
3 τ,T
2 A
fore, (ii) –(ii) imply that y → y strongly inL2(τ,T;H) asm → ∞.Thus, there exists asubsequence
1 5 mk
{y } ⊆ {y } suchthat(2.9)holds.
kj j=1,2,... mk k=1,2,...
Due to convergence (ii) –(ii) and (2.9), if we pass to the limit in (2.8) as m → ∞, then we obtain
1 5 kj
thaty satisfiestheinequality
1 t t 1
ky(t)k2 +ν ky(ξ)k2 dξ− hf(ξ),y(ξ)idξ ≤ ky(τ)k2, (2.10)
2 Z V Z 2
s s
fora.e. t ∈ (s,T),a.e. s ∈ (τ,T)ands= τ.
∞ ∗ ∗
Sincey ∈ L (τ,T;H)∩C([τ,T];V )andH ⊂ V withcontinuousembedding,theny ∈ C([τ,T];H ).
w
R
Thus,equality (2.10)yields
1 t t 1
ky(t)k2 +ν ky(ξ)k2 dξ− hf(ξ),y(ξ)idξ ≤ ky(τ)k2,
2 Z V Z 2
s s
foreacht ∈ [τ,T],a.e. s ∈ (τ,T)ands = τ. Therefore, y satisfiesinequality (1.5).
3 Proof Theorem 1.1
D
Inthissectionweestablish theproofofTheorem1.1. LetΠ betherestriction operator tothefinitetime
t1,t2
subinterval [t ,t ]⊆ [τ,T];ChepyzhovandVishik[5].
1 2
ProofofTheorem1.1. Let −∞ < τ < T < +∞, y ∈ H, f ∈ L2(τ,T;V∗)+L1(τ,T;H), and y be a
τ
weaksolutionofProblem(1.1)on[τ,T].
Let us prove statement (a). Fix an arbitrary s ∈ [τ,T). Since (Π y,Π f,y(s)) ∈ U , then
s,T s,T s,T
∞
Theorem2.2yieldsthatΠ y ∈ L (s,T;H)anditsatisfiesthefollowinginequality:
s,T
V (y(t)) ≤ V (y(s)) forallt ∈ [s,T],
τ τ
whereV isdefinedinformula(1.6). Sinces ∈ [τ,T)beanarbitrary, thenstatement (a)holds.
τ
Letusprovestatement(b). Statement(a)yields
1 t t 1
ky(t)k2 +ν ky(ξ)k2 dξ− hf(ξ),y(ξ)idξ ≤ ky(s)k2, (3.1)
2 Z V Z 2
s s
6
T
foreacht ∈ [s,T],foreachs ∈[τ,T). Inparticular, limsupt→s+ky(t)k ≤ ky(s)kforalls ∈ [τ,T),and
y(t)→ y(s)strongly inH ast → s+ foreachs ∈ [τ,T), (3.2)
becausey ∈C([τ,T];H ).
w F
Letusprovestatement(c). Sincey ∈ L2(τ,T;V)∩L∞(τ,T;H)andf ∈ L2(τ,T;V∗)+L1(τ,T;H),
thenstatements (a)and(b)implythatthemappingt → ky(t)k2 isofbounded variation on[τ,T].
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