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23 pages
Niveau: Supérieur, Licence, Bac+3
STRONG SOLUTIONS TO THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS IN THE HALF-SPACE Marco Cannone U.F.R. Mathematiques, Universite Paris 7, 75251 Paris Cedex 05, France, Fabrice Planchon Laboratoire d'Analyse Numerique, Universite Paris 6, 75252 Paris Cedex 05, France, Maria Schonbek Department of Mathematics, University of California at Santa Cruz, Santa Cruz CA 95064-1099, USA Abstract We derive an exact formula for solutions to the Stokes equations in the half-space with an external forcing term. This formula is used to establish local and global existence and uniqueness in a suitable Besov space for solutions to the Navier-Stokes equations. In particular, well- posedness is proved for initial data in L3(R3+).

  • extended over

  • studied via semi-group techniques

  • ukai's

  • space

  • divergence-free vectors

  • semi-group approach

  • t?2 ?e

  • taken over all

  • space using

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Marco Cannone
U.F.R. Math´ematiques,
Universit´e Paris 7,
75251 Paris Cedex 05, France,
Fabrice Planchon
Laboratoire d’Analyse Num´erique,
Universit´e Paris 6,
75252 Paris Cedex 05, France,
Maria Schonbek
Department of Mathematics,
University of California at Santa Cruz,
Santa Cruz CA 95064-1099, USA
We derive an exact formula for solutions to the Stokes equations in
the half-space with an external forcing term. This formula is used to
establish local and global existence and uniqueness in a suitable Besov
space for solutions to the Navier-Stokes equations. In particular, well-
3 3posedness is proved for initial data inL ( ).+
RIntroduction and definitions
tionof the velocityu(x,t)=(u (x,t),u (x,t),u (x,t)) and the pressurep(x,t)1 2 3
3of an incompressible fluid filling all of is described by the system+

∂u = Δu−∇·(u⊗u)−∇p, ∂t
∇·u = 0, (1)
 u(x,0) = u (x),0 ′ ′ 3u(x,0,t) = 0, x =(x,x )∈ , t≥ 0,3 +
′where x = (x ,x ). Strong solutions of this system are traditionally studied1 2
via semi-group techniques. Mild solutions, that is, strong solutions to the
integral equation derived from (1), which are continuous in time with value
in some Banach space, have been constructed in the half-space in [21], in
Lebesgue spaces. Ukaigave in [19] anexact formula forsolutions to the Stokes
probleminthehalf-space, andremarked thatthisallowstoconstruct solutions
thesystem[11,2,4,16,6]. TheseresultsreliedheavilyontheuseoftheFourier
transform, and on the systematic use of various scales of Besov spaces, which,
unlike Lebesgue ones, do not have local versions. Thus, a priori it is unclear
if such results can be easily extended over other domains, such as bounded or
exterior domains. However, the half-space turns out to be a particular case
of a domain, where, as originally remarked by Ukai, it is possible to obtain
an exact representation formula. In this case the corresponding Besov spaces
are well-defined, and related in an easy way to their whole space counterpart.
Therefore itis possible to extend the theory developed in [2, 16], andto obtain
various existence and uniqueness results, with very rough initial data. We
should note that in [8] a similar direct approach is used to obtain estimates for
the Stokes flow in Hardy spaces. Let us also remark here that Ukai’s formula
was also successfully used in some previous papers by H. Kozono [12, 13].
This paper is organized as follows : in the first section the definition of
Besov spaces is recalled, on both the whole space and the half-space, and the
few results that will be used later are summarized. In the second section,
following Ukai’s celebrated paper [19], an exact representation formula for the
Stokes system in the half-space with an external force is derived. This differs
from the semi-group approach as we don’t need to introduce the projection
operator on divergence-free vectors, and only use the heat kernel in the half-
Rspace. Ourresult differs from[19]since we have anexternal force, andthus we
have to adapt Ukai’s estimates to handle the new term. The third section is
devoted to the Navier-Stokes equations, where solutions are constructed with
3 3initial data u ∈L ( ). More precisely we obtain global solutions for small0 +
initial data in a Besov space, and local in time solutions for arbitrary large
3 3data. Moreover, we have uniqueness of such solutions in C([0,T),L ( )).+
31 Besov spaces in .+
zationwithacontinuous parameterinsteadofthemoreusualdyadicone. This
form of the characterization will be helpful in the following sections to relate
the special structure of the bilinear term in the Navier-Stokes equations, with
bounds in Besov spaces. For other definitions of Besov spaces see [15, 18, 1].
Definition 1
∞Let ψ(x)∈C be such that
22 −|ξ|ˆψ(ξ)=|ξ| e .
s,q˙Let p,q∈ (1,+∞], s∈ , s<1. Then, f ∈B if and only ifp
1 Z ∞ qdtq−skt ψ ∗fk <+∞, (2)pt L t0
1 ·where ψ is the rescaled function ψ( ), and this norm is equivalent to thet 3t t
qusual dyadic norm. If q =∞, we replace the L norm by sup .t
The reader familiar with both scales of spaces will note that we can replace
ˆ ˆψ(ξ) with any φ(ξ) in the Schwartz class, whose support in disjoint from 0.
Theusualcharacterizationinvolves suchafunction,withacompactsupportin
aring. Following Triebel [18], we candefine theBesov spaces onthehalf-space
as restrictions (in the distributional sense) of the Besov spaces in the whole
RDefinition 2
s,q 3˙Let p,q∈ (1,+∞],s∈ . Then B ( ) is the collection of all restrictions ofp +
s,q 3 3˙elements of B ( ). If f is the restriction of g on , its norm is defined byp +
kfk s,q = infkgk s,q (3)3 3˙ ˙B ( ) B ( )p p+
where the infimum is to be taken over allg whose restriction coincides withf.
This definition is not well-suited to any practical purpose. For positive
regularity indices, itturns out thatdirect definitions canbe given, butno such
definitions exist for negative regularity indices. Since our main interest lies
3−(1− ),∞
q 3˙in the Besov spaces B ( ) with q > 3, this could present a seriousq +
problem. However, forthisrangeofindices, theusualextension operatore,i.e.
the extension by zero, is continuous from the Besov space on the half-space to
its counterpart on the whole space. Specifically, if f is a function defined on
3 , we set+

f(x) for x ≥ 03e(f)= (4)
0 for x < 0.3
kefk −(1−3/p),∞ ≤Ckfk −(1−3/p),∞ . (5)˙ 3 ˙ 3B ( ) B ( )p p +
This is a consequence of the characterization of Fourier multipliers on Besov
spaces, and we refer the reader to Triebel ([18] p 167,168) for a complete
Let S(t) denote the heat semi-group in the whole space. We recall the
Proposition 1
′ 3Take α>0, γ ≥ 1, f ∈S ( ) a tempered distribution, then
2 γkfk=supt kS(t)fk (6)L
−α,∞ 3˙is a norm in B ( ) equivalent to the usual dyadic one.γ
RWe remark first that in the whole space it is very useful to use this definition
together with estimates of the heat kernel ([2, 16]). In what follows, we want
to adapt these estimates to the half-space. The following proposition will be
needed. Here E(t) denotes the heat operator in the half-space
Proposition 2
′ 3Take 0<α<1, γ ≥ 1, f ∈S ( ), then+
2supt kE(t)fk γ 3 ≤Ckfk −α,∞ . (7)3˙L ( ) B ( )+ γ +
We recall that E(t) can be easily represented using S(t). Let f be defined on
thehalf-space, lete˜f beitsextension to thewholespace inthefollowing sense:

f(x) for x ≥ 03e˜(f) = (8)′−f(x,−x ) for x < 0.3 3
That is, e˜completes f by the opposite of its mirror image with respect to the
hyperplan x =0. Then, it is well known thatn
E(t)f =rS(t)e˜f. (9)
Where r is the restriction from the whole space to the half-space.
3rf =f . (10)|
We may consider the caseα =1−3/p,γ =p since this is the one we will need.
−α/2 −α/2supt kE(t)fk p 3 = supt krS(t)e˜fk p 3L ( ) L ( )+ +
t t
p 3≤ supt kS(t)e˜fkL ( )
−α/2≤ 2supt kS(t)efk p 3L ( )
≤ Ckefk −(1−3/p),∞ 3˙B ( )p
≤ Ckfk −(1−3/p),∞ .˙ 3 )B (p +
R−α,∞ 3˙Thus, the initial data is in such a Besov spaceB ( ). Expressions of theγ +
type (6) for the half-space will be bounded. To conclude the section we recall
some useful relations between Besov and Sobolev spaces. Let p≤q:
3 3
−1,∞ −1,∞
p 3 q 3˙ ˙B ( )֒→B ( ) ,p q
and, for p> 3
3−1,∞1/2 3 3 3 p 3˙ ˙H ( )֒→L ( )֒→B ( ).p
s 3˙Here H ( ) denotes the usual homogeneous Sobolev space. These inclusions
3are in turn true for all spaces over . From now, we will drop the space+
p3 p 3reference for spaces over like L and write L for spaces over .+ +
2 The Stokes system with an external force
In this section we intend to obtain an exact formula for the solution of the
Stokes system in the half-space,

∂u = Δu+f −∇p, ∂t ∇·u = 0,
(11)u(x,0) = u (x),0 ′ u(x,0,t) = 0,
′ ′ 3f(x,0,t) = 0, x =(x,x )∈ , t≥ 0.3 +
Existence of exact formulas, without the use of semi-group techniques, was
first obtained in [17]. In [19] Ukai gave a complete formulation of the problem
under different boundary conditions. For our purposes we need to obtain a
formula when an external force is present. We will assume that the boundary
value of this external force is zero. If not additional terms would appear.
To obtain the exact expression of the solution we proceed following the steps
in Ukai’s paper. For details we refer the reader to [19]. A careful use of a
combination of Ukai’s results would lead to the same formula but rederiving it
presence of the boundary. For convenience the notations are kept the same as
(1) (2) (3) ′ (1) (2)in Ukai’s paper. Therefore, denote by u = (u ,u ,u ), u = (u ,u ), Rj
2the Riesz transforms defined by R =∂ (−Δ) and S the Riesz transformsj j j
2 3in , which can also be extended in a natural way to . Define
′ 3V u = −S·u +u1 0 0 0
′ 3V u = u +Su2 0 0 0
Here V is acting on vectors to give scalars, and V is vectorial. Note that Sf1 2
stands for (S f,S f). Let U be defined by1 2
′ ′Uf =rR ·S(R ·S+R )ef, (12)3
where r and e are respectively the restriction and the extension to the half-
space defined earlier. We first establish the following formula,
Theorem 1
The solution to the Stokes system (11) is given by
′u = E(t)V u −SUE(t)V u (13)2 0 1 0
t t
˜ ˜− E(t−s)SMfds−SU E(t−s)Nfds,
0 0
(3) ˜u =UE(t)V u +U E(t−s)Nfds. (14)1 0
˜ ˜M and N are two pseudo-differential operators of order 0 defined below.
Proof: take the divergence of the first equation in (11), to get
Δp = ∇·f (15)
′ ′p(x,0) = b(x)
where b is the pressure on the boundary. Recall that e˜f is the extension of f
over the whole space with the opposite of its mirror image, thus the solution
Rof (15) is
p(x) = ⋆e˜(∇·f)+Db, (16)
where D is the single layer potential solution of the Laplace equation when
there is no source:
C ′ ′Db = ∂ p b(y )dy. (17)3
2′ ′ 22 |x −y| +x3
At this stage, it is useful to recall the following important lemma ([19]), which
results from simple manipulations on the symbols of the operators:
Lemma 1
• Both operators V and V commute with any partial derivatives in the1 2
space variable.
• The operator U commutes with ∂ and ∂ , as does E(t).1 2
• For∂ we have3
′∂ U = (I−U)|∇|. (18)3
Note U∂ = ∂ U in the very particular case where it is applied to a function3 3
which has a null boundary value ([19]). Next apply the pseudo-differential
′operator ∂ +|∇| to p, to get3
1′ ′(∂ +|∇|)p =(∂ +|∇|) ⋆e˜(∇·f)=Me˜(∇·f). (19)3 3
Note that the second term disappeared, specifically the operator was applied
to annihilate such a term. This follows since
′′ −|ξ |x ′3F ′(Db)(ξ ) =e F ′(b)(ξ ). (20)x x
Hencewecanapplyourpseudo-differentialoperatortothekernel. Thustaking
′the Fourier transform with respect to x yields
′′ −|ξ |x3(∂ +|ξ|)(e ) =0.3
RUsing the same pseudo-differential operator we define
′ (3)z(x,t) = (∂ +|∇|)u (x,t). (21)3
Thus z is a solution of
′ (3) ′∂z−Δz = (∂ +|∇|)f −(∂ +|∇|)∂ pt 3 3 3
′ (3)= (∂ +|∇|)f −∂ Me˜(∇·f)3 3
= Nf
′z(x,0,t) = 0
′z(x,0) = |∇|V u .1 0
Note that due to the boundary condition the projection onto the hyperplane
commutes with ∂ and ∂ for the second term. The divergence-free property2 3
is used to commute the projection and the first term. The initial condition is
a consequence of the divergence free property of the velocity field. As before
let E(t) be the heat operator for the half-space, thus
Z t
′z =|∇|E(t)V u + E(t−s)Nf(s)ds. (22)1 0
(3)The term u can be recovered from (21), (as in [19])
x3 1(3) ′ ′pu = z(y,y )dydy , (23)3 3
′ ′ 2 22 |x −y| +|x −y |0 3 3
to yield
t1(3)u =UE(t)V u +U E(t−s)Nf(s)ds. (24)1 0 ′|∇| 0
AsinUkai,defineU from(23)andnotethatthisdefinitionofU coincides with
the previous one. In our case, unlike in [19], there is an additional term due to
Uthe external force. Here the operator cannot be seen as a composition of′|∇|
′ −1 3U and|∇| , since this last operator makes no a priori sense in . Therefore
Rit is necessary to rewrite the operator N in a more convenient way. For this
we rewrite the second term in N:
′∂ +|∇|3
∂ M = ∂3 3 2|∇|
′∂ −|∇|3′= 1+|∇|
then, indeed
2 ′∂ +∂ |∇|33 (1) (2)∂ Me˜(∇·f) = (∂ e˜f +∂ e˜f )3 1 22|∇|
2 ′∂ +∂ |∇|33 (3)+ ∂ (e˜(f ))32|∇|
′ ′|∇|(∂ −|∇|)3(1) (2) (3)˜Mf = Q(f ,f )+(1+ ∂ e˜(f ).32|∇|
This newexpression “isolates” thenormalcoordinate, andallows usto express
N in a more suitable way (recallN is actually defined onthe half-space, which
(3) (3)allows to cancel the ∂ f with ∂ e˜f ):3 3
′ (3)Nf = (∂ +|∇|)f −∂ Me˜(∇·f)3 3
2 ′∂ +∂ |∇|3′ (3) 3 (1) (2)= |∇|f − (∂ e˜f +∂ e˜f )1 22|∇|
′∂ −|∇|3′ (3)−|∇| e˜(∂ f ).32|∇|
′Given this last formula, commute E(t) and |∇| to obtain
Z t
3 ˜u =UE(t)V u +U E(t−s)Nf(s)ds, (25)1 0
˜where N is defined on vectors as
′ ′|∇| |∇|2 (1) (2) 2 (3)˜Nf =−(R +R )(S e˜(f )+S e˜(f ))+(1−R +R )eˇ(f ), (26)3 1 2 33 3|∇| |∇|