BULLETIN OF THE TRANSILVANIA UNIVERSITY OF BRAŞOV
ON THE SOLVABILITY OF NAVIER-STOKES
EQUATIONS
CRISTINA SBURLAN
Abstract: In this paper we study the solvability of non-stationary Navier-Stokes
equations for the flow of an incompressible viscous fluid, using the abstract
Fourier Method. Both linear and nonlinear cases are treated. By this method, we
obtain the existence and the uniqueness of the strong solution of the problem,
which is the classical solution of the abstract problem. We construct effectively the
solution as a Fourier series, using the sequence of orthonormal eigenfunctions of
the duality mapping between a Hilbert space and its topological dual, which, in
our case, is Stokes operator.
Keywords: Fourier Method, Navier-Stokes Equations
1. Fourier Method for Stokes System
Let   R N , 2 ≤ N ≤ 3, be a bounded domain with enough smooth boundary to apply
Green’s formula and Sobolev-Kondrashov embedding theorem, and denote
Q    (0, ) ,     (0, ) . Consider the Navier-Stokes system for the flow of an
incompressible fluid
(  u )( x, t )  0 (the incompressibility condition)
~
ut ( x, t )  (u  )u( x, t ) u( x, t )  p( x, t )  f ( x, t ), ( x, t )  Q
u  0 on .
~
Denoting f  f  p, equation (1) becomes
ut ( x, t )  (u  )u ( x, t ) u ( x, t )  f ( x, t )
1
where  is the dynamic viscosity ( 
is the inverse of the Reynolds number).
Re
(1)
(2)
The velocity u = (u1, ..., uN) and the pressure p are not known, and they must be
determinated from the system (1).
We will study the case of dynamical viscous fluids with big  . In this case we have
u ( x, t )  (u  )u ( x, t ) , and we can approximate equation (2) with
ut ( x, t ) u( x, t )  f ( x, t ) (the Stokes system)
(3)
__________ 

Denote C0, ()  { y  (C0 ()) N |   y  0} and H  C0, ()
L2
. We have (see [5],
p. 83) that H  { y  ( L2 ()) N |   y  0, n  y  0 on } . We call this Hilbert space
the space of incompressible fluids and let V  { y  ( H 01 ()) N |   y  0} be

Faculty of Mathematics and Informatics, Ovidius University of Constantza, Romania
On the Solvability of Navier-Stokes Equations
322
subspace of H. Let P : ( L2 ()) N  H be the Helmholz projection. From [5], we have
~
~
~
that P( f )  P( f  p)  P( f )  P(p)  P( f ) , because P (p )  0 .
Denote by A  L(V , V ) the Stokes operator
N
( Ay, w)    yi ( x)  wi ( x)dx, y, w  V .
i 1 
Then we can formulate problem (3) as an evolution equation
~
dy
 Ay  P( f )
dt
(4)
where A is symmetric ( (Ay,w)  (y,Aw) ) and strongly monotone ( ( Ay, w)  y ), and
2
V is the energetic space of this operator. Suppose that we have also the initial
u (0)  u 0  H , which becomes y(0)  y 0 , where y 0  P(u 0 ) .
We have the following theorem (see [7], p. 67)
Theorem 1.1 Let X be a real Hilbert space and L : D( L)  X  X
symmetric and strongly monotone operator, with the domain D (L )
dimensional. Consider the energetic space E of the operator L
A : D( A)  X  X be the Friedrichs extension of the operator L,
Au := Ju, u  D ( A) , D( A)  {u  E | Ju  X }
(where J : E  E * ,
condition
a linear,
infinitely
and let
Ju, v  (u, v) E , u, v  E is the duality mapping from E to
E * ). If the embedding E  X is compact, then there exist the sequences (en ) nN  E
and (n ) nN  (0, ) , which are the eigenfunctions and the eigenvalues of A, i.e.,
(Aen, v) = λn(en, v), v  X , n  N
(5)
and they have the properties
i) (en ) nN is an orthonormal basis in E;
ii) ( n en ) nN is an orthonormal basis in X;
iii) (n en ) nN is an orthonormal basis in E * ;
iv) (n ) nN is an increasingly sequence to +∞.
Consider en  V and n  (0, ) from the above theorem (applied for the Hilbert
spaces V  H ) and bn (t )  y 0 n e
n t
t
  Pn ( s)e n ( s t ) ds , where y 0 n , Pn (s ) are the
0
~
Fourier coefficients of y 0 and, respectively, P ( f ) , in V . Now we can apply for the
problem (4) the Fourier method presented in [1] and we can formulate the following
theorem
~
Theorem 1.2 If P( f )  L2 (0, T ; H ) , then the function y (t ) 

 b (t )e
n 1
the space C (0, T ;V )  H 1 (0, T ;V * ) ,
problem (4), i.e., y(0) = y0 and
n
n
belongs to
and this is the unique weak solution of the
~
( y ' (t ), v)   ( Ay(t ), v)  ( P( f )(t ), v), v  V .
Bulletin of the Transilvania University of Braşov Vol 13(48) - 2006
323
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Moreover, if P ( f )  H 1 (0, T ; H ) , then y  C (0, T ;V )  C 1 (0, T ; H ) is the classical
solution of the Cauchy problem (4).
2. Fourier Method for Navier-Stokes system
Consider the Navier-Stokes nonlinear system (1) and the Hilbert spaces
H  { y  ( L2 ()) N |   y  0, n  y  0 on }
and V  { y  ( H 01 ()) N |   y  0} .
Consider again the Stokes operator A  L(V , V ) ,
N
( Ay, w)    yi ( x)  wi ( x)dx, y, w  V ,
i 1 
and define the threelinear form b : V × V × V → R,
b( y, z, w) 
N
  y D z w dx ,
i , j 1
i
i
j
j
which determines the nonlinear operator C : V → V ,
C ( y, w)  b( y, y, w), y, w  V .
We have
(u  )u  (u1 D1  u 2 D2    u N DN )u  (u1 D1u k  u 2 D2 u k    u N DN u k )1k  N .
Remark 2.1 The scalar product in H between h  H and v  V is in fact the product
between h and v in the duality from V * and V .
Consider the weak formulation of the problem, which for f  L2 (0, T ;V * ) and
u 0  H (given) asks for finding the function u  L2 (0, T ;V ) such that
u t  L2 (0, T ;V * ) and
du
 Au  C (u )  f on (0, T )
dt
u (0)  u 0
(6)
(7)
A weak solution which belongs to C (0, T; V ) ∩ C1 (0, T;H) is named strong solution.
Remark 2.2 From the Remark 2.1, we have that equation (6) can be reformulated as
follows
d
u, v
dt
H
  u, v
H
 (u  )u , v
H
 f , v , v  V
(8)
Remark 2.3 We can write
d
u, v
dt
H
 f  Au  C (u ), v
H
, v  V .
Because A is linear and continuous from V to V * , if u  L2 (0, T ;V ) we have that
Au  L2 (0, T ;V * ) . On the other side, C (u )  L1 (0, T ;V * ) (see [8] p. 281). So
f Au  C (u )  L1 (0, T ;V * ) and then u t  L1 (0, T ;V * ) . From here we have (see [8],
p. 250) that u is a.e. equal with a continuous function from [0, T) to V * , so, after a
modification on a set of null measure, u will be a continuous function from [0, T) to V * .
On the Solvability of Navier-Stokes Equations
324
~
~
~
Consider now f  L2 (0, T ; H ) , f t  L1 (0, T ; H ) , u 0  H given, where T  0 . We
~
have (see [8] p. 250) that f is a.e. equal with a continuous function from [0, T ] to H. We
will search for the weak solution of the Navier-Stokes equation as the Fourier series in V

u ( x, t )   bn (t )  en ( x)  lim
n 
n 0
n
b
k 0
k
(t )  ek ( x)
(9)
where (en ) nN is the orthonormal basis in V formed with eigenfunctions of the duality
mapping J : V  V * ( J   , see [3], p. 127-128), obtained from Theorem 1.1,  n are
the corresponding eigenvalues, and (bn ) nN are the corresponding Fourier coefficients.
Denoting en  (enm )1 m N and vn ( x, t ) 
n
b
p 0
p
(t )  e p ( x) , n  N , the kth component
(1 ≤ k ≤ N) of the vector (v n  )vn will be
N  n
n

v1n D1vnk  vn2 D2 vnk    vnN DN vnk     b p e mp   b p Dm e kp  .
m 1  p 0
p 0

We have (using Mertens theorem)
(u  )u  lim (v n  )v n ;
n 
((vn  )vn )(, t ), e j

H
N   N  n
n


        b p (t )e mp ( x)   b p (t )Dm e kp ( x)    e kj ( x)dx  

 

k 1   m 1  p  0
p 0



p
n
N
N




  b p l (t )bl (t )      e mp l ( x) Dm elk ( x)   e kj ( x)dx 
p 0 l 0

 k 1  m1

denote
 c j (b0 (t ), b1 (t ), , bn (t )), j  0 .
Taking in (8) v = ej ( j  N ), replacing formally (9) in this relation and using the fact that
< ei, ej >V = δij , i  N , Je j   e j   j e j and < ej , v >V =λj < ej , v >H , v  V ,
we obtain that bj(t) verify the scalar nonlinear Cauchy problem
b 'j  j b j   j c j (b0 , b1 ,, bn )  f j (t ), b j (0)  u0 j ,0  j  n,
(10)
where u 0 j and fj(t) are the corresponding Fourier coefficients in V for u0 and f (, t ) .
n  N . We have that the solution of homogeneous linear system
 t
b  j b j  0,0  j  n , has the form b j (t )  e j  k j , k j  R,0  j  n . We apply
Let
'
j
the variation of constants method and we search the components of the solution
b(t )  (b0 (t ), b1 t ,, bn (t )) of system (10) to be of the form
b j (t )  e
 j t
 h j (t ),0  j  n,
(11)
and replacing in (10) we obtain, for all 0 ≤ j ≤ n,
 j t
h 'j (t )  e
 j t
f j (t )   j e
Denoting h = (hj)0≤j≤n and
c j (e 0t h0 (t ), e 1t h1 (t ),, e nt hn (t ))
(12)
Bulletin of the Transilvania University of Braşov Vol 13(48) - 2006
 j t
F (t , h)  (e
 j t
f j (t )   j e
325
c j (e 0t h0 , e 1t h1 ,, e nt hn )) 0 j n ,
the system (10) leads us to
h'  F (t , h), h(0)  (u 00 , u 01 , u 02 ,u 0 n )
(13)
We have that cj is the sum of quadratic terms in h0, h1, ...hn, and from here it results that
F is continuous with respect to (t, h), and F is Lipschitzian in h uniformly with respect to
t (being of class C1 with respect to h) on a relative compact neighborhood of the point
(0, u 00 , u 01 , u 02 , u 0 n )  R n  2 . Applying now Picard’s theorem, it results that the
~
system (10) has a unique solution h =h(t), h  C 1 ([0, T ]), 0  T  T . We will prove that

the series
 b (t )e ( x) ,with
n 0
n
bn given by (11) and (13), converges uniformly to
n
u  C (0, T ;V ) , the series

 b (t )e
n 0
'
n
n
( x) converges uniformly to u t  C (0, T ;V * ) , and,
moreover, this is the unique strong solution of the Navier-Stokes system (1) (for a fixed
 ). For this, we need the following Lemma:

Lemma 2.1 The series

n 0
 ( N 1)
n
is convergent in R.
Proof. We have 0  0  1  2    n     .
For   R , denote by N(λ) the cardinal number of the set { j  N |  j  } .
N(λ) is given by Weyl’s law (see [9]): N ( ) 
 () N N
  R ( ) ,
(2 ) N
where R(λ) = O(λN−1),  ( ) is the volume of the domain , and ωN is the volume of the
unity sphere from RN. As R(λ) = O(λN−1), there exist m > 0 and λ′ > 0 such that
 () N N
  mN 1 ,    ' , so
N
(2 )
 () N N
 () N N
 mN 1 
  N ( )  mN 1 
 ,    ' .
N
(2 )
(2 ) N
M n  { j  N | n   j  n  1} 
For all n  N , consider the sets
N ( ) 
={ j  N |
We
1
1 1

 } (finite sets) and denote by mn the cardinal number of this set.
n 1  j n
have
that
M n  { j  N |  j  n  1} \ { j  N |  j  n}, n  N * ,
so
mn  N ( n  1)  N ( n ), n  N * and m0 = N(1).
On the other hand, there exists n0  N such that
N 1
N
N 1
N
 () N
 () N
 m n 1 
n

1

N
(
n

1
)

m
n

1

n

1
(2 ) N
(2 ) N
N 1
N
N 1
N
 () N
 () N
and  m n

n  N( n)  m n

n , n  n0 .
N
N
(2 )
(2 )
From here we obtain that
On the Solvability of Navier-Stokes Equations
326
mn  m  n
N 1
2
N 1
N


 denote
N 
2
2

(

)

1
1




N
2 
1  

1 
 n 1    1  An .
 n 

 n 

(2 ) N




Furthermore, we have that
N 1


An
 () N
1  1  2
1
 m  N 3  1  
 1 
 N 2
N 1
N
 n 

n
(2 )
n 2 
n 2


1

Because for N ≥ 2 the series
n 1
N


2
1



 1    1 .
 n 



An
N 1
 0 , applying the
is convergent and lim n
n
1
n
N 2
2
N 2
2
n
An
limit comparison criterion it results that the series  N 1 is convergent. We have now
n 1 n

 
m
A
1 
1

. But M0 is finite,  N 1  N n1  Nn1 ,





N 1
N 1
N

1


n
n
jM n  j
n 0  n
jM 0  j
n 1  jM n  j


A
for all n  N * , and because the series  Nn1 is convergent, by the comparison
n 1 n

1

that
1
1

criterion it results that the series

n 0
N 1
converges. □
n
Coming back to our problem, denote M 1  sup sup hn (t ) and M 2  sup sup hn' (t ) .
nN t[ 0 ,T ]
nN t[ 0 ,T ]
We have that M1 < ∞ and M2 < ∞ (because hn, h′n are continuous on the compact interval
[0, T], for all n  N ).
2
 t
 2 t
We have bn (t )  e n  hn (t )  e n  hn2 (t ) . But for all δ > 0, δ < T, using the
2
fact that λn → ∞, we have that there exists n  N such that e
n  n , t  [ , T ] , so bn (t )  M 12
1
2

N 1
n
 2n t

Then we have
n 0
V
 bn (t )en

  bn (t )   , so
n 0
2
 b (t )
n 0
for all t  [ , T ] , where δ is arbitrary in (0, T).
2
1

N 1
n
,
, n  n , t  [ , T ] . From here, using the
comparison criterion and Lemma 2.1, we obtain that the series

 e  2n 
n
2
is convergent

 b (t )e
n 0
n
n
converges uniformly
for t  [0, T ] to u  C ([0, T ]; V ) .
 t
 t
Furthermore, for n  N we have bn' (t )  n e n hn (t )  e n hn' (t ) ,and we obtain
2
bn' (t )  2 2 M 12 2n e  2nt  2 M 22 e  2nt .
Bulletin of the Transilvania University of Braşov Vol 13(48) - 2006
2
bn' (t )
From here it results that
 2 2 M 12 e 2nt  2M 22
n
327
e 2nt
.
2n
But, similarly with the above method, for any δ > 0, δ < T, using the fact that λn → ∞,
we have that there exist n' , n''  N such that
1
e  2nt  e  2n 
e
so

2
n
1


N 1
n

N 1
n
, n  n' , t  [ , T ] ,
1
e  2nt  e  2n 
2n t

N 1
n
, n  n'' , t  [ , T ] ,
, n  n'' , t  [ , T ].
Then for n  n  max {n' , n'' } , we have
2
bn' (t )
1
 2 2 M 12
n

N 1
n
1
 2M 22

N 1
n
, n  n , t  [ , T ] .
Using again the comparison criterion and Lemma 2.1, it results that the series


bn' (t )
2
 b (t )e
is convergent. But
n
n 0
2

'
n
n 0

so the series
 b (t )e
'
n
n 0



n
V
n 0
*
bn' (t )
n
2


 n en
V
*
bn' (t )
n
n 0
2
 ,
converges to u~  C (0, T ;V * ) . Also, it is very easy to see that
n
u
u~ 
(as distributions on (0, T)).
t
We will prove now that u 

b e
is a weak solution for problem (1). For t = 0, we
n n
n 0


n 0
n 0
 bn (0)en ( x)   u 0n en ( x)  u 0 ( x) .
have u ( x,0) 
We obtain that
n
b e ,e
'
j
j 0
j

k
f
j 0
j
j 0

k
n
 c
j 0
H
n
  J  b j e j , ek
j 0
H
n
j 0
j
j
j
(b0 , b1 ,, bn )  e j , ek

H
H
 b 'j e j , ek
f
j
,0  k  n , so, furthermore, using that Jej = λjej , we have
e j , ek
n

j
j 0
H
n

n
 b e , e
j

H
n
 c
j 0
j
j
(b0 , b1 ,, bn )  e j , ek

H
,0  k  n .
e j , ek
H
For n → ∞, we obtain:
u
, eh
t
  Ju, ek
H
H
 (u  )u, ek  f , ek
H
, k  N .
On the Solvability of Navier-Stokes Equations
328
But (ek ) kN is an orthonormal basis in V , therefore
u
,w
t
  Ju, w
H

H
 (u  )u, w  f , w H , w V , so u is a weak solution of problem (1).
Remark 2.4 Under the above conditions, we have that the weak solution u belongs to
C (0, T ;V )  C 1 (0, T ; H ) , so it is a strong solution of problem (1).
Remark 2.5 We can prove the uniqueness of the strong solution. For this, if we suppose
that u1 and u2 are two strong solutions of the Navier-Stokes system (1), and we consider
the Fourier series developments in V


n 0
n 0
u1 ( x, t )   p n (t )en ( x), u 2 ( x, t )   q n (t )en ( x) ,
we can prove that p n (t )  q n (t ), n  N , so u1  u2 .
Now we can formulate the following theorem
~
~
~
Theorem 2.1 If f  L2 (0, T ; H ) , f t  L1 (0, T ; H ) , u 0  H , T  0 , then the function

u ( x, t )   bn (t )en ( x) given in (9) is the unique strong solution of the Navier-Stokes
n 0
nonlinear system (6)-(7).
References
1. Sburlan, C., Sburlan, S., Fourier Method for Evolution Problems, In: BAM2038CI/2002, Technical University Budapest, 2002, p. 27–35.
2. Sburlan, C., On the solvability of Navier-Stokes Equations, In: Mathematical
Modeling of Environmental and Life Science Problems, Ed. Acad. Rom., Bucarest,
2006, p. 223–230.
3. Pascali, D., Sburlan, S., Nonlinear Mappings of Monotone Type, Ed. Acad. Rom. Sijthoff & Noordhoff Int.Publ., 1978.
4. Sburlan, C., Sburlan, S., Fourier Method for Stokes Equation, BAM2038-CI/2002,
Technical University Budapest, 2002, p. 159–163.
5. Sohr, H., The Navier-Stokes Equations. An Elementary Functional Analytic
Approach, Birkhauser Verlag, 2001.
6. Zeidler, E., Applied Functional Analysis. Applications to Mathematical Physics,
Springer Verlag, 1995.
7. Sburlan. S., Morosanu, G., Monotonicity Methods for PDE’s, MB-11/PAMM,
Budapest, 1999.
8. Temam, R., Navier-Stokes Equations, North-Holland Publ. Comp., 1977.
9. Hörmander, L., The Spectral Function of an Elliptic Operator, Acta Math., 1968, p.
93–218.
Asupra rezolvării sistemului Navier-Stokes
Rezumat: In aceastǎ lucrare studiem rezolvarea sistemului Navier-Stokes
nestaţionar pentru curgerea unui fluid vâscos incompresibil, folosind metoda
Fourier abstractǎ. Este tratat atât cazul liniar, cât şi cel neliniar. Prin aceastǎ
metodǎ, obţinem existenţa şi unicitatea soluţiei tari a problemei, care este
Bulletin of the Transilvania University of Braşov Vol 13(48) - 2006
soluţia clasicǎ a problemei abstracte.Vom construi efectiv soluţia ca o serie
Fourier, folosind şirul ortonormal de funcţii proprii ale aplicaţiei de dualitate
între un spaţiu Hilbert şi dualul său topologic, care, în cazul nostru, este
operatorul lui Stokes.
Cuvinte cheie: Metoda Fourier , Ecuaţiile Navier-Stokes
329