Generalized solutions of the Cauchy problem for the Navier-Stokes system and diffusion processes

0.36 MB·English

by  S. Albeverio

Generalized solutions of the Cauchy problem for
the Navier-Stokes system and diffusion processes
S. Albeverio
Institut fu¨r Angewandte Mathematik, Universita¨t Bonn,
Wegelerstr. 6, D-53115 Bonn, Germany SFB 611, Bonn, BiBoS,
Bielefeld - Bonn
CERFIM, Locarno and USI (Switzerland)
8
0
Ya. Belopolskaya,
0
2 St.Petersburg State University for Architecture and Civil
Engineering, 2-ja Krasnoarmejskaja 4,
n
190005, St.Petersburg, Russia
a
J

9 Abstract
2
We reducethe construction of aweak solution of theCauchy prob-

] lem for the Navier-Stokes system on R3 to the construction of a solu-
R
tion toastochastic problem. Namely, weconstructdiffusionprocesses
P
. which allow us to obtain a probabilistic representation of a weak (in
h
t distributional sense) solution to the Cauchy problem for the Navier-
a
Stokes system on a small time interval. Strong solutions on a small
m
time interval are constructed as well
[

AMS Subject classification : 60H10, 60J60 , 35G05, 35K45
2
v
Key words: Stochastic flows, diffusion process, nonlinear para-
8
0 bolic equations, Cauchy problem.
0
1
. Introduction
9
0
7
The main purpose of this article is to construct both strong and weak
0
: solutions (in certain functional classes) of the Cauchy problem for
v
i the Navier-Stokes (N-S) system in R3. To this end we consider a
X
stochastic problem and show that the solution of the Cauchy problem
r
a for the Navier-Stokes system can be constructed via the solution of
this stochastic problem.
The approach we develop in this article is based on the theory
of stochastic equations associated with nonlinear parabolic equations
started by McKean [1] and Freidlin [2],[3] and generalized by Belopol-
skaya and Dalecky [4], [5] on one hand and on the theory of stochastic
1
flows due to Kunita [6] on the other hand. In our previous paper [7]
we have constructed a stochastic process that allows us to prove the
existence and uniquenessof alocal intime classical (C2-smooth in the
spatialvariable)solutionoftheCauchyproblemfortheNavier -Stokes
system. Inthepresentpaperweconstructaprocesswhichallows usto
obtain construction of solutions of the both weak and strong Cauchy
problem for this system. Later we plan to apply a similar approach
for the Navier-Stokes equation for compressible fluids extending the
results from [9], [10].
A close but different approach is the Euler-Lagrange approach to
incompressible fluids which was developed by Constantin [11] and
Constantin and Iyer [12]. Shortly, the main differences in these ap-
proaches are the following: we use a probabilistic representation for
the Euler pressure instead of the Leray projection and obtain differ-
ent formulas for the stochastic representation of the velocity field. We
discuss these differences with more details in the last section of the
present work.
Thestructureof the presentarticle is as follows. Inthefirstsection
we give some preliminary information concerning different analytical
approaches totheNavier-Stokes system. Herewerecall somecommon
ways to eliminate the pressure and to obtain a closed equation for the
velocity.
The classical approaches here are based on the so called Leray
(Leray-Hodge)-projection that is a projection of the space of square
integrablevectorfieldstothespaceofdivergencefreesquareintegrable
vector fields. Applying such a projection to the velocity equation
one can eliminate the pressure p and get the closed equation for the
velocity u. This operator is used both in numerous analytical papers
(see [15] for references) and in papers wherethe N-S system is studied
fromtheprobabilisticpointofview [16],[17], [12]. Finally thepressure
is reconstructed from the Poisson equation.
One more possibility to eliminate the pressure appears when one
considers the equation for the vorticity of the velocity field u and uses
the Biot-Sawart law to obtain a closed system. From the probabilistic
point of view this approach was investigated in [18].
Inourpreviouspaper[7]wedonotusetheLerayprojectionbutin-
stead westartwith consideration ofa systemconsisting of theoriginal
velocity equation and the Poisson equation for the pressure and con-
struct their probabilistic counterpart. The probabilistic counterpart
of the N-S system was presented in the form of a system of stochas-
tic equations. Furthermore we prove the existence and uniqueness
of a solution to this stochastic system and show that in this way we
construct a unique classical (strong) solution of the Cauchy problem
for the N-S system defined on a small time interval depending on the
Cauchy data.
In the present paper we also reduce the N-S system to the sys-
tem of equations consisting of the original velocity equation and the
Poisson equation for the pressure but then an associated stochastic
problem considered here allows to construct a generalized (distribu-
tional) solution to the Cauchy problem for the N-S system. The as-
2
sociated stochastic problem is studied in section 5. In sections 1-4
we expose auxiliary results used in section 5. Namely, in section 1 we
giveanalytical preliminariesandrecallthenotionsofstrong,weakand
mild solutions to the Cauchy problem for the Navier-Stokes system.
More detail can be found the recent book by Lemarie-Rieusset [15].
In section 2 we give a short review of probabilistic approaches to the
investigation of the Navier-Stokes system [7], [16] -[18]. In section 3
we study a probabilistic representation of the solution to the Poisson
equation, while in section 4 we recall some principal fact of the Ku-
nita theory of stochastic flows and apply the results from [19], [20] to
construct a solution of the Cauchy problem for a nonlinear parabolic
equation (see also [21]). Finally all these preliminary results are used
toconstructtheprobabilisticcounterpartoftheNavier-Stokes system,
prove that there exists a unique local solution to the corresponding
stochastic system and apply the results to construct both the strong
and weak (and simultaneously mild) solutions to the Cauchy problem
for the Navier-Stokes system.
1 Preliminaries
As it was mentioned in the introduction the main purpose of this
article is to construct both strong and weak solutions (in certain func-
tionalclasses) oftheCauchyproblemfortheNavier-Stokes systemvia
diffusion processes.
Consider the Cauchy problem for the Navier-Stokes system
∂u
+(u, )u = ν∆u p, u(0,x) = u (x), x R3, (1.1)
0
∂t ∇ −∇ ∈
divu = 0. (1.2)
Here u(t,x) R3,x R3,t [0, ) is the velocity of the fluid at the
∈ ∈ ∈ ∞
position x at time t and ν > 0 is the viscosity coefficient and p(t,x)
is a scalar field called the pressure which appears in the equation to
enforce the incompressibility condition (1.2). Later we set ν = σ2 for
2
reasons to be explained below.
By eliminating the pressure from (1.1),(1.2) one gets a nonlinear
pseudo-differentialequationwhichistobesolved. Thereexistdifferent
ways to do it and we consider now some of them.
Given a vector field f let Pf be given by
Pf = f ∆ 1 f. (1.3)
−
−∇ ∇·
Here and below we denote by u v the inner product of vectors u and
·
v valued in R3.
The map P called the Leray projection is a projection of the space
L2(R3) L2(R3)3 of square integrable vector fields to the space of
≡
divergence free vector fields and we discuss its properties below. A
quitedirectdefinition ofPis connected with theRiesz transformation
Rj. Recall that Rk = √∇k∆ which means that for f ∈ L2 we have
−
3
(R f) = iξjfˆ(ξ) where (f) = fˆ is the Fourier transform of f.
F j ξ F
Then P is de|fi|ned on L2(R3) as P = Id+R R or
⊗
3
(Pf) = f + R R f .
j j j k k
k=1
X
Since R R is a Calderon-Zygmund operator, Pf may be defined on
k j
many Banach spaces.
Set
3
γ(t,x) = u u = Tr[ u]2 (1.4)
k j j k
∇ ∇ ∇
k,j=1
X
and note that γ can be presented as well in the form
γ = u u = (u u ).
k j k j
∇·∇· ⊗ ∇ ∇
j,k
X
.
By computingthedivergence of bothpartsof (1.1) andtaking into
account (1.2) we derive the equation
∆p(t,x) = γ(t,x) (1.5)
−
thus arriving at the Poisson equation. The formal solution of the
Poisson equation is given by
p = ∆ 1γ = ∆ 1 u u (1.6)
− −
∇·∇· ⊗
since divu= 0 and finally we present p in the form
∇
p = ∆ 1 u u.
−
∇ ∇ ∇·∇· ⊗
Substituting this expression for p into (1.1) we obtain the following
∇
Cauchy problem
∂u
= ν∆u P (u u), u(0) = u . (1.7)
0
∂t − ∇· ⊗
There are a number of ways to define a notion of a solution for
the Cauchy problem (1.7). We will appeal mainly to the Leray weak
solution [13] or to the Kato mild solution [14].
1.1 Leray and Kato approaches to the solu-
tions of the Navier-Stokes equations
Let = (R3) = C be the space of all infinitely differentiable
c∞
D D
functions on R3 with compact support equipped with the Schwartz
topology. Let be the topological dual of and denote by φ,ψ =
′
D D h i
φ(x)ψ(x)dx the natural coupling between φ and ψ . If it
R3 ∈ D ∈ D′
will not lead to misunderstandings we will use the same notation for
R
vector fields u and v as well, that is
3
h,u = h (x)u (x)dx.
k k
h i R3
Z k=1
X
4
We recall that a weak solution of the N-S system on [0,T] R3
×
is a distribution vector field u(t,x) in ( ((0,T) R3))3 where u is
′
D ×
locally square integrable on (0,T) R3, div u = 0 and there exists
×
p ((0,T) R3) such that
′
∈ D ×
∂u
= ν∆u (u u) p, limu(t)= u (1.8)
0
∂t −∇· ⊗ −∇ t 0
→
holds.
The Leray solution to the N-S equations is constructed through a
limiting procedure from the solutions to the mollified N-S equations
∂u = ν∆u ((u q ) u) p,
∂t −∇· ∗ ε ⊗ −∇
u= 0, (1.9)
∇·
limt 0u(t) = u0.
→
Namely it is proved that there exists a function
u L ((0, ),L2) L2((0,T),(H˙ 1))
ε ∞
∈ ∞ ∩
such that (at least for a subsequence u ) strongly converging in
εk
(L2 ((0,T) R3))3 to u which satisfies (1.9).
loHcere H˙ 1×is the homogenous Sobolev space H˙ 1 = f S : f
{ ∈ ′0 ∇ ∈
L2 with norm f = f .
H L
} k k 1 k∇ k 2
On the other hand to construct the Kato solution means to con-
struct a solution u to the following integral equation
t
u(t)= et∆u e(t s)∆P (u u)(s)ds. (1.10)
0 −
− ∇· ⊗
Z0
Note that instead of looking for u(t,x) and p(t,x) one can prefer
to look for their Fourier images uˆ(t,λ) = (2π)−32 R3e−iλ·xu(t,x)dx.
The Leray and Kato approaches stated in terms of the Fourier
R
transformations of the Navier-Stokes system can be described as fol-
lows.
Applying the Fourier transformation to the relation (1.7) written
in the form
t t
h,u(t) = h,u(0) + h,∆u(s) h, (u u)(s)
h i h i h i− h ∇· ⊗ i
Z0 Z0
we derive the relation
t
hˆ,uˆ = hˆ,uˆ hˆ, λ 2uˆ(s) ds (1.11)
0
h i h i− h | | i −
Z0
i t 3
λkhˆl(λ),uˆ (s,λ)u (s,λ λ)dλdλds.
(2π)32 Z0 ZR3ZR3k,l=1 ′ l k − ′ ′
X
Here uˆ corresponds to the Fourier transformation of u.
On the other hand if we are interested in the Kato mild solution
of the N-S system then we may apply the Fourier transformation to
(1.10) and derive the following equation
χ(t,λ) = exp ν λ 2t χ(0,λ)+ (1.12)
{− | | }
5
t 1
ν λ 2e νλ2(t s) (χ(s) χ(s)) (λ)ds
− | | −
| | 2 ◦
Z0 (cid:20) (cid:21)
for the function
3
χ(t,λ) = 2 π 2 λ 2uˆ(t,λ).
ν 2 | |
(cid:18) (cid:19)
Here
i λ dλ
′
χ1◦χ2(λ) = −π3 ZR3(χ1(λ1)·eλ)Π(λ)χ2(λ−λ′)|λ′|2||λ| −λ′|2, (1.13)
e = λ and
λ λ
| |
Π(λ)χ = χ e (χ e ), (1.14)
λ λ
− ·
Coming back to (1.7) we note that the Leray projection allows
to eliminate the pressure p(t,x) from the Navier-Stokes system, to
construct u and finally to look for p defined by the solution of the
auxiliary Poisson equation.
Another way to eliminate p(t,x) from the system (1.1),(1.2) is to
consider the function v(t,x) = curlu(t,x) called the vorticity. Since
curl p(t,x) = 0 one can derive a closed system for u and v. Namely
∇
for u and v we arrive at the system consisting of the equation
∂v
+(u )v = ν∆v+(v )u, (1.15)
∂t ·∇ ·∇
and the so called Biot-Savart law having the form
1 (x y) v(y)
u(t,x) = − × dy. (1.16)
4π R3 x y 3
Z | − |
Here the cross-product u v is given by
×
e e e
1 2 3
u v = det u u u =
1 2 3
×  
v v v
1 2 3
 
(u v u v )e +(u v u v )e +(u v u v )e ,
2 3 3 2 1 3 1 1 3 2 1 2 2 1 3
− − −
where (e ,e ,e ) is the orthonormal basis in R3.
1 2 3
Note that the term (v )u can be written as ( u)v or even as
·∇ ∇
v, where is the deformation tensor defined as the symmetric
u u
D D
part of u
∇ 1
= ( u+ uT),
u
D 2 ∇ ∇
since by direct computation we see that
1
( u)v v = ( u+ uT)v = 0.
u
∇ −D 2 ∇ ∇
To be able to present the precise statements concerning the ex-
istence and uniqueness of solutions to the N-S equations we have to
introduce a number of functional spaces to be used in the sequel.
6
1.2 Functional spaces
We describe here functional spaces which will be used in the sequel.
Let = (R3)bethespaceofallinfinitelydifferentiablefunctions
D D
on R3 with compact supports equipped with the Schwartz topology.
Let be the topological dual to . The elements of are called
′ ′
D D D
Schwartz distributions.
The space of R3-valued vector fields h with components h
k
∈ D
shall be denoted by D(R3) and D shall denote the space dual to
′
D(R3).
Let Lq(R3) denote the Banach space of functions f which are ab-
solutely integrable taken to the q-th power with the norm f =
q
k k
( f(x)qdx)1q;
R3| |
Let Z denote the set of all integers, and suppose that k Z is
R ∈
positive and 1 < q < . Denote by Wk,q = Wk,q(R3) the set of all
∞
real functions h defined on R3 such that h and all its distributional
derivatives α of order α = α k belong to Lq(R3). It is a
j
∇ | | ≤
Banach space with norm
P
h k,p = ( Dαh(x)qdx)1q. (1.17)
k k R3| |
α kZ
|X|≤
We denote the dual space of Wk,q by W k,m where 1 + 1 = 1. El-
− m q
ements of W k,q can be identified with Schwartz distributions. The
−
space W k,q is also a Banach space with norm
−
φ = sup φ,h ,
k,q
k k− khkk,q≤1|h i|
where
φ,h = φ(x)h(x)dx.
h i R3
Z
The spaces Wk,p for k Z and p > 1 are called Sobolev spaces. If
∈
p = 2 we use the notation Hk for the Hilbert spaces Wk,2 . In a
natural way one can define the spaces Wk,q, Hk of vector fields with
components in Wk,p, and Hk and so on.
Set
= v D :divv = 0
V { ∈ }
and let
H = closure of in L2(R3) , V = closure of in H1 .
{ V } { V }
(1.18)
Let Ck(R3,R3) denote the space of k-times differentiable fields
b
with the norm
g = Dβg
k kCbk k k∞
β k
|X|≤
and let Ck,α(R3,R3) be the space of vector fields whose k-th deriva-
b
tives are H¨older continuous with exponent α, 0 < α < 1 with the
norm
g = g +[g]
k kCbk,α k kCbk k+α
7
where
Dβg(x) Dβg(y)
[g] = sup | − |.
k+α x y α
|βX|=kx,y∈R3 | − |
We denote by Lip(R3) the space of bounded Lipschitz continuous
functions with the norm
g(x) g(y)
g = sup | − |.
Lip
k k x y
x,y R3
∈ | − |
Spaces of integrable functions on the whole R3 appear to be not
satisfactory to construct a solution to the N-S equations and one has
to consider spaces of locally integrable functions.
Let f : R3 R1 be a Lebesgue measurable function. A set of
→
functions f : f(x)pdx < for all compact subsets K in R3 is
{ p K| | ∞}
denotedbyL andcalledaspaceoflocallyintegrablefunctions. Note
locR
that L1(R3) L1 (R3) . Although Lp (R3) are not normed spaces
⊂ loc loc
they are readily topologized. Namely a sequence u converges to u
n
inLp (R3)if u uinLp(K)foreachopenK { G}havingcompact
loc { n} → ⊂
closure in R3. Local spaces Wk,p(R3) can be defined to consist of
loc
functions belonging to Wk,p(K) for all compact K R3.
⊂
k,p
A local space W (G) is defined as a space of functions belonging
loc
to Wk,p(G) for all G G with compact closure in G. A function f
′ ′
⊂ ∈
k,p k,p
W (G) with compact support will in fact belong to W (G). Also
loc 0
functions in W1,p(G) which vanish continuously on the boundary ∂G
1,p
will belong to W (G) since they can be approximated by functions
0
with compact support.
In the whole space R3 and with p, q satisfying 1 q p < de-
≤ ≤ ∞
noteby p anonhomogenousMorreyspaceandbyMp ahomogenous
q q
M
Morrey space with norms given respectively by
Mpq = (f ∈ Lqloc :kfkMpq = xs0uRp30s<uRpRp3−q3kfkLq(B(x0,R)) < ∞),
∈
(1.19)
and
Mqp = (f ∈Lqloc : kfkMpq = xs0u∈Rp30<suRp≤1R3p−q3kfkLq(B(x0,R)) < ∞)
(1.20)
where B(x ,R) is a closed ball of R3 with center at x and radius R.
0 0
Respectively the integrable function is said to belong to Mq(G) if
there exists a constant C such that
3(1 1)
f(x)dx CR −q (1.21)
| | ≤
ZG∩BR
for all balls B . The norm in Mq(G) is defined as the minimum of
R
the constants C satisfying (1.21)
Adistributionuon(0,T) R3 issaidtobeuniformlylocallysquare
×
integrable if for all ϕ ((0,T) R3)
∈ D ×
T
sup ϕ(t,x x )u(t,x) 2dxdt < .
0
x0 R3Z0 ZR3|k − k ∞
∈
8
Equivalently u is uniformly locally square integrable if and only
if for all t0 < t1 ∈ (0,T) the function Ut0,t1(x) = ( tt01ku(t,x)k2dt)12
belongs to the Morrey space L2 . In this case we write
uloc R
u (L2 L2((t ,t ) R3)).
∈ ∩0<t0<t1<T uloc t 0 1 ×
For 1 p the Morrey space of uniformly locally integrable
functions≤on R≤3 ∞is the Banach space Lp of Lebesgue measurable
uloc
functions f on R3 such that the norm f is finite, where
p,uloc
k k
f p,uloc = sup ( f(x)pdx)1p.
k k | |
x0∈R3 Zkx−x0k<1
For t < t , 1 p,q the space Lp Lq((t ,t ) R3)) is the
0 1 ≤ ≤ ∞ uloc,x t 0 1 ×
Banach spaceof Lebesguemeasurablefunctions f on(t ,t ) R3 such
0 1
×
that the norm
sup ( ( t1 f(t,x)qdt)pqdx)p1
x0∈R3 Zkx−x0k<1 Zt0 | |
is finite.
A C function f on R3 is called rapidly decreasing if
∞
lim Dαf(x)(1+ x )n = 0
x | | k k
→∞
holds for any multi-index α and any positive integer n. Let =
S
(R3) be the space of rapidly decreasing C functions equipped
∞
S −
with the Schwartz topology and be the topological dual of . Since
′
S S
includes , is a subset of . The elements of are called
′ ′ ′
S D S D S
tempered distributions.
1.3 Weak, strong and mild solutions of the
Navier-Stokes system
Now weare readyto give moreprecisedefinitions andstatements con-
cerningtheexistenceanduniquenessofsolutionsoftheN-Sequations.
Definition 1.1.(Weak solutions)A weak solution of the Navier-
Stokes system on (0,T) R3 is a distribution vector field u(t,x), u
× ∈
( ((0,T) R3)d such that
′
D ×
a) u is locally square integrable on (0,T) R3 ,
×
b) u= 0,
∇·
c) there exists p ((0,T) R3) such that
′
∈ D ×
∂ u= ∆u (u u) p.
t
−∇· ⊗ −∇
The classical results concerning the existence of square integrable
weak solutions are due to Leray [13].
Theorem 1.1. (Leray’s theorem) Let u (L2(R3))3 so that
0
∈
u = 0. Then there exists a weak solution u L ((0, ),(L2)d)
∞
∇· ∈ ∞ ∩
L2((0, ),(H1)3) for the Navier -Stokes equation on (0, ) R3 so
∞ ∞ ×
9
that lim u(t) u = 0. Moreover, the solution u satisfies the
t 0 0 2
→ k − k
energy inequality
t
u(t) 2 +2 u 2dxds u 2, (1.22)
k k Z0 ZR3k∇⊗ k ≤ k 0k2
where
d d d
u(t) 2 = u (t,x)2dx, u = ∂ u 2.
k k2 R3| k | ∇⊗ | k j|
k=1Z k=1j=1
X XX
Definition 1.2. (Mild solution) The Kato mild solution of
(1.1), (1.2) is a solution of (1.7) constructed as a fixed point of the
transform
t
v et∆u (x) e(t s)∆P (v v)(θ,x)dθ =et∆u B(v,v).
0 − 0
7→ − ∇· ⊗ −
Z0
(1.23)
.
Note that the right hand side of (1.7)
t
et∆u (x) e(t s)∆P (u u)(θ,x)dθ = Φ(t,x,u) (1.24)
0 −
− ∇· ⊗
Z0
is a nonlinear map in the corresponding space and the solution u is
obtained by the iterative procedure
u0 = et∆u , un+1 = et∆u B(un,un). (1.25)
0 0
−
Hence to construct a mild solution to (1.1), (1.2) means to find
a suitable functional space for which Φ(t,x,u) given by (1.20) is a
contraction.
To this end one has to find a subspace of L2 L2((0,T)
ET uloc,x t ×
R3) so that the bilinear transformation B(u,v) of the form (1.11) is
bounded as a map . Then one may consider the space
T T T
E ×E → E
E defined by f E iff f and (et∆f) and prove
T ′ ′ 0<t<T T
⊂ S ∈ ∈ S ∈ E
the following result.
Theorem 1.3. The Picard contraction principle.
Let L2 L2([0,T) R3) be such that the bilinear map B is
ET ⊂ uloc,x t ×
bounded on Then:
T
E
(a) If u is a weak solution for the Navier-Stokes equation
T
∈ E
(1.1) (1.2) then the associated initial value belongs to E .
T
(b) There exists a positive constant C such that for all u E
0 T
∈
satisfying u = 0 and et∆u < there exists a weak solution
∇· k 0kET ∞
u of (1.1) (1.2) associated with the initial value u
T 0
∈ E
t
u= et∆u e(t s)∆P (u u)ds. (1.26)
0 −
− ∇· ⊗
Z0
The classical results assert that for sufficiently smooth initial data
for example for u in the Sobolev space Hk , k > d +1, there exists
0 2
a short time strong unique solution to (1.1), (1.2). On the other
hand Leray proved the existence of a global weak solution of finite
energy, i.e. u L2 called the Leray-Hopf solution. Although the
∈
uniquenessandfullregularity ofthissolutionarestillanopenproblem
nevertheless one knows that if a strong solution exists then a weak
solution coincides with it.
10