An a posteriori condition on the numerical approximations of the Navier-Stokes equations for the existence of a strong solution

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by  Masoumeh Dashti

AN A POSTERIORI CONDITION ON THE NUMERICAL
APPROXIMATIONS OF THE NAVIER-STOKES EQUATIONS FOR
THE EXISTENCE OF A STRONG SOLUTION∗
MASOUMEH DASHTI AND JAMES C. ROBINSON †
7
Abstract. In their 2006 paper, Chernyshenko et al prove that a sufficiently smooth strong
0
solutionofthe 3dNavier-Stokes equations isrobustwithrespect tosmallenough changes ininitial
0 conditionsandforcingfunction. Theyalsoshowthatifaregularenoughstrongsolutionexiststhen
2 Galerkinapproximationsconvergetoit. Theythenusetheseresultstoconcludethattheexistenceof
asufficientlyregularstrongsolutioncanbeverifiedusingsufficientlyrefinednumericalcomputations.
n
In this paper we study the solutions with minimal required regularity to be strong, which are less
a
regular than those considered in Chernyshenko et al (2006). We prove a similar robustness result
J
andshowthevalidityoftheresultsrelatingconvergentnumericalcomputationsandtheexistenceof
2 thestrongsolutions.
1
Key words. Navier-Stokesequations, Galerkinmethod
]
P
A 1. Introduction. For the three-dimensional Navier-Stokes equations the exis-
tence of strong solutions over an arbitrary time interval is not known. Therefore the
.
h
validity of the results of the numerical solutions of 3d Navier-Stokes equations is not
t
a obvious. This problem is addressed in Chernyshenko et al (2006) where a rigorous
m relationshipbetweennumericalandsufficiently regularexactsolutions is given. They
[ show for sufficiently smooth initial conditions and forcing functions (data) that al-

though a priori there is no guaranteeof the validity of the numericalsolutions, there
1
is an a posteriori condition that if satisfied by the numerical results guarantees the
v
1 existenceofastrongsolution. Inthispaperweshowthevalidityoftheresultsproved
4 in Chernyshenko et al (2006) for the less regular strong solutions which are not cov-
3 ered there.
1
WewillstudytheNavier-Stokesequationsintheirfunctionalform. Forabounded
0
domainΩ we let be the space ofdivergence-freesmoothvector-valuedfunctions on
7 H
0 Ω with compact support and zero average and define
/
h H =closure of in[L2(Ω)]3,
t H
a V =closure of in[H1(Ω)]3.
m H
: We use the same notation H and V for the similar spaces of periodic functions over
v
i the periodic domain Q. Then the Navier-Stokes equations in their functional form
X
are written as (Constantin and Foias 1988, Robinson 2001)
r
a du
+νAu+B(u,u)=f, with u(0)=u (1.1)
0
dt
whereAu= Π∆,B(u,u)=Π(u. )uwithΠtheorthogonalprojectionfromL2 into
− ∇
H. We will consider this equation with the following cases for the data
(a) ‘minimal regularity’when u V and f L2(0,T;H) L1(0,T;V)
0
∈ ∈ ∩
(b) ‘second order regularity’when u V2 and f L2(0,T;V) L1(0,T;V2),
0
∈ ∈ ∩
∗SUBMITTEDTOSIAMJOURNALONNUMERICALANALYSIS
†Mathematics Institute, University of Wawick, Coventry CV4 7AL. UK
([email protected]), ([email protected]).
1
where Vm = Hm V. For the periodic case we know that D(Am/2) = Hm V
∩ ∩
for all m and therefore we define the normon Vm as u = Am/2u. We denote by
m
k k | |
(, ) and the inner product and norm on H.
· · |·|
We will show that the strong solution of (1.1) with the data introduced in (a) or
(b), ifitexists,remainsstrongifthe changesinthe initial conditionandforcingfunc-
tion are small enough. The exact conditions required for these changes to be ‘small’
aregivenintheorems4.1and4.2. Forexampleinthecaseofminimallyregularstrong
solutions we require
T
D(u v ) + Df(s) Dg(s) ds
0 0
| − | | − |
Z0
1 ν3 1/4 k2 T 27k2 1 1
< exp Du(s)4+ Du(s) Au(s) ds .
k (cid:18)27T(cid:19) − 2 Z0 2 ν3| | ν| || | !
We then, in theorems 6.1 and 6.2, use these robustness results to find an a posteriori
condition that if satisfied by sufficiently refined numerical approximations, implies
the existence of a strong solution. We also show that if a strong solution exists the
Galerkin approximations convergeto it and then use this to prove that the existence
ofastrongsolutioncanbeverifiedbytheGalerkinapproximations. Inthelastsection
weconsiderachannelflowasaphysicalexamplethatcanbedescribedbytheNavier-
Stokesequationswiththeconditionsintroducedabove. Forthisexamplewewillshow
how the results of this paper canbe applied to the Galerkinapproximationsto verify
the existence of a strong solution.
The results we prove here for a strong solution with lowest regularity hold in a
generalbounded domainas wellas inthe absence ofboundaries unlike the results for
more regular solutions which are proved only for the equations in a periodic domain
or the whole space.
2. General ODE lemma. WefirstproveanODElemma whichwillbeusedin
dealing with the differential inequalities that appear in the proofs. We consider the
differential inequality
dy
δ(t)+αyn with y(0)=y >0
0
dt ≤
and find the conditions on y , δ(t) and α that ensure that y(t) exists on a finite time
0
interval [0,T]. This lemma is a generalization of the result obtained for n = 2 in
Chernyshenko et al (2006).
Lemma 2.1. Let T > 0, α > 0 and n > 1 be constants and let δ(t) be a non-
negative continuous function on [0,T]. Let y satisfy the differential inequality
dy
δ(t)+αyn with y(0)=y >0 (2.1)
0
dt ≤
and define
T
η =y + δ(s)ds.
0
Z0
(i) If
1
η < (2.2)
[(n 1)αT]1/(n−1)
−
then y(t) remains bounded on [0,T]
2
(ii) y(t) 0 uniformly on [0,T] as η 0.
→ →
Proof.
We first consider the following differential inequality
z˙ αzn with z(0)=η (2.3)
≤
and show that
sup y(T) sup z(T). (2.4)
≤
S1 S2
where S and S are the sets of all possible solutions of inequalities (2.1) and (2.3)
1 2
respectively.
Since y˙, z˙, y(0)andz(0)arenon-negative,the suprema ofy(T)and z(T)happen
when
y˙=δ(t)+αyn,
z˙=αzn
for all t [0,T]. In this case, for the difference w =y z we have
∈ −
n−1 n 1 t
w˙ =δ(t)+αw − yn−1−k zk with w(0)= δ(s)ds.
k −
k=0(cid:18) (cid:19) Z0
X
Since y(t) and z(t) are greater than zero and assuming they remain finite over [0,T],
there exists some M >0 such that
w˙ αMw+δ(t)
≤
and therefore by Gronwall’s inequality
t
w(t) w(0)eαMt+ δ(s)ds
≤
Z0
t
=(1 eαMt) δ(s)ds.
−
Z0
This implies w(t) 0 for any t [0,T] and (2.4) follows.
≤ ∈
The inequality (2.3) can be written as
z˙
α
zn ≤
Integrating both sides from 0 to t yields
η
z(T) . (2.5)
≤ ( αT(n 1)(η)n−1+1)1/n−1
− −
Since y(t) y(T) z(T), y(t) remains finite on [0,T] provided that
≤ ≤
αT(n 1)(η)n−1 <1
−
which yields (2.2).
Furthermore, it is clear from (2.5) that z(T) 0 as η 0, from which it follows
→ →
that y(t) 0 uniformly on [0,T].
→
3
3. The inequalities for the nonlinear term. For the nonlinear operator,we
will use the following inequalities in this paper
(B(u,v),Aw) k Du Dv 1/2 Av 1/2 Aw (3.1)
| |≤ | || | | | | |
(B(w,u),A2w) c u w 2 (3.2)
| |≤ k k3k k2
(B(u,w),A2w) c′ u w 2, (3.3)
| |≤ k k3k k2
The first inequality holds for both Dirichlet and periodic boundary conditions (Con-
stantinandFoias1988)whiletheothertwoarevalidonlyintheabsenceofboundaries
with c and c′ independent of the size of the domain (Kato 1972). The constant k for
the periodic and no-slip domain and also the whole of R3, is independent of the do-
main Ω. In a general domain with nonzero boundary conditions however, it depends
on the regularity properties of the boundary of Ω. To see this we use the Ho¨lder
inequality to write
3
B(u,v,Aw) u D v Aw .
i L6 i j L3 j
| |≤ k k k k | |
i,j=1
X
By the Sobolev inequality
u c Du (3.4)
L6 s
k k ≤ | |
we have
Dv Dv 1/2 Dv 1/2 c1/2 Av 1/2 Dv 1/2.
k kL3 ≤k kL6 | | ≤ s | | | |
Therefore we can write
3
B(u,v,Aw) u D v Aw
i L6 i j L3 j
| |≤ k k k k | |
i,j=1
X
9 u Dv Aw
L6 L3
≤ k k k k | |
9c3/2 Du Dv 1/2 Av 1/2 Aw
≤ s | || | | | | |
and so k = 9c3/2. For no-slip boundary conditions and the whole of R3, c , the
s s
constant of the Sobolev inequality (3.4), does not depend on Ω (Ziemer 1989). For
a bounded domain with non-zero boundary conditions, c depends on the regularity
s
properties of ∂Ω but is independent of the size of Ω (Adams 1975). Therefore for the
cubicdomainofthe periodicboundaryconditionsalso,itdoesnotdependonthesize
of the domain. In fact, the cubic domain has the strong Lipschitz property and for
suchadomainAdams(1975,Lemma5.10)hasshownthatc =4√2. Similarly,cand
s
c′ depend on the constant of the Sobolev inequality kukL6/(3−2k) ≤ cs,kkukHk which
is again independent of the size of the domain (Adams 1975 and Ziemer 1989).
In obtaining the robustness conditions in the next section, we keep track of the
constants k, c and c′ when they appear, so that the value of the constant coefficients
in the robustness conditions could be computed if desired.
The inequalities (3.1)–(3.3) are not as elegant as the inequalities available for
the more regular solutions considered in Chernyshenko et al (2006) and this is why
different proofs are needed in the less regular cases we are studying here.
We note here that assuming a strong solution u L∞(0,T;V) L2(0,T;V2)
∈ ∩
exists, from (3.1) we have B(u,u) L2(0,T;H) and therefore (1.1) implies that
∈
4
du/dt L2(0,T;H). Having u V2 and f L2(0,T;V) in case (b), from the
0
∈ ∈ ∈
regularityresultforperiodicdomainsprovedinConstantinandFoias(1988)weknow
that the strong solution u is in fact more regular and an element of L∞(0,T;V2)
∩
L2(0,T;V3).
4. Robustness of strong solutions. Here we show that as long as a strong
solution exists for some specific initial data and forcing function, the equations with
sufficiently close data also have a strong solution. A similar result about the robust-
ness of strong solutions with respect to changes in the forcing function is proved by
Fursikov in his 1980 paper. For the three-dimensional Navier-Stokes equations with
initial condition u Vm where m 1/2, he shows that the set of forcing functions
0
∈ ≥
for which a strong solution exists is open in Lq(0,T;Vm−1) with q 2.
≥
Forbothminimallyregularsolutions(a)andthe moreregularcase(b)thecondi-
tion we obtain here for the data depends explicitly on the viscosity coefficient unlike
themoreregularcasesconsideredinChernyshenkoetal(2006). Therobustnessresult
we prove for the minimally regular strong solutions holds in a general bounded do-
main. For the secondorderregularsolutionshowever,we need to restrictthe domain
to be periodic or the whole of R3 since the inequalities (3.2) and (3.3) only hold in a
periodic domain (Constantin and Foias 1988) or the whole of R3 (Kato 1972).
(a) Strong solutionswithminimalregularity. Toprovetherobustnesswith
respect to the data we write the governing equation for the difference between the
solution of the equations with nearby data and the strong solution and find a bound
on the norm of this difference.
Theorem 4.1. Let f L1(0,T,V) L2(0,T;H), u V and u L∞(0,T;V)
0
∈ ∩ ∈ ∈ ∩
L2(0,T;V2) be a strong solution of
du
+Au+B(u,u)=f with u(0)=u .
0
dt
If
T
D(u v ) + Df(s) Dg(s) ds
0 0
| − | | − |
Z0
1 ν3 1/4 k2 T 27k2 1 1
< exp Du(s)4+ Du(s) Au(s) ds (4.1)
k (cid:18)27T(cid:19) − 2 Z0 2 ν3| | ν| || | !
then the solution of
dv
+νAv+B(v,v)=g with v(0)=v
0
dt
is also a strong solution on [0,T] with the same regularity as u.
Proof. By the local existence of strong solutions (Constantin and Foias 1988,
Temam 1995) we know that there exists T∗ > 0 such that v L∞(0,T′;V)
∈ ∩
L2(0,T′;V2) for any T′ < T∗. We consider T∗ the maximal time of existence of
the strongsolutionv meaning that limsup Dv = . In the followingargument
t→T∗| | ∞
we assume T∗ T and deduce a contradiction.
≤
We consider w =u v which satisfies
−
dw
+νAw+B(u,w)+B(w,u) B(w,w)=f g.
dt − −
5
Over the time interval t [0,T′) for any T′ < T∗ we have dv/dt L2(0,T′;H) and
∈ ∈
since T∗ T, du/dt L2(0,T′;H). Therefore taking the inner product of the above
≤ ∈
equation with Aw and using (3.1) we can write
1 d
Dw2+ν Aw2 k Du Dw1/2 Aw3/2+k Dw Du1/2 Au1/2 Aw
2dt| | | | ≤ | || | | | | || | | | | |
+k Dw3/2 Aw3/2+ D(f g) Dw.
| | | | | − || |
We then use Young’s inequality to remove Aw (which causes the appearence of ν in
| |
the coefficients) and obtain
1 d 27k4 k2 27k4
Dw2 Du4+ Du Au Dw2+ Dw6+ D(f g) Dw.
2dt| | ≤ 4ν3 | | 2ν| || | | | 4ν3 | | | − || |
(cid:18) (cid:19)
Dividing by Dw (assuming Dw is nonzero. If not, replacing Dw by Dw +e
0
| | | | | | | |
with e >0 will leadus to the same result after tending e to zeroat the end) we get
0 0
d 27k4 k2 27k4
Dw Du4+ Du Au Dw + Dw5+ D(f g).
dt| |≤ 4ν3 | | 2ν| || | | | 4ν3 | | | − |
(cid:18) (cid:19)
Now letting
k2 t 27k2 1 1
β(t)= Du4+ Du Au ds
2 2 ν3| | ν| || |
Z0 (cid:18) (cid:19)
and setting y(t)= Dw(t)e−β(t) the above inequality can be written as
| |
dy
αy5+δ(t),
dt ≤
where α= 27k4e4β(T) and δ(t)= Df(t) Dg(t).
4ν3 | − |
By lemma 2.1, if the condition(4.1) is satisfied, Dw(t) is uniformly bounded on
| |
[0,T∗). This implies that Dv(T∗) < which contradicts the maximality of T∗. It
| | ∞
follows that v(t) is a strong solution on [0,T].
(b) Strong solutions with second order regularity. We follow the same
approachasintheminimallyregularcasetoshowtherobustnessforastrongsolution
withsecondorderregularity. Herealsotheinequalitiesweneedtouseforthenonlinear
operator to find a bound on the V2-norm of the difference between nearby solutions
depend on the V3-norm of this difference and therefore the robustness condition on
the data depends explicitly on the viscosity coefficient. However, its dependence is
weaker in this case.
Theorem4.2. Letf L1(0,T,V2) L2(0,T;V),u V2 andu L∞(0,T;V2)
0
∈ ∩ ∈ ∈ ∩
L2(0,T;V3) be a strong solution of
du
+Au+B(u,u)=f with u(0)=u .
0
dt
If
T 1 2ν T
A(u v ) + A(f g) dt < exp (c+c′) u dt , (4.2)
0 0 3
| − | Z0 | − | crT −Z0 k k !
6
then the solution of
dv
+Av+B(v,v)=g with v(0)=v
0
dt
is also a strong solution on [0,T] with the same regularity as u.
Proof. As before, the difference w=u v satisfies
−
dw
+νAw+B(u,w)+B(w,u)+B(w,w)=f g.
dt −
Taking the inner product of the above equation with A2w and using (3.2) and (3.3)
we obtain
1 d
w +ν w 2 (c+c′) u 2 w 2+c w w 2+ f g w .
2dtk k2 k k3 ≤ k k3k k2 k k3k k2 k − k2k k2
We apply Young’s inequality to the second term on the right hand side and get
1 d c2
w 2 (c+c′) u w 2+ w 4+ f g w .
2dtk k2 ≤ k k3k k2 4νk k2 k − k2k k2
Dividing the above inequality by w yields
2
k k
d c2
w (c+c′) u w + w 3+ f g .
dtk k2 ≤ k k3k k2 4νk k2 k − k2
Now we consider
y(t)= w e−β(t)
2
k k
with β(t)= t(c+c′) u dt. The equation with this new variable becomes
0 k k3
R dy
δ(t)+αy3,
dt ≤
where α = c2e2β(T) and δ = f g . Using lemma 2.1 w is bounded if the
4ν k − k2 k k2
condition (4.2) is satisfied.
5. Convergence of Galerkin approximations. Hereweshowthatifastrong
solutionwithminimalorsecondorderregularityexists,Galerkinapproximationscon-
verge to it. Similar results about convergence of various numerical method assuming
the existence of a strong solution, are given for finite element methods by Heywood
andRannacher(1982),foraFouriercollocationmethodbyE(1993)andforanonlin-
ear Galerkinmethod by Devulder, Marion and Titi (1993). Here as in Chernyshenko
etal(2006),wemakenoassumptionontheregularityoftheGalerkinapproximations.
For the minimally regular case the result we prove here holds for the solution of
the equations overa generalbounded domain. For the second orderregular solutions
we need to use inequality (3.3) which is valid only in a periodic domain or the whole
of R3.
In the following theorems we let P be the orthogonal projection in H onto the
n
space spanned by the first n eigenfunctions of the Stokes operator A, w n , or-
{ j}j=1
dered so that their corresponding eigenvalues satisfy 0 < λ λ ... . Since the
1 2
≤ ≤
eigenfunctions of A are smooth (Constantin and Foias 1988), for any u Vm, m 0
(with V0 =H) we have u =P u= n (u,w )w Vm. ∈ ≥
n n j=1 j j ∈
We notethatwhatweobtainhereaboutthe convergenceofGalerkinapproxima-
P
tions would be useful even if the existence of regular solutions was known.
7
(a) Minimal regularity. Thefollowingtheorem,liketherobustnesstheoremin
the minimal regularity case, holds in sufficiently smooth bounded domains as well as
in the absence of boundaries.
Theorem 5.1. Let u V, f L2(0,T;H) and u L∞(0,T;V) L2(0,T;V2)
0
∈ ∈ ∈ ∩
be a strong solution of the Navier-Stokes equations
du
+νAu+B(u,u)=f(t) with u(0)=u .
0
dt
Then u , the solution of Galerkin system
n
du
n
+νAu +P B(u ,u )=P f(t) with u (0)=P u , (5.1)
n n n n n n n 0
dt
converges strongly to u in both L∞(0,T;V) and L2(0,T;V2) as n .
→∞
Proof. We consider w =u u which satisfies
n n
−
dw
n
+νAw +P B(u,w )+P B(w ,u) P B(w ,w )=Q f(t) Q B(u,u).
n n n n n n n n n n
dt − −
Letting q =u P u and taking the inner product of the above equation with Aw
n n n
−
while noting that (P B(u,v),Aw )=b(u,v,P Aw ) we obtain
n n n n
1 d
Dw 2+ν Aw 2 k Du Dw 1/2 Aw 3/2
n n n n
2dt| | | | ≤ | || | | |
+(cid:16)Dw Du1/2 Au1/2 Aw + Dw 3/2 Aw 3/2
n n n n
| || | | | | | | | | |
+Du3/2 Au1/2 Aq + f Aq .
n n
| | | | | | | || |
(cid:17)
Applying Young’s inequality and redefining the constant k gives
d 1 1 1
Dw 2 k Du4+ Du Au Dw 2+ Dw 6
dt| n| ≤ ν3| | ν| || | | n| ν3| n|
(cid:26)(cid:18) (cid:19)
+Du3/2 Au1/2 Aq + f Aq .
n n
| | | | | | | || |
o
Now letting
t 1 1
β(t)=k Du4+ Du Au ds
ν3| | ν| || |
Z0 (cid:18) (cid:19)
and
y (t)=e−β(t) Dw 2
n n
| |
the above inequality can be written as
dy
n αy3 +δ (t),
dt ≤ n n
with α= k e2β(T) and δ (t)= f Aq +k Du3/2 Au1/2 Aq .
ν3 n | || n| | | | | | n|
Using the Ho¨lder inequality we have
1/2 1/2 1/2
T T T T
δ (s)ds f 2ds +k Du3 Auds Aq 2ds
n n
Z0 ≤ Z0 | | ! Z0 | | | | !  Z0 | | !
 8 
Since u L2(0,T;V2), u(s) V2 for almost every s [0,T] and therefore q (s) =
n
∈ ∈ ∈
Q u(s) 0 in V2 as n and Aq (s)2 Au(s)2 for a.e. s [0,T]. Hence the
n n
→ →∞ | | ≤| | ∈
Lebesgue dominated convergence theorem for each s implies that T Aq 2ds tends
0 | n|
to zero as n and therefore
→∞ R
T
δ (s)ds 0
n
→
Z0
as n . Since y (0) 0 as n , lemma 2.1 shows that y , and hence Dw 2,
n n n
→∞ → →∞ | |
converges to zero uniformly on [0,T] as n .
→∞
(b) Second order regularity. Here also,in a similar way to the previous case,
we try to find a bound on the V2-norm of the difference between a second order
regularstrong solutionand Galerkinapproximations. Howeverhere the more regular
solution space makes the proof easier.
Theorem5.2. Letu V2,f L2(0,T;V) L1(0,T;V2)andu L∞(0,T;V2)
0
∈ ∈ ∩ ∈ ∩
L2(0,T;V3) be a strong solution of the Navier-Stokes equations
du
+νAu+B(u,u)=f(t) with u(0)=u .
0
dt
Then u , the solution of Galerkin system
n
du
n
+νAu +P B(u ,u )=P f(t) with u (0)=P u , (5.2)
n n n n n n n 0
dt
converges strongly to u in both L∞(0,T;V2) and L2(0,T;V3) as n .
→∞
Proof. Let w =u u . Then w satisfies
n n n
−
dw
n
+νAw +P B(u,w )+P B(w ,u)+P B(w ,w )=Q f Q B(u,u)
n n n n n n n n n n
dt −
Taking the inner product of the above equation with A2w and using (3.2) and (3.3)
n
we obtain
1 d
w 2+ν w 2 (c+c′) u w 2+c w w 2
2dtk nk2 k nk3 ≤ k k3k nk2 k nk3k nk2
+ Q f Q B(u,u) w
n n 2 n 2
k − k k k
Now we use Young’s inequality to remove the dependence on w and then divide
n 3
k k
by w to obtain
2
k k
d c2
w (c+c′) u w + w 3+ Q f Q B(u,u) ,
dtk nk2 ≤ k k3k nk2 4νk nk2 k n − n k2
in which the coefficient of w does not depend on n.
n 2
k k
Letting
y = w e−β(t),
n n 2
k k
with β(t)=exp t(c+c′) u(s) ds yields
0 k k3
(cid:16)R (cid:17)
y˙ δ(t)+αy3,
n ≤ n
9
where α= c2e2β(T) and δ(t)= Q f Q B(u,u) . So by lemma 2.1 if
4ν k n − n k2
T
η = Q u + Q f Q B(u,u) ds 0 as n ,
n 0 m n n 2
k k k − k → →∞
Z0
then y (t) 0 uniformly on [0,T] as n .
n
→ →∞
By (5.2) Q u 0 as n . We know that
n 0 m
k k → →∞
B(u,u) c u u (5.3)
2 2 3
k k ≤ k k k k
(Kato 1972, Constantin and Foias 1988) and therefore f(s) B(u(s),u(s)) V2
− ∈
for almost every s [0,T]. So since w n form a basis for V2 as well as H,
∈ { j}j=1
Q (f(s) B(u(s),u(s))) 0 and
n 2
k − k →
Q (f(s) B(u(s),u(s))) f(s) B(u(s),u(s))
n 2 2
k − k ≤k − k
foralmosteverys [0,T]. ThereforebytheLebesguedominatedconvergencetheorem
∈
it follows that
T
Q (f(s) B(u(s),u(s))) 0
n 2
k − k →
Z0
and the result follows.
6. Numerical verification of the existence of a strong solution. Here we
show that the existence of minimal and second order regular strong solutions can be
verified via computations using sufficiently refined Galerkin approximations.
Theorem 6.1. i) Consider the Navier-Stokes equations
du
+νAu+B(u,u)=f with u(0)=u (6.1)
0
dt
with u V, f L2(0,T;H) L1(0,T;V), that hold in a bounded domain Ω with
0
∈ ∈ ∩
sufficiently smooth boundary or in a periodic domain.
Let v L∞(0,T;V) L2(0,T;V2) be a numerical approximation of u satisfying
∈ ∩
dv
+νAv+B(v,v) L1(0,T;V) L2(0,T;H)
dt ∈ ∩
and
T dv(s)
Dv(0) Du + +νAv(s)+B(v(s),v(s)) f(s) ds
0 1
| − | k ds − k
Z0
1 ν3 1/4 k2 T 27k2 1 1
< exp Dv(s)4+ Dv(s) Av(s) ds . (6.2)
k (cid:18)27T(cid:19) − 2 Z0 2 ν3| | ν| || | !
Then the Navier-Stokes equations (6.1) have a strong solution u L∞(0,T;V)
∈ ∩
L2(0,T;V2).
ii) Let u be a strong solution of (6.1). Then there exists an N such that the
Galerkin approximations u satisfy the inequality (6.2) for all n > N. Therefore in
n
viewofpart(i), upasses the aposterioritestasasolutionapproximated byu i.e. the
n
10