On the regularity of the bilinear term for solutions to the incompressible

Marco Cannone

Description

Niveau: Supérieur, Licence, Bac+3
On the regularity of the bilinear term for solutions to the incompressible Navier-Stokes equations Marco Cannone U.F.R. Mathematiques, Universite Paris 7, 2 place Jussieu, 75251 Paris Cedex 05, France, e-mail and Fabrice Planchon Program in Applied and Computational Mathematics, Princeton University, Princeton NJ 08544-1000, USA e-mail Abstract We derive various estimates for strong solutions to the Navier- Stokes equations in C([0, T ), L3(R3 )) that allow us to prove some reg- ularity results on the kinematic bilinear term. Introduction and definitions The Cauchy problem for the Navier-Stokes equations governing the time evolution of the velocity u(x, t) = (u1(x, t), u2(x, t), u3(x, t)) and the pressure 1

• besov spaces


• differential operator


• h˙sp ??


• ?? l3


• definition involving


• various estimates


• space variable


• sobolev ones


• reg- ularity results

Sujets

• Planchon
• Princeton University
• Differential operator

Informations

• Publié par: vehot
• Nombre de lectures: 15
• Langue: English

Extrait

On the regularity of the bilinear term
for solutions to the incompressible
Navier-Stokes equations
Marco Cannone
U.F.R. Math´ematiques,
Universit´e Paris 7,
2 place Jussieu,
75251 Paris Cedex 05, France,
e-mail [e-mail]
and
Fabrice Planchon
Program in Applied and Computational Mathematics,
Princeton University,
Princeton NJ 08544-1000, USA
e-mail [e-mail]
Abstract
We derive various estimates for strong solutions to the Navier-
3 3Stokes equations in C([0,T),L ( )) that allow us to prove some reg-
ularity results on the kinematic bilinear term.
Introduction and definitions
The Cauchy problem for the Navier-Stokes equations governing the time
evolution of the velocityu(x,t)= (u (x,t),u (x,t),u (x,t))and thepressure1 2 3
1
R3p(x,t) of an incompressible fluid filling all of is described by the system

∂u = Δu−∇·(u⊗u)−∇p,
∂t
(1) ∇·u = 0, 3u(x,0) = u (x), x∈ , t≥ 0.0
The existence of local solutions to this system which are strongly continuous
p 3in time and take value in Lebesgue spaces L ( ) is a well known result for
p ≥ 3 (see [2]). In the critical case, p = 3, for which solutions of (1) are
invariant by rescaling, one can construct strong solutions in a subclass of
3 3 3C (L ) = C([0,T),L ( )) (see [5, 20, 7, 9]), but their uniqueness withint
the natural class was proved only recently ([6]). The key tool in obtaining
3−1,∞
q˙this uniqueness result was the use of the Besov spaces B , for q < 3.q
These spaces have been used previously, but mainly withq≥ 3, in obtaining
global existence results (see [12],[2],[18]). In addition, it was already noticed
invariouscontexts (see[2],[17])howthebilinearterm, which isthedifference
between the solution and the solution to the linear heat equation (with same
initialdata),behavesbetterthanthesolutionitself. Weimprovetheseresults
in the present paper, and show how this gain in regularity is related to the
uniqueness problem, the main estimates involved being of the same kind.
Moreover, this allows to extend the decay estimates on the gradient of the
solution to (1) obtained by T. Kato in [9].
In order to simplify our study let us introduce the projection operator
onthedivergence freevectorfields. Weremarkthat isapseudo-differential
operator of order 0 which will be continuous on all spaces subsequently used
p(primarilybecauseitiscontinuousonallLebesguespacesL ,for1<p<∞).
A common method solving (1) is to reduce the system to an integral
equation,
Z t
(2) u(x,t) =S(t)u (x)− S(t−s)∇·(u⊗u)(x,s)ds,0
0
tΔwhereS(t)=e is the heat kernel, and then to solve it via a fixed point ar-
gument in a suitable Banach space (see [2],[9],[10]). Following [2], we remark
that the bilinear term in (2) can be reduced to a scalar operator,
Z
t 1 ·
√(3) B(f,g)= G ∗(fg)ds,
2(t−s) t−s0
2
R
R
R
P
P
R
Pwhere G is analytic, such that
C
(4) |G(x)| ≤
41+|x|
C
(5) |∇G(x)| ≤ .
41+|x|
This can be derived easily from the study of the operator under the integral
sum, S(t−s)∇·, since its symbol consists of terms like
ξ ξ ξ 2j k l −(t−s)|ξ|(6) − e
2|ξ|
2−(t−s)|ξ|outside the diagonal, with another term ξ e on it. For the sake ofj
2−|ξ|simplicity, we will take G as the inverse Fourier transform of|ξ|e .
As we mentioned previously, Besov spaces are a useful tool in studying
the bilinear operator B. In what follows we will use spaces of functions on
3, so henceforth the reference to the domain space will be omitted. Let us
recall the following definition. The reader will find equivalent definitions of
Besov spaces in [1], [16],[19].
Definition 1
∞Let θ(x)∈C be such that
2−|ξ|ˆθ(ξ)=|ξ|e .
s,q˙Let p,q∈ (1,+∞),s∈ , s< 1. Then, f ∈B if and only ifp
1Z ∞ qdt−s q(7) kt θ ∗fk <+∞,pt L t0
1 ·where θ is the rescaled function θ( ), and this norm is equivalent to thet 3t t
usual dyadic norm.
We will also make use of the homogeneous version of the Sobolev-Bessel
ss p˙ 2spaces,definedsimplybyf ∈H ifandonlyifΔ f ∈L . Thereaderfamiliarp
22 −|ξ|ˆ ˆwith Besov spaces will note that by replacing θ(ξ) with θ(ξ) =|ξ| e one
obtains the usual characterization via the Gauss-Weierstrass kernel. We will
use this fact further in the paper. Among various embeddings between these
p 0,2˙spaces and the Lebesgue and Sobolev ones, we recall that L = H , forp
s,2 s s˙ ˙ ˙1<p<∞, and B =H =H , the usual homogeneous Sobolev space.2 2
3
R
R
PTheorems and proofs
Let us start with the aforementioned result on the regularity of the solution.
Theorem 1
3 3Let u(x,t) be a solution of (2) in C([0,T),L ), with initial data u ∈L and0
denote by w the function w =u−S(t)u , then0
1˙(8) w∈C([0,T),H ).3/2
3/2In other words, the gradient of w is continuous in time with value in L .
1 3˙Ofcourse,thisestimatemakessense,for,viaSobolevembedding,H ֒→L .
3/2
This regularity result can be seen in connection with an estimate derived by
T. Kato in [9] that assures that u(x,t), the solution of Theorem 1, is such
1−3/2q qthat t ∇u(x,t)∈L , for q≥ 3. Therefore the estimate (8) extends this
last estimate to the value q ≥ 3/2 for the bilinear term alone, as if u only0
3belongs to L , the linear part in general doesn’t verify (8).
Let’s postpone the proof of the theorem for a moment, and comment
furtheronthemeaningofthisresult. In[17],itwasshownthatforself-similar
3
−(1− ),∞
q3 ˙solutions (forwhich theinitialdatawasn’t inL , butinsomeB withq
1,2˙q > 3), the bilinear term was in B , and it is a simple matter to obtain
3/2
1 s s,2˙ ˙ ˙H instead. This is slightly better, as H ֒→ B for p < 2. Now, in3/2 p p
order to obtain this result, one makes use of the special structure of a self-
similar solution. For such solutions, the time regularity is intimately related
√ √
to the space regularity because of the scaling u(x,t) =1/ tU(x/ t). On
p qthe other hand, using L (L ) estimates, it was proved in [17] that for at x
0,23 3 3˙solution in C (L ) with initial data u ∈ L the function w ∈ B ֒→ L .t 0 3
0,21˙ ˙One remarks that H ֒→B . The proof of that result was a consequence33/2
of the following proposition applied with q = 6.
Proposition 1
Let 3 ≤ q ≤ 6. Then the bilinear operator B(f,g) is bicontinuous from
q2 2 6−1,
1−3/q 1−3/q q q−3q q ∞ ˙L (L )×L (L ) into L (B ).t tx x t q/2
∞ 3 ∞ 3In particular, if q = 3, B(f,g) is bicontinuous from L (L )×L (L ) intot x t x
1,∞∞ ˙L (B ). This last estimate for q = 3 was used in [6]. The proof we are3t
2
4giving here for 3 ≤ q ≤ 6 is nothing but paraphrasing the case q = 6 dealt
with in [17]. More precisely, we will prove the estimate by duality. To this
end, let 0<T <∞. By hypothesis,
Z T 1
1−3/q
(9) kfg(x,t)k dt<∞,
q/2
0
where the integral in time is replaced by a sup = sup if q = 3. Thent t∈[0,T]
∞for an arbitrary test function ϕ(x)∈C , we consider the functional0
(10) I =hB(f,g),ϕi.t
We find
Z tD E1 ·
(11) I = G √ ∗(fg),ϕ dst 2(t−s) t−s0
√Z tD E1 ·2 ˇ= 2 fg(t−s ), G ∗ϕ ds
3s s0
ˇwhere G(x) = G(−x), and we made a change of variable. Applying H¨older
inequality both in time and space variables, we get
Z 1−3/qt 1
1−3/q
(12) |I| ≤ kfg(t−s)k dsqt L
0
!√ 3/qZ t dsq/36/q−1ˇ× ks G (·)∗ϕ ks q/(q−2) s0

1 ·ˇ ˇwhere G = G . Using Definition 1, the second integral is found to bes 3s s
6 q 6 q1− , −1,
q 3 q q−3˙ ˙lessthanthenormofϕinB ,whichisexactlythedualofB (Theq/(q−2) q/2
restriction q ≤ 6 is mainly because we are interested in positive regularity
indices). We see that with this proposition we are far from the actual result
pof Theorem 1, because the third index is greater than 2. Nevertheless, if
p−3
we think of uniqueness, we can make a parallel with a recent result, proved
(among other things, and in a more general framework) in [8]. We state it
here in a pure analytical frame instead of a stochastic one and applying the
Besov formalism once again. In order to proceed we need to introduce the
following definition involving pseudo-measures.
Definition 2
3A tempered distribution ψ is called a pseudo-measure on if
ˆ