Wellposedness and Stability Results for the Navier-Stokes Equations in <span class="mathjax-formula">$\mathbf {R}^3$</span>

www.elsevier.com/locate/anihpc

Wellposedness and stability results for the Navier–Stokes equations in R ³

Jean-Yves Chemin

, Isabelle Gallagher

aLaboratoire J.-L. Lions, UMR 7598, Université Paris VI, 175, rue du Chevaleret, 75013 Paris, France bInstitut de Mathématiques de Jussieu, UMR 7586, Université Paris VII, 175, rue du Chevaleret, 75013 Paris, France

Received 30 March 2007; accepted 31 May 2007 Available online 1 February 2008

Abstract

In [J.-Y. Chemin, I. Gallagher, On the global wellposedness of the 3-D Navier–Stokes equations with large initial data, Annales Scientifiques de l’École Normale Supérieure de Paris, in press] a class of initial data to the three dimensional, periodic, incom- pressible Navier–Stokes equations was presented, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The aim of this article is twofold. First, we adapt the construction of [J.-Y. Chemin, I. Gallagher, On the global wellposedness of the 3-D Navier–Stokes equations with large initial data, Annales Scientifiques de l’École Normale Supérieure de Paris, in press] to the case of the whole space: we prove that if a certain nonlinear function of the initial data is small enough, in a Koch–Tataru [H. Koch, D. Tataru, Well-posedness for the Navier–Stokes equations, Advances in Mathematics 157 (2001) 22–35] type space, then there is a global solution to the Navier–Stokes equations. We provide an example of initial data satisfying that nonlinear smallness condition, but whose norm is arbitrarily large inC⁻¹. Then we prove a stability result on the nonlinear smallness assumption. More precisely we show that the new smallness assumption also holds for linear superpositions of translated and dilated iterates of the initial data, in the spirit of a construction in [H. Bahouri, J.-Y. Chemin, I. Gallagher, Re- fined Hardy inequalities, Annali di Scuola Normale di Pisa, Classe di Scienze, Serie V 5 (2006) 375–391], thus generating a large number of different examples.

©2008 Published by Elsevier Masson SAS.

Keywords:Navier–Stokes equations; Global wellposedness

1. Introduction

1.1. On the global wellposedness of the Navier–Stokes system

We consider the three dimensional, incompressible Navier–Stokes system inR³, (NS)

⎧⎨

⎩

∂tu−u+u· ∇u= −∇p, divu=0,

u_|t=0=u0.

* Corresponding author.

E-mail addresses:[email protected] (J.-Y. Chemin), [email protected] (I. Gallagher).

0294-1449/$ – see front matter ©2008 Published by Elsevier Masson SAS.

doi:10.1016/j.anihpc.2007.05.008

Hereuis a three-component vector fieldu=(u₁, u₂, u₃)representing the velocity of the fluid, pis a scalar denoting the pressure, and both are unknown functions of the space variablex∈R³and of the time variablet∈R⁺. We have chosen the kinematic viscosity of the fluid equal to one for simplicity – a comment on the dependence of our results on viscosity is given further down in this introduction.

It is well known that(NS)has a global, smooth solution if the initial data is small enough in the scale invariant spaceH˙¹², where we recall thatH˙^s is the set of tempered distributionsf with Fourier transformfˆinL¹_loc(R³)and such that

fH˙^s def=

R³

|ξ|^2sf (ξ )ˆ ²dξ ¹

2

is finite. We recall that the scaling of(NS)is the following: for any positiveλ, the vector fielduis a solution associated with the datau0ifu_λis a solution associated withu0,λ, where

u_λ(t, x)=λu λ²t, λx

and u_0,λ(x)=λu₀(λx).

The result inH˙¹² is due to H. Fujita and T. Kato in [10] (see also [22] for a similar result, where the smallness of u₀ is measured byu_0L²∇u_0L²). Since then, a number of works have been devoted to proving similar wellposedness results for larger classes of initial data; one should mention the result of T. Kato [17] where the smallness is measured inL³(see also [15]) and the result of M. Cannone, Y. Meyer and F. Planchon (see [4]) where the smallness is measured in the Besov spaceB˙⁻¹⁺

3 p

p,∞ . Let us recall that, for positiveσ, u_B˙−σ

p,r

def=t^σ2S(t )u

L^p

L^r(R⁺,^dt_t )

where S(t )=e^t denotes the heat flow. The importance of this result can be illustrated by the following example: ifφ is a function in the Schwartz spaceS(R³), let us introduce the family of divergence free vector fields

φ_ε(x)=cos x3

ε

(∂₂φ,−∂₁φ,0).

Then, for smallε, the size ofφ_εB˙^−σ_p,r isε^σ.

Let us also mention the result by H. Koch and D. Tataru in [18] where the smallness is measured in the spaceBMO⁻¹, defined by

u_BMO−1

def= u_B˙−1

∞,∞+ sup

x∈R³ R>0

R⁻³²

P (x,R)

S(t )u(y)²dy dt ¹

2

,

whereP (x, R)= [0, R²] ×B(x, R)andB(x, R)denotes the ball of radiusRcentered at zero.

As observed by H. Koch and D. Tataru, this norm seems to be the ultimate norm for the initial data for which the classical Picard’s iterative scheme can work. Indeed the first iterate, S(t )u0 must be inL² locally inR⁺×R³. In particular, S(t )u0must be inL²([0,1] ×B(0,1)). Then considering the norm of the space must be invariant by translation as well as by the scaling of the equation, we get the norm · _BMO−1. Moreover, let us notice that we have

sup

x∈R³ R>0

R⁻³²

P (x,R)

S(t )u(y)²dy ¹

2 S(t )u

L²(L^∞)

and thusu_B˙−1

∞,∞u_BMO−1u_B˙−1

∞,2.

Moreover the space C˙⁻¹= ˙B_∞⁻¹_,∞ seems to be optimal independently of the method of resolution, due to the following argument (see [1] for instance). LetBbe a Banach space continuously included in the spaceSof tempered distributions onR³. Let us assume that, for any(λ, a)∈R⁺×R³,

f λ(· −a)

B=λ⁻¹fB.

Then we have that|f, e^−|·|2 |CfB. By dilation and translation, we deduce that f_C˙−1=sup

t >0

t¹²S(t )f

L^∞CfB.

We have proved that any Banach space included inS, translation invariant and which has the right scaling is included inC˙⁻¹.

Let us point out that none of the results mentioned so far are specific to the Navier–Stokes equations, as they do not use the special structure of the nonlinear term in(NS).

Our aim in this paper is to go beyond the smallness condition on the initial data and to exhibit arbitrarily large initial data inC˙⁻¹which generate a unique, global solution. We refer among others to [9,11,13,16,19,20,23] or [24] where different types of examples are provided. This was also performed in [8] in the periodic case, where we presented a new, nonlinear smallness assumption on the initial data, which may hold despite the fact that the data is large.

That result uses the structure of the nonlinear term, as it is based on the fact that the two dimensional Navier–Stokes equation is globally well posed.

The first theorem of this paper consists in a result of global existence under a nonlinear smallness hypothesis (The- orem 1 below). The proof consists mainly in introducing an idea of [6] in the proof of the Koch and Tataru Theorem.

The nonlinear smallness hypothesis is, roughly speaking, that the first iterate S(t )u0· ∇S(t )u0is exponentially small with respect tou₀.

Then we exhibit an example of a family of initial data with very largeC˙⁻¹ norm which satisfies the nonlinear smallness hypothesis. This example fits the structure of the nonlinear termu· ∇u.

Then, we study the stability of this nonlinear smallness condition, but not in the usual sense of a perturbation by a small vector field. This problem of a small perturbation was solved by I. Gallagher, D. Iftimie and F. Planchon in [13]

and by P. Auscher, S. Dubois and P. Tchamitchian in [1].

Our purpose is different. Once constructed an initial data generating a global solution, we want to generate a large family of global solutions that may not be close to the one we start with, in theC˙⁻¹norm. This is done with a fractal type transform (see the forthcoming Definition 1.3). Roughly speaking, this is the linear superposition of an arbitrarily large number of dilated and translated iterates of the initial data, and we will see that the initial data so-transformed still satisfies the nonlinear smallness assumption. That of course enables one to construct a very large class of initial data satisfying that smallness assumption; the transformation is based on a construction of [2].

1.2. Definitions

Before presenting more precisely the results of this paper, let us give some definitions and notation. We shall be using Besov spaces, which are defined equivalently using the Littlewood–Paley decomposition or the heat operator.

As both definitions will be useful in the following, we present them both in the next definition.

Definition 1.1.Letϕ∈S(R³)be such that ϕ(ξ )ˆ =1 for|ξ|1 and ϕ(ξ )ˆ =0 for |ξ|>2. Define, forj ∈Z, the functionϕ_j(x)^def=2^3jϕ(2^jx), and the Littlewood–Paley operatorsS_j^def=ϕ_j∗·and_j^def=S_j+1−S_j.Letfbe inS(R³).

Thenf belongs to the homogeneous Besov spaceB˙_p,q^s (R³)if and only if

• The partial sum_m

−mjf converges towardsf as a tempered distribution;

• The sequenceε_j^def=2^{j s}_jfL^pbelongs to^q(Z).

In that case f_B˙s

p,q

def=

j∈Z

2^{j sq}_jf^q_Lp

¹

q

and ifs <0, the one has the equivalent norm f_B˙^s

p,q ∼t^−2sS(t )f

L^p

L^q(R⁺;^dt_t ). (1.1)

Let us notice that the above equivalence comes from the inequality, proved for instance in [6], S(t )ja

L^pCe^−C−122jtjaL^p. (1.2)

Note that the following Sobolev-type continuous embeddings hold:

B˙_p^s1

1,q1⊂ ˙B_p^s2

2,q2, as soon as s₁− d

p₁=s₂− d

p₂ withp₁p₂andq₁q₂. We shall denote byPthe Leray projector onto divergence free vector fields

P=Id−∇⁻¹div.

Before stating the first result of this paper, let us introduce the following space.

Definition 1.2.We shall denote byEthe space of functionsf inL¹(R⁺; ˙B_∞⁻¹_,1)such that

j∈Z

2^−j_jf (t )

L^∞

L²(R⁺;t dt )<∞ equipped with the norm

fE

def= f_L1(R⁺; ˙B_∞⁻¹_,1)+

j∈Z

2^−j_jf (t )

L^∞

L²(R⁺;t dt ).

Let us remark that, for any homogeneous functionσ of order 0 smooth outside 0, we have

∀p∈ [1,∞], σ (D)_jf

L^pC_jfL^p.

Thus the Leray projectionPonto divergence free vectors fields maps continuouslyEintoE.

1.3. Statement of the results 1.3.1. Global existence results

The first result we shall prove is a new global wellposedness result, under a nonlinear smallness assumption on the initial data.

Theorem 1.There is a constantC0such that the following result holds. Letu0∈ ˙H¹²(R³)be a divergence free vector field. Suppose that

P S(t )u0· ∇S(t )u0

EC₀⁻¹exp −C₀u₀⁴_B˙−1

∞,2

. (1.3)

Then there is a unique, global solution to(NS)associated with u₀, satisfying u∈Cb R⁺; ˙H¹²

∩L² R⁺; ˙H³² .

Remark.For the sake of simplicity, we state the theorem for initial data inH˙¹², but it obviously works for initial data inB˙_∞⁻¹_,2.

The proof of Theorem 1 is given in Section 2 below; it consists in writing the solutionu(which exists for a short time at least), asu=S(t )u0+Rand in proving a global wellposedness result for the perturbed Navier–Stokes equation satisfied byR, under assumption (1.3). While the proof follows the lines of that of Koch and Tataru (see [18]), a small modification of classical Picard’s argument is needed to control the linear term, which is not small.

In fact, the main point of the paper is to exhibit examples of applications of this theorem which go beyond the assumption of smallness ofu0_B˙−1

∞,2. The problem we have to solve is the construction of large initial datau0such that a quadratic functional, namelyP(S(t )u0· ∇S(t )u₀)is small. This demands a careful use of the structure of this quadratic functional.

The situation here is different from our previous work (see [8]), devoted to the periodic case, where the structure of the equation was used through the fact that the bidimensional Navier–Stokes equation is globally wellposed.

Now let us state the theorem that ensures that Theorem 1 is relevant.

Theorem 2.Let φ∈S(R³)be a given function, and consider two real numbersεandαin]0,1[. Define ϕε(x)=(−logε)¹⁵

ε^1−α cos x₃

ε

φ

x1,x₂ ε^α, x3

.

There is a constantC >0such that forεsmall enough, the smooth, divergence free vector field u_0,ε(x)= ∂₂ϕ_ε(x),−∂₁ϕ_ε(x),0

satisfies

C⁻¹(−logε)¹⁵ u0,ε_B˙−1

∞,∞C(−logε)¹⁵, and S(t )u0,ε· ∇S(t )u0,ε

ECε^α3(−logε)²⁵. (1.4)

Thus forεsmall enough, the vector field u_0,εgenerates a unique, global solution to(NS).

The proof of Theorem 2 is the purpose of Section 3.

Remark.One can also write this example in terms of the Reynolds number of the fluid: let Re>0 be the Reynolds number, and define the rescaled velocity fieldv(t, x)=νu(νt, x)where ν=1/Re. Thenvsatisfies the Navier–Stokes equation

∂tv+P(v· ∇v)−νv=0

and Theorem 2 states the following: the vector field v_0,ν(x)=(−logν)¹⁵cos

x3

ν

(∂₂φ)

x₁,x2

ν^α, x₃

, ν^α(−∂₁φ)

x₁,x2

ν^α, x₃

satisfies v_0,νB˙−1

∞,∞∼Cν(−logν)¹⁵

and generates a global solution to the Navier–Stokes equations ifνis small enough. Compared with the usual theory of global existence for the Navier–Stokes equations, we have gained a (power of a) logarithm in the smallness assumption in terms of the viscosity, since classically one expects the initial data to be small with respect toν.

1.3.2. Stability results

The second aim of this paper is to give some stability properties of global solutions. It is known since [13] that any initial data inB˙⁻¹⁺

3 p

p,∞ giving rise to a unique global solution is stable: a small perturbation of that data also generates a global solution (see [1] for the case ofBMO⁻¹). Here we present a stability result where the perturbation is as large as the initial data but has a special form: it consists in the superposition of dilated and translated duplicates of the initial data, in the spirit of profile decompositions of P. Gérard (see [14]). This transform is a version of the fractal transform used in [2] in the study of refined Sobolev and Hardy inequalities. Let us be more precise and define the transformation. We shall only be considering compactly supported initial data for this study, and up to a rescaling we shall suppose to simplify that the support of the initial data is restricted to the unit cubeQofR³centered at 0.

Definition 1.3.LetX=(x₁, . . . , x_K)be a set ofKdistinct points inR³. ForΛ∈2^N, let us define TΛ,X

S→S

f →T_Λ,Xf^def=

J∈{1,...,K}T_Λ^Jf with T_Λ^Jf (x)^def=Λf (Λ(x−xJ)).

It can be noted that this is a generalization of the fractal transformationT^k studied in [2].

The next statement is quite easy to prove: it shows that this transformation on the initial data preserves global wellposedness, as soon as the scaling parameter Λis large enough (the thresholdΛbeing unknown as a function of the initial data). The theorem following that statement gives a quantitative approach to that stability: if the initial datau0satisfies the smallness assumption (1.3) of Theorem 1, then so doesTΛ,Xu0as soon asΛis large enough (the threshold being an explicit function of norms ofu0).

More precisely we have the following results.

Proposition 1.1. Let u₀ be a divergence free vector field in H˙¹²(R³)generating a unique, global solution to the Navier–Stokes equations andXbe a finite sequence of distinct points. There isΛ0>0such that, for anyΛΛ0, the vector fieldTΛ,Xu0also generates a unique, global solution.

Remarks.

• Using the global stability of global solutions proved in [13], a global solution associated to an initial data inH˙¹² is always inL^∞(R⁺; ˙H¹²)∩L²(R⁺; ˙H³²).

• As the proof of that result in Section 4.1 will show (see Proposition 4.1), Proposition 1.1 can be generalized to the case where the vector fieldu0is replaced by any finite sequence of vector fields inH˙¹² generating a global solution.

• As we shall see in the proof of Theorem 3 stated below, the functionsT_Λ,Xu₀andu₀have essentially the same norm inC˙⁻¹.

Now let us state the quantitative stability theorem, in particular in the case of an initial data satisfying the as- sumptions of Theorem 1. In order to avoid excessive heaviness, we shall assume from now on that the initial data is compactly supported, and after scaling, supported in the unit cubeQ= ]−¹₂,¹₂[^d. We shall consider sequencesX such that

inf

(J,J)∈{1,...,K}² J=J

d(x_J, x_J);d x_J,^cQ

δ >0. (1.5)

We shall prove the following theorem.

Theorem 3.Letu₀be a smoothH˙¹² divergence free vector field, compactly supported in the cubeQ. Suppose thatu₀ satisfies(1.3)in the following slightly looser sense:there isη∈ ]0,1[such that

P S(t )u0· ∇S(t )u0

EC₀⁻¹exp −C0 u0_B˙−1

∞,2+η4

−η. (1.6)

Then there is a positiveΛ₀(depending only onη, K, δ,u_0H˙−1andu_0B˙−3

∞,∞)such that for anyΛΛ₀, the vector fieldTΛ,Xu0satisfies(1.3)and in particular generates a global solution to(NS). Moreover, for allrin[1,∞],

u_0B˙−1

∞,r−ηT_Λ,Xu_0B˙−1

∞,ru_0B˙−1

∞,r+η.

Remarks.

• The factorη appearing in (1.6) means that u0 must not saturate the nonlinear smallness assumption (1.3) of Theorem 1.

• The proof of this theorem is based on the fact that the Besov norm of index−1 as well asP(S(t )u0· ∇S(t )u0)E

are invariant under the action ofT_Λ,X, up to some small error terms.

As a conclusion of this introduction, let us state the following result, which describes the action ofT_Λ,X on the familyu_0,εintroduced in Theorem 2.

Theorem 4.Letu_0,εbe the family introduced in Theorem2. For anyKandδ, a constantΛ₀exists, which is indepen- dent ofε, such that the following result holds. For any familyXand anyΛΛ₀, there is a global smooth solution of(NS)with initial dataT_Λ,Xu_0,ε.

Remark.Let us point out that as opposed to Proposition 1.1, Theorem 3 (or rather Lemmas 4.1 and 4.2 which are the key to its proof) provides precise bounds onΛ₀so that the constantΛ₀appearing in Theorem 4 may be chosen independently ofε.

2. Proof of Theorem 1

2.1. Main steps of the proof

Let us start by remarking that in the case whenu₀is small then there is nothing to be proved, so in the following we shall suppose thatu_0B˙−1

∞,2 is not small, sayu_0B˙−1

∞,21.

We follow the method introduced by H. Koch and D. Tataru in [18] in order to look for the solutionuunder the formuF +R, whereuF(t )^def=S(t )u0. Let us denote byQthe bilinear operator defined by

Q(a, b)(t )^def= −1 2 t 0

S(t−t)P a(t)· ∇b(t)+b(t)· ∇a(t) dt.

ThenRis the solution of

(MNS) R=Q(u_F, u_F)+2Q(u_F, R)+Q(R, R).

To prove the global existence ofu, we are reduced to proving the global wellposedness of(MNS); that relies on the following easy lemma, the proof of which is omitted.

Lemma 2.1.LetXbe a Banach space, letLbe a continuous linear map fromXtoX, and letB be a bilinear map fromX×XtoX. Let us define

L_L(X) def= sup

x=1

Lx and B_B(X)

def= sup

x=y=1

B(x, y).

IfL_L(X)<1, then for anyx₀inXsuch that x₀X<(1− L_L(X))²

4B_B(X)

,

the equation

x=x0+Lx+B(x, x)

has a unique solution in the ball of center0and radius ¹₂⁻_B^LL(X)

B(X) ·

Let us introduce the functional space for which we shall apply the above lemma. We define the quantity U (t )^def=u_F(t )²

L^∞+tu_F(t )⁴

L^∞, which satisfies

∞ 0

U (t ) dtCu0²_B˙−1

∞,2+Cu0⁴_B˙−1

∞,4

Cu0⁴_B˙−1

∞,2

(2.1) recalling that we have supposed thatu_0B˙−1

∞,21 to simplify the proof.

For allλ0, let us denote byX_λthe set of functions onR⁺×R³such that vλ

def=sup

t >0

t¹²v_λ(t )

L^∞+ sup

x∈R³ R>0

R⁻³²

P (x,R)

v_λ(t, y)²dy ¹₂

<∞, (2.2)

where

v_λ(t, x)^def=v(t, x)exp

−λ t 0

U (t) dt

whileP (x, R)= [0, R²] ×B(x, R)andB(x, R)denotes the ball ofR³of centerxand radiusR. Let us point out that, in the case whenλ=0, this is exactly the space introduced by H. Koch and D. Tataru in [18], and that for anyλ0 we have due to (2.1),

vλv0Cvλexp Cλu0⁴_B˙−1

∞,2

. (2.3)

From Lemmas 3.1 and 3.2 of [18] together with the above equivalence of norms, we infer that Q(v, w)

λCvλwλexp Cλu0⁴_B˙−1

∞,2

. (2.4)

Theorem 1 follows from the following two lemmas.

Lemma 2.2. There is a constantC >0 such that the following holds. For any nonnegativeλ, for anyt 0 and anyf ∈E, we have

t 0

S(t−t)f (t) dt λ

CfE.

Lemma 2.3.Letu₀∈ ˙B_∞⁻¹_,2be given, and defineu_F(t )=S(t )u0. There is a constantC >0such that the following holds. For anyλ1, for anyt0and anyv∈X_λ, we have

Q(u_F, v)(t )

λ C λ¹⁴vλ.

End of the proof of Theorem 1. Let us apply Lemma 2.1 to Eq.(MNS)satisfied byR, in a spaceXλ. We chooseλ so that according to Lemma 2.3,

Q(u_F,·)(t )

L(Xλ)1 4·

Then according to Lemma 2.1, there is a unique solutionRto(MNS)inX_λas soon asQ(u_F, u_F)satisfies Q(u_F, u_F)

X_λ 1

16Q_B(Xλ)

.

But (2.4) guarantees that

Q_B(Xλ)Cexp Cλu0⁴_B˙−1

∞,2

,

so it is enough to check that for some constantC, Q(uF, uF)

XλC⁻¹exp −Cλu0⁴_B˙−1

∞,2

.

By Lemma 2.2, this is precisely condition (1.3) of Theorem 1, so under assumption (1.3), there is a unique, global solution R to (MNS), in the space Xλ. This implies immediately that there is a unique, global solution u to the Navier–Stokes system inXλ. The fact thatubelongs toCb(R⁺; ˙H¹²)∩L²(R⁺; ˙H³²)is then simply an argument of propagation of regularity (see for instance [21]). 2

2.2. Proof of Lemma 2.2

Thanks to (2.3), it is enough to prove Lemma 2.2 forλ=0.

Let us start by proving that_t

0S(t−t)f (t) dtbelongs toL²(R⁺;L^∞); that will give in particular the boundedness of the second norm entering in the definition ofXλ.

Using (1.2), we get

t 0

_jS(t−t)f (t) dt L^∞

C

t 0

e^{−C−122j(t−t)}_jf (t)

L^∞dt.

Young’s inequality then gives

t 0

_jS(t−t)f (t) dt

L²(R⁺;L^∞)

C2^−j_jf_L¹_(R+;L^∞), thus the series(_jt

0S(t−t)f (t) dt)_j∈Zconverges inL²(R⁺;L^∞), and

t 0

S(t−t)f (t) dt

L²(R⁺;L^∞)

CfE.

This implies in particular that sup

x∈R³ R>0

R⁻³²

P (x,R)

t

0

S(t−t)f (t) (y) dt

2

dy ¹₂

CfE. (2.5)

The second part of the norm defining · X_λin (2.2) is therefore controlled by the norm off inE.

To estimate the first part of that norm, let us write that for anyt0 and anyj∈Z, t¹²_j

t 0

S(t−t)f (t) dt=G⁽¹⁾_j (t )+G⁽²⁾_j (t ) with

G⁽¹⁾_j (t )^def=t¹²

t

2

0

S(t−t)_jf (t) dt and

G⁽²⁾_j (t )^def=t¹² t

t 2

S(t−t)_jf (t) dt.

Using again (1.2) we have, sincet2(t−t), G⁽¹⁾_j (t )

L^∞C

t

2

0

(t−t)¹²2^je^{−C−122j(t−t)}2^−jjf (t)

L^∞dt

2^−j_jfL¹(R⁺;L^∞).

In order to estimateG⁽²⁾_j (t )L^∞, let us write, sincet2t, G⁽²⁾_j (t )

L^∞C t 0

e^{−C−122j(t−t)}t¹²_jf (t)

L^∞dt.

Using the Cauchy–Schwarz inequality, we get G⁽²⁾_j (t )

L^∞C2^−jt¹²_jf (t )

L²(R⁺;L^∞).

Then using (2.5) and summing overj∈Zconcludes the proof of Lemma 2.2. 2 2.3. Proof of Lemma 2.3

We have (see for instance [18] or [7]) that

Q(v, w)(t, x)= t 0

R³

k(t−t, y)v(t, x−y)w(t, x−y) dy dt

=k (vw)(t, x) withk(τ, ζ ) C (√

τ+ |ζ|)⁴· The proof relies now mainly on the following proposition.

Proposition 2.1.Letu₀∈ ˙B_∞⁻¹_,2be given, and defineu_F(t )=S(t )u0. There is a constantC such that the following holds. Consider, for any positiveRand for(τ, ζ )∈R⁺×R³, the following functions:

K_R⁽¹⁾(τ, ζ )^def=1_|ζ|R

1

|ζ|⁴ and K_R⁽²⁾(τ, ζ )^def=1_|ζ|R

1 (√

τ+ |ζ|)⁴· Then for anyλ1and anyR >0,

e^{−λ0tU (t) dt}K_R⁽¹⁾ (u_Fv)

L^∞([0,R²]×R³) C

λ¹²Rvλ. (2.6)

Moreover, for anyλ1and anyR >0, e^{−λ0tU (t) dt}K_R⁽²⁾ (u_Fv)

L^∞([R²,2R²]×R³) C

λ¹⁴Rvλ. (2.7)

Proof of Proposition 2.1. Let us write that V_λ⁽¹⁾(t, x)^def=e^−λ

t

0U (t) dtK_R⁽¹⁾ (u_Fv)(t, x)

t 0

cB(0,R)

1

|y|⁴e^−λ

t

tU (t) dtuF(t,·)

L^∞vλ(t, x−y)dtdy.

By the Cauchy–Schwarz inequality and by definition ofU, we infer that

V_λ⁽¹⁾(t, x) t

0

cB(0,R)

1

|y|⁴e^{−2λttU (t) dt}u_F(t,·)²

L^∞dtdy ¹

2t 0

cB(0,R)

1

|y|⁴v_λ(t, x−y)²dtdy ¹

2

C

λR ¹

2

t 0

cB(0,R)

1

|y|⁴v_λ(t, x−y)²dtdy ¹₂

. (2.8)

Now let us decompose the integral on the right on rings; this gives

t 0

cB(0,R)

1

|y|⁴v_λ(t, x−y)²dtdy=^∞

p=0

t 0

B(0,2^p+1R)\B(0,2^pR)

1

|y|⁴v_λ(t, x−y)²dtdy

1

R ∞ p=0

2^−p+3 2^p+1R₋3

× t 0

B(0,2^p+1R)

v_λ(t, x−y)²dt dy.

AstR²andpis nonnegative, we have t

0

cB(0,R)

1

|y|⁴v_λ(t, x−y)²dtdyC R

∞ p=0

2^−p 2^p+1R₋3

P (x,2^p+1R)

v_λ(t, z)²dt dz

C

R ∞ p=0

2^−p sup

R>0

1 R³

P (x,R)

vλ(t, z)²dt dz.

By definition of · λ, we infer that t

0

cB(0,R)

1

|y|⁴vλ(t, x−y)²dtdyC Rv²λ. Then, using (2.8), we conclude the proof of (2.6).

In order to prove the second inequality, let us observe that e^−λ

t

0U (t) dt K_R⁽²⁾ (u_Fv)(t, x)K⁽²¹⁾_R (t, x)+K⁽²¹⁾_R (t, x) with K_R⁽²²⁾(t, x)^def=

t

2

0

B(0,R)

1 (√

t−t+ |y|)⁴e^−λ

t

tU (t) dtuF(t,·)

L^∞vλ(t, x−y)dtdy, K_R⁽²²⁾(t, x)^def=

t

t 2

B(0,R)

1 (√

t−t+ |y|)⁴e^−λ

t

tU (t) dtu_F(t,·)

L^∞v_λ(t, x−y)dtdy.

Using the Cauchy–Schwarz inequality, ast∈ [R²,2R²]andt2(t−t), we infer that K_R⁽²¹⁾(t, x)

^t2

0

B(0,R)

1 (√

t−t+ |y|)⁸e^−2λ

t

tU (t) dtu_F(t,·)²

L^∞dtdy ¹₂

× 2^t

0

B(0,R)

v_λ(t, x−y)²dtdy ¹

2

C

λ¹²

B(0,R)

dy (R+ |y|)⁸

¹

22^t

0

B(0,R)

vλ(t, x−y)²dtdy ¹

2

C

(t λ)¹² R⁻³²

R² 0

B(0,R)

v_λ(t, x−y)²dtdy ¹₂

,