Local and global well-posedness results for
density-dependent incompressible fluids
Boris Haspot ∗
Abstract
This paper is dedicated to the study of the initial value problem for density dependent incompressible viscous fluids in RN with N ≥ 2. We address the question
of well-posedness for large data having critical Besov regularity and we aim at stating well-posedness in functional spaces as close as possible to the ones imposed in the incompressible Navier Stokes system by Cannone, Meyer and Planchon in [7] where u0∈ B
N p−1
p,∞ with 1 ≤ p < +∞. This improves the analysis of [13], [14] and [2] where
u0 is considered belonging to B
N p−1
p,1 with 1 ≤ p < 2N . Our result relies on a new
a priori estimate for transport equation introduce by Bahouri, Chemin and Danchin in [5] when the velocity u is not considered Lipschitz.
1
Introduction
In this paper, we are concerned with the following model of incompressible viscous fluid with variable density:
∂tρ + div(ρu) = 0,
∂t(ρu) + div(ρu ⊗ u) − div(2µ(ρ)Du) + ∇Π = ρf,
divu = 0,
(ρ, u)/t=0 = (ρ0, u0).
(1.1)
Here u = u(t, x) ∈ RN stands for the velocity field and ρ = ρ(t, x) ∈ R+ is the density,
Du = 12(∇u +t ∇u) is the strain tensor. We denote by µ the viscosity coefficients of the fluid, which is assumed to satisfy µ > 0. The term ∇Π (namely the gradient of the pressure) may be seen as the Lagrange multiplier associated to the constraint divu = 0. We supplement the problem with initial condition (ρ0, u0) and an outer force
f . Throughout the paper, we assume that the space variable x ∈ RN or to the periodic box TN
a with period ai, in the i-th direction. We restrict ourselves the case N ≥ 2.
The existence of global weak solution for (1.1) under the assumption that ρ0 ∈ L∞ is
nonnegative, that divu0, and that
√
ρ0u0 ∈ L2 has been studied by different authors. It
is based on the energy equality:
k√ρu(t)k2L2 + Z t 0 kpµ(ρ)Du(τ )k2L2dτ = k √ ρ0u0k2L2 + Z 2(ρf · u)(τ, x)dτ dx. (1.2) ∗
Karls Ruprecht Universit¨at Heidelberg, Institut for Applied Mathematics, Im Neuenheimer Feld 294, D-69120 Heildelberg, Germany. Tel. 49(0)6221-54-6112
Using (1.2) and the fact that the density is advected by the flow of u so that the Lp norms of ρ are (at least formally) conserved during the evolution, it is then possible to use compactness methods to prove the existence of global weak solution. This approach has been introduced by J. Leray in 1934 in the homogeneous case (i.e ρ = 1) and no external force. For the non-homogeneous equation (1.1), we refer to [3] and to [28] for an overview of results on weak solution. Some recent improvements have been obtained by B. Desjardins in [19],[20] and [21].
The question of unique resolvability for (1.1) has been first addressed by O. Ladyzhen-skaya and V. Solonnikov in the late seventies ( see [27]). The authors consider system (1.1) in a bounded domain Ω with homogeneous Dirichlet boundary conditions for u. Under the assumption that u0 ∈ W2−
2
q,q (q > N ) is divergence-free and vanishes on ∂Ω
and that ρ0 ∈ C1( ¯Ω) is bounded away from zero, the results are the following:
• Global well-posedness in dimension N = 2
• Local well-posedness in dimension N = 3. If in addition u0 is small in W2−2q,q,
then global well-posedness holds true.
Let us mention by passing that for the dimension N = 2, O. Ladyzhenskaya and V. Solonnikov use a quasi-conservation law for the H1 norm of the velocity and get global H1 solutions.
The case of unbounded domains has been investigate by S. Itoh and A. Tani in [24]. In this framework, system (1.1) has been shown to be locally well-posed. In the present paper, we aim at proving similar qualitative results in the whole space RN or in the torus TN under weaker regularity assumptions.
Guided in our approach by numerous works dedicated to the incompressible Navier-Stokes equation (see z.g [29]):
(N S)
(
∂tv + v · ∇v − µ∆v + ∇Π = 0,
divv = 0,
we aim at solving in the case where the data (ρ0, u0, f ) have critical regularity for the
scaling of the equations and in particular when the initial velocity is in the same Besov spaces than Cannone, Meyer and Planchon in [7] for the incompressible Navier-Stokes system. It means that we want get strong solutions when u0 belongs to B
N p−1
p,∞ with 1 ≤
p < +∞. By critical, we mean that we want to solve the system functional spaces with norm in invariant by the changes of scales which leaves (1.1) invariant. That approach has been initiated by H. Fujita and T. Kato in [23]. In the case of incompressible fluids, it is easy to see that the transformations:
v(t, x) −→ lv(l2t, lx), ∀ l ∈ R have that property.
For density-dependent incompressible fluids, one can check that the appropriate trans-formation are
(ρ0(x), u0(x)) −→ (ρ0(lx), lu0(lx)), ∀ l ∈ R.
The use of critical functional frameworks led to several new well-posedness results for incompressible fluids (see [7], [26]). In the case od the density dependent incompressible fluids we want cite recent improvements by R. Danchin in [11], [12], [13], [14], [15], [16], [17] and of H. Abidi in [1] and Abidi, Paicu in [2]. All these works give results of strong solutions in finite time in critical spaces for the scaling of the equations. More precisely R. Danchin show the existence of strong solution in finite time in [16], [17], [18] when the initial data check (ρ−10 − 1, u0) ∈ B
N 2 2,∞∩ L∞ × B N 2−1 2,1 or (ρ −1 0 − 1, u0) ∈ B N p p,1× B N p p,1 with
1 ≤ p ≤ N . Moreover R. Danchin need of a condition of smallness on the initial data, it means that kρ−10 − 1k
B N
p p,1
is assumed small. More recently H. Abidi and M. Paicu in [2]
have improved this result by working with initial data in B
N p1 p1,1× B N p2−1 p2,1 with p1 and p2
good choosen and checking some relations. In particular they show that we have strong solution for initial data u0 in B
N p2−1
p2,1 with 1 ≤ p2 < 2N which improves the results of
R. Danchin. All these results use of crucial way the fact that the solution are Lipschitz. In particular, it explains the choice of the third index for the Besov space where r = 1, indeed it allows a control of ∇u in L1T(B
N p
p,1) which is embedded in L1T(L
∞). This control
is imperative in these works to get estimate via the transport equation on the density. However the scaling of (1.3) suggests choosing initial data (ρ0, u0) in B
N p1 p1,r0 ×B N p2 p2,r(see the
definition of Besov spaces in the section 2) with (p1, p2) ∈ [1, +∞)2and (r, r 0
) ∈ [1, +∞]2. Indeed it seems that it is not necessary by the study of the scaling of the equation to impos that r, r0 = 1 as in the works of H. Abidi, R. Danchin and M. Paicu. The goal of this article is to reach the critical case with a general third index for the Besov spaces r and r0. More precisely in the sequel we will restrict our study to the case where the initial data (ρ0, u0) and external force f are such that, for some positive constant ¯ρ:
(ρ0− ¯ρ) ∈ B N p1+ε p1,∞ ∩ L ∞ , u0∈ B N p2 p2,r and f ∈ eL 1 loc(R+, ∈ B N p2−1 p2,r ).
with r ∈ [1, +∞], ε > 0 and p1, p2 will verify appropriate inequalities ( for a definition
of eL1 see section 2).
In this article we improve the result of H. Abidi, R. Danchin and M. Paicu by working with initial data in B
N p2
p2,r with r ∈ [1, +∞]. In particular we generalize to the case of the
Navier-Stokes incompressible dependent density the result of Cannone-Meyer-Planchon in [7]. For making we introduce a new idea to control the density via the transport equation when the velocity is not Lipschitz. We use some new a priori estimates on the transport equation when the velocity is only log Lipschitz. The difficulty is to deal with the loss of regularity on the density, that is why we need to ask more regularity on ρ0
to compense this loss of regularity. The crucial point is that the density stay in a good multiplier space. Moreover we improve the results of H. Abidi, R. Danchin and M. Paicu because we do not need to assume some condition of smallness on the initial density. In [11], [12], [13], [14], [15], [16], [17], [1] and [2] , we need to make the additional assumption that ρ − ¯ρ is small in B
N p
p,1. The reason why is that we handled the elliptic operator in
the momentum equation of (1.1) as a constant coefficient second order operator plus a perturbation introduced by ρ− ¯ρ which, if sufficiently small, may be treated as a harmless
source term. For smoother data however, the additional regularity can compensate large perturbations of the constant coefficient operator. This fact has been used in [14] leads to local well-posedness results for smooth enough data with density bounded away from zero. The price to pay however is that assuming extra smoothness precludes from using a critical framework. In fact to treat the case of large initial data on the density, we follow a idea of R. Danchin in [?] use for the case of Navier-Stokes compressible where to get good estimate on the elliptic operator in the momentum equation, we need to split the behavior high frequencies and low-middle frequencies of the viscosity terms.
In the present paper, we address the question of local well-posedness in the critical functional framework under the sole assumption that the initial density is bounded away from 0 and tends to some positive constant ¯ρ at infinity (or has average ¯ρ if we consider the case of periodic boundary conditions). To simplify the notation, we assume from now on that ¯ρ = 1. Hence as long as ρ does not vanish, the equations for (a = ρ−1− 1,u) read: ∂ta + u · ∇a = 0,
∂tu + u · ∇u + (1 + a)(∇Π − µ∆u) = f,
divu = 0,
(a, u)/t=0= (a0, u0).
(1.4)
One can now state our main result. The first theorem generalize the work of Can-none,Meyer, Planchon in [7] when the third index of the Besov space for the initial data is in [1, +∞[.
Theorem 1.1 Let 1 ≤ r < ∞, 1 ≤ p1< ∞, 1 < p2 < ∞ and ε > 0 such that:
N p1 + ε < N p2 + 1 and N p2 − 1 ≤ N p1 . Assume that u0 ∈ B N p2−1 p2,r with divu0 = 0, f ∈ eL 1 loc(R+, B N p2−1 p2,r ) and a0 ∈ B N p1+ε p1,∞ ∩ L ∞,
with 1 + a0 bounded away from zero and it exists c > 0 such that:
ka0k B N p1 p1,∞ ≤ c. If p1 1 + 1 p2 > 1
N, there exists a positive time T such that system (1.4) has a solution (a, u)
with 1 + a bounded away from zero and:
a ∈ eC([0, T ], B N p1+ ε 2 p1,∞ ), u ∈ C([0, T ]; Be N p2−1 p2,r ) ∩ eL 1(0, T, B N p2+1 p2,r ) N and ∇Π ∈ eL1(0, T, B N p2−1 p2,r ).
This solution is unique when N2 ≤ 1 p1 +
1 p2.
Remark 1 This theorem can be applied for non degenerate viscosity, but for the sim-plicity in the sequel the viscosity is assumed to be a constant µ > 0.
Remark 2 As in the work of H. Abidi and M. Paicu in [2], we are able to get strong solution when u0 ∈ B
N p2−1
[16] and [17].
Moreover we get weak solution with initial data very close from (a0, u0) ∈ BN,∞1+ε × B∞,r−1
and (a0, u0) ∈ B∞,∞ε × BN,r0 . It means that in the first case we are not very far from the
Koch-Tataru in [26] initial data for the velocity u0, and in the second case we are close
to just impose a condition of norm L∞ on a0 which is of great interest for the system
bifluid.
Remark 3 In the previous theorem, we need of a condition of smallness, because when p2 6= 2, we have extra term in our proposition 3.5 which requires a condition of smallness
on a. It seems that there is a specific structure in the scalar case when p2= 2.
In the following theorem, we improve the previous result in the specific case where p2 = 2.
In this case we don’t need to impose condition of smallness on the initial data. Theorem 1.2 Let 1 ≤ r < ∞, 1 ≤ p1< ∞ and ε > 0 such that:
N p1 + ε < N 2 + 1 and N 2 ≤ 1 + N p1 . Assume that u0 ∈ B N 2−1
2,r with divu0 = 0, f ∈ eL1loc(R+, B N 2−1 2,r ) and a0 ∈ B N p1+ε p1,∞ ∩ L ∞,
with 1 + a0 bounded away from zero. There exists a positive time T such that system
(1.4) has a solution (a, u) with 1 + a bounded away from zero and:
a ∈ eC([0, T ], B N p1+ ε 2 p1,∞ ), u ∈ C([0, T ]; Be N 2−1 2,r ) ∩ eL 1(0, T, BN2+1 2,r ) N and ∇Π ∈ eL1(0, T, B N 2−1 2,r ).
This solution is unique when N2 ≤ 1 p1 +
1 2.
In the following theorem we choose r = +∞, we have to treat the case of a linear loss of regularity on the density ρ which depends of the solution u. This fact come from your estimate on the transport equation when ∇u ∈ eL1T(B
N p2
p2,∞), in this case we have a loss of
regularity on the density which depends of this quantity.
Theorem 1.3 Let 1 ≤ p1 < ∞, 1 < p2 < ∞, and ε > 0 such that:
N p1 + ε < N p2 + 1 and N p2 ≤ 1 +N p1 . Assume that u0 ∈ B N p2−1 p2,∞ with divu0 = 0, f ∈ eL 1 loc(R+, B N p2−1 p2,∞) and a0 ∈ B N p1+ε p1,∞ ∩ L ∞,
with 1 + a0 bounded away from zero and it exists c > 0 such that:
ka0k B N p1 p1,∞ ≤ c. If p1 1 + 1 p2 > 1
N, there exists a positive time T such that system (1.4) has a solution (a, u)
with 1 + a bounded away from zero and:
a ∈ eC([0, T ], Bpσ(T )1,∞), u ∈ C([0, T ]; Be N p2−1 p2,∞ ∩ eL 1(0, T, B N p2+1 p2,∞) N and ∇Π ∈ eL1(0, T, B N p2−1 p2,∞), .
with: σ(T ) = N p1 + ε − λkuk e L1 T(B N p2+1 p2,∞) + λ Z T 0 W (t0)dt0,
for any λ > 0 and any nonegative integrable function W over [0, T ] such that σ(T ) ≥ −1 − N min(1
p1, 1
p01). This solution is unique when 2 N ≤ 1 p1 + 1 p2.
In the following theorem, we generalize the previous result with large initial data for the density when p2= 2.
Theorem 1.4 Let 1 ≤ p1 < ∞ and ε > 0 such that:
N p1 + ε < N 2 + 1 and N 2 ≤ 1 + N p1 . Assume that u0 ∈ B N 2−1
2,∞ with divu0 = 0, f ∈ eL1loc(R+, B N 2−1 2,∞ ) and a0 ∈ B N p1+ε p1,∞ ∩ L ∞,
with 1 + a0 bounded away from zero. There exists a positive time T such that system
(1.4) has a solution (a, u) with 1 + a bounded away from zero and:
a ∈ eC([0, T ], Bpσ(T )1,∞), u ∈ C([0, T ]; Be N 2−1 2,∞ ∩ eL1(0, T, B N 2+1 2,∞ ) N and ∇Π ∈ eL1(0, T, B N 2−1 2,∞ ), . with: σ(T ) = N p1 + ε − λkuk e L1 T(B N 2+1 2,∞ ) + λ Z T 0 W (t0)dt0,
for any λ > 0 and any nonegative integrable function W over [0, T ] such that σ(T ) ≥ −1 − N min(1
p1, 1
p01). This solution is unique when 2 N ≤ 1 p1 + 1 2.
The key of the theorems 1.1, 1.2, 1.3 and 1.4 is new estimate for transport equation for u which is not considered Lipschitz. In this case we have to pay a loss of regularity on the density. ρ. The basic idea is to deal with this loss of regularity by conserving a in eCT(B
N p1
p1,+∞) ∩ L
∞ which have good properties of multiplier. Moreover to avoid as in
the work s of H. Abidi, R. Danchin and M. Paicu conditions of smallness on the initial density, the basic idea is that having coefficients in a in eCT(B
N p1
p1,+∞) provides us with
some uniform decay on the high frequencies of the variable coefficients so that the elliptic operator may be considered as an operator with smooth coefficients (of the type which has been investigated in [15]) plus a small error term.
Our paper is structured as follows. In the section 2, we give a few notation and briefly introduce the basic Fourier analysis techniques needed to prove our result. Section 3 and 4 are devoted to the proof of key estimates for the linearized system (1.4) in particular the elliptic operator of the momentum equation with variable coefficients and the transport equation when the velocity is not assumed Lipschitz. In section 5, we prove the existenceof solution for theorem 1.1 and 1.2 whereas section 6 is devoted to the proof of uniqueness. Finally in section 7, we briefly show how to prove theorem 1.3 and 1.4. Elliptic and technical estimates commutator are postponed in an appendix.
2
Littlewood-Paley theory and Besov spaces
Throughout the paper, C stands for a constant whose exact meaning depends on the context. The notation A . B means that A ≤ CB. For all Banach space X, we denote by C([0, T ], X) the set of continuous functions on [0, T ] with values in X. For p ∈ [1, +∞], the notation Lp(0, T, X) or LpT(X) stands for the set of measurable functions on (0, T ) with values in X such that t → kf (t)kX belongs to Lp(0, T ).
2.1 Littlewood-Paley decomposition
Littlewood-Paley decomposition corresponds to a dyadic decomposition of the space in Fourier variables. Let α > 1 and (ϕ, χ) be a couple of smooth functions valued in [0, 1], such that ϕ is supported in the shell supported in {ξ ∈ RN/α−1 ≤ |ξ| ≤ 2α}, χ is supported in the ball {ξ ∈ RN/|ξ| ≤ α} such that:
∀ξ ∈ RN, χ(ξ) +X
l∈N
ϕ(2−lξ) = 1.
Denoting h = F−1ϕ, we then define the dyadic blocks by: ∆lu = 0 if l ≤ −2,
∆−1u = χ(D)u = eh ∗ u with eh = F−1χ,
∆lu = ϕ(2−lD)u = 2lN
Z
RN
h(2ly)u(x − y)dy with h = F−1χ, if l ≥ 0, Slu =
X
k≤l−1
∆ku .
Formally, one can write that: u =P
k∈Z∆ku . This decomposition is called
nonhomoge-neous Littlewood-Paley decomposition.
2.2 Nonhomogeneous Besov spaces and first properties
Definition 2.1 For s ∈ R, p ∈ [1, +∞], q ∈ [1, +∞], and u ∈ S0(RN) we set: kukBs p,q = ( X l∈Z (2lsk∆lukLp)q) 1 q.
The Besov space Bp,qs is the set of temperate distribution u such that kukBs
p,q < +∞.
Remark 4 The above definition is a natural generalization of the nonhomogeneous Sobolev and H¨older spaces: one can show that B∞,∞s is the nonhomogeneous H¨older space Cs and
that B2,2s is the nonhomogeneous space Hs. Proposition 2.1 The following properties holds:
1. there exists a constant universal C such that: C−1kukBs
p,r ≤ k∇ukBp,rs−1 ≤ CkukBsp,r.
2. If p1< p2 and r1≤ r2 then Bps1,r1 ,→ B
s−N (1/p1−1/p2)
3. (Bs1
p,r, Bp,rs2)θ,r0 = B
θs1+(1−θ)s2
p,r0 .
Let now recall a few product laws in Besov spaces coming directly from the paradifferen-tial calculus of J-M. Bony (see [6]) and rewrite on a generalized form in [2] by H. Abidi and M. Paicu (in this article the results are written in the case of homogeneous sapces but it can easily generalize for the nonhomogeneous Besov spaces).
Proposition 2.2 We have the following laws of product: • For all s ∈ R, (p, r) ∈ [1, +∞]2 we have:
kuvk e Bs p,r ≤ C(kukL ∞kvkBs p,r + kvkL∞kukBp,rs ) . (2.5)
• Let (p, p1, p2, r, λ1, λ2) ∈ [1, +∞]2 such that:1p ≤ p11 + p12, p1 ≤ λ2, p2 ≤ λ1, 1p ≤ 1 p1 + 1 λ1 and 1 p ≤ 1 p2 + 1
λ2. We have then the following inequalities:
if s1+ s2+ N inf(0, 1 − p11 −p12) > 0, s1+λN2 < pN1 and s2+λN1 < pN2 then:
kuvk Bs1+s2−N( 1 p1+ 1p2− 1p ) p,r . kukBp1,rs1 kvkBp2,∞s2 , (2.6)
when s1+λN2 = pN1 (resp s2+λN1 = pN2) we replace kukBs1
p1,rkvkBs2p2,∞ (resp kvkBp2,∞s2 ) by kukBs1 p1,1kvkBs2p2,r (resp kvkBp2,∞s2 ∩L∞), if s1+ N λ2 = N p1 and s2+ N λ1 = N p2 we take r = 1. If s1+ s2= 0, s1 ∈ (λN1 −pN2,pN1 − λN2] and p11 +p12 ≤ 1 then: kuvk B −N ( 1 p1+ 1p2− 1p ) p,∞ . kukBp1,1s1 kvkBp2,∞s2 . (2.7) If |s| < Np for p ≥ 2 and −N p0 < s < N p else, we have: kuvkBs p,r ≤ CkukBsp,rkvk B N p p,∞∩L∞ . (2.8)
Remark 5 In the sequel p will be either p1 or p2 and in this case λ1 = p11 −p12 if p1 ≤ p2,
resp 1λ = p1
2 − 1
p1 if p2 ≤ p1.
Corollary 1 Let r ∈ [1, +∞], 1 ≤ p ≤ p1 ≤ +∞ and s such that:
• s ∈ (−N p1, N p1) if 1 p+ 1 p1 ≤ 1, • s ∈ (−Np 1 + N ( 1 p + 1 p1 − 1), N p1) if 1 p+ 1 p1 > 1,
then we have if u ∈ Bp,rs and v ∈ B
N p1 p1,∞∩ L ∞: kuvkBs p,r ≤ CkukBp,rs kvk B N p1 p1,∞∩L∞ .
The study of non stationary PDE’s requires space of type Lρ(0, T, X) for appropriate Banach spaces X. In our case, we expect X to be a Besov space, so that it is natural to localize the equation through Littlewood-Payley decomposition. But, in doing so, we ob-tain bounds in spaces which are not type Lρ(0, T, X) (except if r = p). We are now going to define the spaces of Chemin-Lerner in which we will work, which are a refinement of the spaces LρT(Bs
p,r).
Definition 2.2 Let ρ ∈ [1, +∞], T ∈ [1, +∞] and s1 ∈ R. We set:
kuk e LρT(Bp,rs1)= X l∈Z 2lrs1k∆ lu(t)krLρ(Lp) 1r .
We then define the space eLρT(Bs1
p,r) as the set of temperate distribution u over (0, T ) × RN
such that kuk
e
LρT(Bs1p,r)< +∞.
We set eCT( eBp,rs1) = eLT∞( eBsp,r1) ∩ C([0, T ], Bsp,r1). Let us emphasize that, according to
Minkowski inequality, we have: kuk e LρT(Bs1p,r)≤ kukL ρ T(Bp,rs1) if r ≥ ρ, kukLe ρ T(Bs1p,r)≥ kukL ρ T(Bp,rs1) if r ≤ ρ.
Remark 6 It is easy to generalize proposition 2.2, to eLρT(Bs1
p,r) spaces. The indices s1,
p, r behave just as in the stationary case whereas the time exponent ρ behaves according to H¨older inequality.
Here we recall a result of interpolation which explains the link of the space Bs
p,1 with the
space Bp,∞s , see [11].
Proposition 2.3 There exists a constant C such that for all s ∈ R, ε > 0 and 1 ≤ p < +∞, kuk e LρT(Bs p,1) ≤ C1 + ε ε kukLeρ T(Bsp,∞) 1 + log kuk e LρT(Bs+εp,∞) kuk e LρT(Bs p,∞) .
Definition 2.3 Let Γ be an increasing function on [1, +∞[. We denote by BΓ(RN) the
set of bounded real valued functions u over RN such that:
kukBΓ = kukL∞+ sup j≥0
k∇SjukL∞
Γ(2j) < +∞.
We give here a proposition concerning these spaces showed by J-Y. Chemin, see [5].
Proposition 2.4 Let ε > 0 and u ∈ eL1T(B
N p+1
p,r ) then we have u ∈ L1T(BΓ(RN)) with
3
Estimates for parabolic system with variable coefficients
In this section, the following linearization of the momentum equation is studied: ∂tu + b(∇Π − µ∆u) = f + g, divu = 0, u/t=0= u0. (3.9)
where b, f , g and u0 are given. Above u is the unknown function. We assume that
u0∈ Bp,rs and f ∈ eL1(0, T ; Bp,rs ), that b is bounded by below by a positive constant b and
that a = b − 1 belongs to eL∞(0, T ; B
N p1+α p1,∞ ) ∩ L
∞. In the present subsection, we aim at
proving a priori estimates for (3.9) in the framework of nonhomogeneous Besov spaces. Before stating our results let us introduce the following notation:
AT =1 + b−1k∇bk e L∞(B N p1+α−1 p1,∞ ) with α > 0. (3.10)
Proposition 3.5 Let ν = bµ and (p, p1) ∈ [1, +∞].
• If p1 > p we assume that s ∈ (−pN1,pN1) if 1p+p11 ≤ 1 and s ∈ (−pN1+N (1p+p11−1),Np1)
if 1p +p1
1 > 1.
• If p1≤ p then we suppose that s ∈ (−N p, N p) if p ≥ 2 and s ∈ (− N p0, N p) if p < 2.
If p 6= 2 we need to assume than there exists c > 0 such that: kak e L∞(B N p1+α−1 p1,∞ ) ≤ c.
Let m ∈ Z be such that bm = 1 + Sma satisfies:
inf
(t,x)∈[0,T )×RNbm(t, x) ≥
b
2. (3.11)
There exist three constants c, C and κ (with c, C, depending only on N and on s, and κ universal) such that if in addition we have:
ka − Smak e L∞(0,T ;B N p1 p1,∞)∩L∞ ≤ cν µ (3.12) then setting: Zm(t) = 22mαµ2ν−1 Z t 0 kak2 B N p1 p1,∞∩L∞ dτ,
Let α0 > 0 checking α0 ≤ min(1, α,s−2+
2 m
2 ). We have for all t ∈ [0, T ] and κ = s α0: kuk e L∞ T(Bp,rs )+ κνkukLe1T(Bp,rs+2)≤ e CZm(T ) ku0kBs p,r + A κ T(kPf kLe1T(Bsp,r) + µm1kPgk e Lm T(B s−2+ 2m p,r ) + µm1(ν(p − 1) p )ATkukLe1T(Bs+2−α 0 p,r ) . (3.13)
Moreover we have ∇Π = ∇Π1+ ∇Π2 with: bk∇Π1kLe1 T(Bsp,r)≤ A κ TkPf kLe1 T(Bp,rs ), bk∇Π2kLe1 T(Bsp,r)≤ A κ T kQgk e Lm T(B s−2+ 2m p,r ) + µkak e L∞ T(B N p1+α p1,+∞) k∆uk e Lm T(B s−2+ 2m p,r ) . (3.14)
Remark 7 Let us stress the fact that if a ∈ eL∞((0, T ) × B
N p1
p1,∞) then assumption (3.11)
and (3.12) are satisfied for m large enough. This will be used in the proof of theorem 1.2 and 1.4. Indeed, according to Bernstein inequality for m large enough 93.11) and (3.12) are satisfied.
Proving proposition 3.5 in the case b = cste is not too involved as one can easily get rid of the pressure by taking advantage of the Leray projector P on solenoidal vector-fields. Then system (3.9) reduce to a linear ψDO which may be easily solved by mean of energy estimates. In our case where b is not assumed to be a constant, getting rid of the pressure will still be an appropriate strategy. This may be achieved by applying the operator div to (3.9). Indeed by doing so, we see that the pressure solves the elliptic equation:
div(b∇Π) = divF. (3.15)
with F = f + g + µa∆u. Therefore denoting by Hb the linear operator F → ∇Π, system
(3.9) reduces to a linear ODE in Banach spaces. Actually, due to the consideration of two forcing terms f and g with different regularities, the pressure has to be split into two parts, namely Π = Π1+ Π2 with:
div(bΠ1) = divf (3.16)
div(bΠ2) = divH and H = g + µa∆u. (3.17)
Proof of proposition 3.5:
Let us first rewrite (3.9) as follows: ∂tu − bmµ∆u + b∇Π = f + g + Em, divu = 0, ut=0 = u0. (3.18)
with Em = µ∆u (Id − Sm)a and bm = 1 + Sma. Note that by using corollary 1 and as
−Np 1 < s < N p1 for p ≥ 2 or N p01 < s < N
p1 else, the error term Em may be estimated by:
kEmkBs p,r . ka − Smak B N p1 p1,∞∩L∞ kD2uk Bs p,r. (3.19)
Now applying operator ∆q and next operator of free divergence yield P to momentum
equation (3.18) yields: d
where we denote by uq = ∆qu and with:
e
Rq= eR1q+ eRq2.
where:
e
R1q = µ P∆q(bm∆u) − Pdiv(bm∇uq),
e
R2q = µ Pdiv(bm∇uq) − div(bm∇uq) = −µQdiv(Sma∇uq).
where Q is the gradient yield projector.
case p 6= 2
Next multiplying both sides by |uq|p−2uq, and integrating by parts in the second, third
and last term in the left-hand side, we get by using Bony decomposition (for the notation see [11]): 1 p d dtkuqk p Lp+ µ Z RN bm|∇uq|2|uq|p−2dx + µ Z RN bm|uq|p−4|∇|u|2|2dx ≤ kuqkp−1Lp (kPfqkLp+ kPgqkLp+ k eRqkLp+ k∆q(T∇aΠ)kLp+ 2qk∆q(TaΠ)kLp + k∆q(T 0 ∇Πa)kLp+ kP∆qEmkLp). (3.21)
Indeed we have as divu = 0 and by using Bony’s decomposition and by performing an integration by parts: Z RN ∆q(a∇Π)|uq|p−2uq= Z RN ∆q(T 0 ∇Πa)|uq|p−2uqdx − Z RN ∆q(T∇aΠ)|uq|p−2uqdx − Z RN ∆q(TaΠ)div(|uq|p−2uq)dx. Next we have: ∇(|uq|p−2) · uq= (p − 2)|uq|p−4 X i,k ukq∂iukquiq,
and by H¨older’s and Berstein’s inequalities: k∇(|uq|p−2) · uqk L p p−1 ≤ C(p − 2)2 qku qkp−1Lp .
Next from inequality (3.21), we get by using lemma A5 in [16]: 1 p d dtkuqk p Lp+ ν(p − 1) p2 2 2qku qkpLp ≤ kuqkp−1Lp kPfqkLp+ kPgqkLp+ kP∆qEmkLp + k∆q(T∇aΠ)kLp+ 2qk∆q(TaΠ)kLp+ k∆q(T 0 ∇Πa)kLp+ k eRqkLp,
Therefore, elementary computation yield (at least formally):
e− ν(p−1) p2 2 2qtd dt e ν(p−1) p2 2 2qt kuqkLp . kPfqkLp+ kPgqkLp+ kP∆qEmkLp + k∆q(T∇aΠ)kLp+ 2qk∆q(TaΠ)kLp+ k∆q(T 0 ∇Πa)kLp+ k eRqkLp.
We thus have: kuq(t)kLp. e− ν(p−1) p2 2 2qt k∆qu0kLp+ Z t 0 e− ν(p−1) p2 2 2q(t−τ ) kPfqkLp+ kPgqkLp+ kP∆qEmkLp+ k∆q(T∇aΠ)kLp+ 2qk∆q(TaΠ)kLp+ k∆q(T 0 ∇Πa)kLp+ k eRqkLp(τ )dτ,
which leads for all q ≥ −1, after performing a time integration and using convolution inequalities to: (ν(p − 1) p2 ) 1 m2 2q mkuqk LmT(Lp) . k∆qu0kLp+ kPfqkL1 T(Lp)+ k∆q(T∇aΠ1)kL 1 T(Lp) + 2qk∆q(TaΠ1)kL1 T(Lp)+ k∆q(T 0 ∇Π1a)kL1T(Lp)+ k eRqkLT1(Lp)+ kP∆qEmkL1T(Lp) + (ν(p − 1) p2 ) 1 m−12q( 2 m−2) k∆q(T∇aΠ2)k L1T(Lp)+ 2qk∆q(TaΠ2)kL1 T(Lp) + k∆q(T 0 ∇Π2a)kL1T(Lp)+ kPgqkLmT(Lp), (3.22)
We are now interested by treating the commutator term eR1q, we have then by using lemma 1 in the appendix the following estimates with α < 1:
k eR1qkLp . cqν2¯ (−1+α)qskSmak B N p1+α p1,∞ kDukBs p,r, (3.23)
where (cq)q∈Z is a positive sequence such that cq ∈ lr, and ¯ν = µ. Note that, owing to
Bernstein inequality, we have: kSmak B N p1+α p1,∞ . 2mαkak B N p1 p1,∞
Next we have by corollary 1:
k eR2qkLp . cqν2¯ −qskSmak B N p1 p1,r∩L∞ kukBs+2 p,r , (3.24)
Hence, plugging (3.23), (3.24) and (3.19) in (3.22), then multiplying by 2qs and summing up on q ∈ Z in lr, we discover that, for all t ∈ [0, T ]:
kuk e L∞T(Bs p,r)+ ( ν(p − 1) p ) 1 mkuk e LmT(Bp,rs+ 2m) ≤ ku0kBs p,r + kPf kLe1T(Bp,rs ) + kTaΠ1kLe1 T(B s+1 p,r )+ kT∇aΠ1kLe1 T(Bp,rs )+ kT 0 ∇Π1akLe1 T(Bsp,r) + C ¯νka − Smak e L∞(B N p1 p1,∞∩L∞) kuk e L1(Bs+2 p,r )+ 2 mα Z T 0 kak B N p1 p1,∞ (τ )kukBs+1 p,r (τ )dτ + ν(p − 1) p2 ) 1 m(kPgk e Lm T(B s−2+ 2m p,r ) + kTaΠ2k e Lm T(B s−1+ 2m p,r ) + kT∇aΠ2k e Lm T(B s−2+ 2m p,r ) + kT∇Π0 2ak e Lm T(B s−2+ 2m p,r ) , (3.25)
for a constant C depending only on N and s. With our assumption on α, α0 and s, the terms kT∇aΠ1kLe1
T(Bp,rs ) and kT 0
∇Π1akLe1
T(Bp,rs ) may be bounded by:
kak e L∞(B N p1+α p,∞ ∩L∞) k∇Π1k e L1 T(B s−α0 p,r )
whereas kT∇aΠ2k e LmT(Bp,rs−2+ 2m) and kT∇Π0 2ak e LmT(Bp,rs−2+ 2m)
may be bounded by:
kak e L∞(B N p1+α p,∞ ∩L∞) k∇Π2k e Lm T(B s−2+ 2m −α0 p,r ) .
Moreover we control kTaΠ1kLe1T(Bp,rs+1) and kTaΠ2kLem T(B s−1+ 2m p,r ) by respectively: kak e L∞(B N p1 p,∞∩L∞) k∇Π1kLe1 T(Bp,rs ). kak e L∞(B N p1 p,∞∩L∞) k∇Π2k e Lm T(B s−2+ 2m p,r ) .
Hence in view of proposition 8.8 and provided that 0 < α0 < min(1, α,s2) and s < pN
1
(which is assumed in the statement of proposition 3.5) and α00∈ [0, α0],
bk∇Π1k e L1 T(B s−α00 p,r ). A s−α00 α0 T kQf kLe1(Bp,rs ). (3.26)
On the other hand, by virtue of proposition 2.2, and of assumption on α, α0, α00 and s, we have: kQHk e Lm T(B s−2+ 2m −α00 p,r ) . kQgk e Lm T(B s−2+ 2m −α00 p,r ) + µkak e L∞(B N p1+α p1,∞ ∩L∞) k∆uk e Lm T(B s−2+ 2m −α00 p,r ) . As α0 ≤ min(1, α,1 2(s − 2 + 2 m)), proposition 8.8 with σ = s − 2 + 2 m − α 00 (here comes s > 2 − m2) applies, from which we get for all ε > 0 (ε = 0 does if m ≥ 2),
bk∇Π2k e Lm T(B s−2+ 2m −α00 p,r ) . A s−2+ 2m +ε α0 T kQgk e Lm T(B s−2+ 2m −α00 p,r + µkak e L∞(B N p1+α p1,∞ ∩L∞) k∆uk e Lm T(B s−2+ 2m −α00 p,r ) . (3.27) Let X(t) = kukL∞
t (Bsp,r)+ νbkukL1t(Bs+2p,r ). Assuming that m has been chosen so large as
to satisfy: C ¯νka − Smak L∞T(B N p1 p1,∞∩L∞) ≤ ν,
and by interpolation we get:
C ¯ν2mαkak B N p1 p1,∞ kukBs+2 p,r ≤ κν + C2ν¯222mα 4κν kak 2 B N p1 p1,∞ kukBs p,r, (3.28)
Plugging (3.26), (3.27) and (3.28) in (3.25), we end up with:
X(T ) ≤ ku0kBs p,r+ A s α0 T kPf k e L1 t(Bp,rs )+ kPgkLemt (B s−2+ 2m p,r ) + C Z t 0 ¯ ν2 ν 2 2mαkak2 B N p1 p1,∞ (τ ) × X(τ )dτ + (ν(p − 1) p ) 1 mATkuk e L1 T(B s+2−α0 p,r ) .
Case p = 2
In this case we don’t need of condition of smallness on kak
B N p1 p1,∞∩L∞
, indeed the bad terms as eR2
q or 2qk∆q(TaΠ)kL2 disapear in the integration by parts as divuq = 0. So we
can follow the same procedure and conclude.
4
The mass conservation equation
4.1 Losing estimates for transport equation
We now focus on the mass equation associated to (1.1): (
∂ta + v · ∇a = g,
a/t=0 = a0.
(4.29)
We will precise in the sequel the regularity of a0, v and g. In this section we intend to
recall some result on transport equation associated to vector fields which are not Lipschitz with respect to the space variable. Since we still have in mind to get regularity theorems, those vector field cannot be to rough. In order to measure precisely the regularity of the vector field v, we shall introduce the following notation:
Vp01,α(t) = sup j≥0 2j N p1k∇Sjv(t)kLp1 (j + 1)α < +∞. (4.30)
Let us remark that if p1 = +∞ then V 0
p1,α is exactly the norm kBΓk of definition 2.3.
4.1.1 Limited loss of regularity
In this section, we make the assumption that there exists some α ∈]0, 1[ such that the function Vp01,αdefined in (4.30) be locally integrable. We will show that in the case α = 1, then a linear loss of regularity may occur. In the theorem below, Bahouri, Chemin and Danchin show in [5] that if α ∈]0, 1[ then the loss of regularity in the estimate is arbitrarily small.
Theorem 4.5 Let (p, p1) be in [1, +∞]2 such that 1 ≤ p ≤ p1 and σ satisfying σ >
−1 − N min(1 p1, 1 p0). Assume that σ < 1 + N p1 and that V 0 p1,α ∈]0, 1[ is in L 1([0, T ]).
Let a0 ∈ Bp,∞σ and g ∈ eL1T(Bσp,∞). Then the equation (4.29) has a unique solution
a ∈ C([0, T ], ∩σ0 <σBσ
0
p,∞) and the following estimate holds for all small enough ε:
kak e L∞ T(B σ−ε p,∞)≤ C ka0kBp,∞σ + kgkLe1 T(Bσp,∞) exp C ε1−αα (Vp1,α(T )) 1 1−α,
where C depends only on α, p, p1, σ and N .
In the following proposition, we are interested in showing a control of the high frequencies on the density when u is not Lipschitz. Indeed we recall that in the proposition 3.5 when p = 2, we need to control the high frequencies of the density. In particular the following proposition is useful only in the case of theorem 1.2.
Proposition 4.6 Let (p, p1) be in [1, +∞]2 such that 1 ≤ p ≤ p1 and σ satisfying σ > −1 − N min(p1 1, 1 p0). Assume that σ < 1 + N p1 and that V 0 p1,α ∈]0, 1[ is in L 1([0, T ]).
Let a0 ∈ Bp,∞σ and g ∈ eL1T(Bp,∞σ ), the equation (4.29) has a unique solution a ∈
C([0, T ], ∩σ0 <σBσ
0
p,∞) and the following estimate holds for all small enough ε:
X l≥m 2(σ−ε)lk∆la(t0)kL∞(Lp). X l≥m (2σlk∆la0kLp) + Cη α 1−α Z t 0 Vp01,α(t0) × (ka0kBσ p,∞ + kgkLe1T(Bσp,∞)) exp C ε1−αα (Vp1,α(t 0 ))1−α1 dt0.
where C depends only on α, p, p1, σ and N .
Proof:
By using the proof of Bahouri, Chemin and Danchin in [5] one can write:
2(2+j)σtk∆ la(t)kLp ≤ 2(2+l)σk∆lf0kLp+ C( 2C η log 2) α 1−α Z t 0 Vp01,α(t0)ka(t0)k B σ t0 p,∞ dt0. (4.31)
Whence taking the supremum over l ≥ m, we get
sup t0∈[0,t] sup l≥m (2σt0lk∆la(t 0 )kLp) . sup l≥m (2σlk∆la0kLp) + Cη α 1−α Z t 0 Vp01,α(t0)ka(t0)k B σ t0 p,∞ dt0.
We apply Gr¨onwall inequality in (4.31) and we insert in previous inequality:
sup t0∈[0,t] sup l≥m (2σt0lk∆la(t 0 )kLp) . sup l≥m (2σlk∆la0kLp) + Cη α 1−αkf0kBσ p,∞ Z t 0 Vp01,α(t0) × exp C ε1−αα (Vp1,α(t 0 ))1−α1 dt0.
which leads to the proposition.
Remark 8 In the sequel, we will use the theorem 4.5 and the proposition 4.6 when p1 = ∞ and α = 1 + ε
0
−1
r. Indeed we will have u is in eL 1(B
N p2+1
p2,r ) (with 1 ≤ r < ∞)
and according proposition 2.4 and definition 2.3 we get: Z t 0 V∞,α0 (t 0 )dt0 . kuk e L1(B N p2+1 p2,r ) .
So it will allow to get estimates on the density with an arbitrarly small loss of regularity.
4.1.2 Linear loss of regularity in Besov spaces
Tis section is devoted to the estimates with linear loss of regularity. Remind that u is log-lipschitz only if only if there some constant C such that:
k∇SjkL∞ ≤ C(j + 1) for all j ≥ −1.
Theorem 4.6 Let 1 ≤ p ≤ p1 ≤ ∞ and s1 ∈ R satisfies σ > −N min(p11,p10). Let σ in
]s1, 1 +pN1[ and v a vector field. There exists a constant C depending only on p, p1, σ, s1
and N such that for any λ > C, T > 0 and any nonnegative integrable function W over [0, T ] such that if σT ≥ s1 with:
σt= σ − λ
Z t
0
Vp01,1(t0) + W (t0)dt0 then the following property holds true.
Let a0 ∈ Bσp,∞ and g = g1+ g2 with, for all t ∈ [0, T ], g1(t) ∈ Bp,∞σt and:
∀j ≥ −1, k∆jg2(t)kLp ≤ 2−jσt(2 + j)W (t)kf (t)kBσt p,∞.
Let a ∈ C([0, T ]; Bs1
p,∞) be a solution of (4.29). Then the following estimate holds:
sup t∈[0,T ] ka(t)kBσt p,∞ ≤ λ λ − C ka0kBσp,∞+ Z T 0 kg1(t)kBσ p,∞dt.
In the following proposition, we generalize this result to the high frequencies of the density ρ.
Proposition 4.7 Let 1 ≤ p ≤ p1 ≤ ∞ and s1 ∈ R satisfies σ > −N min(p11,p10). Let σ
in ]s1, 1 + Np1[ and v a vector field. There exists a constant C depending only on p, p1,
σ, s1 and N such that for any λ > C, T > 0 and any nonnegative integrable function W
over [0, T ] such that if σT ≥ s1 with:
σt= σ − λ
Z t
0
Vp01,1(t0) + W (t0)dt0 then the following property holds true.
Let a0 ∈ Bσp,∞ and g = g1+ g2 with, for all t ∈ [0, T ], g1(t) ∈ Bp,∞σt and:
∀j ≥ −1, k∆jg2(t)kLp ≤ 2−jσt(2 + j)W (t)kf (t)kBσt p,∞.
Let a ∈ C([0, T ]; Bs1
p,∞) be a solution of (4.29). Then the following estimate holds:
sup t∈0,T X l≥m 2lσtk∆ lakLp≤ λ λ − C ka0kBσp,∞+ Z T 0 kg1(t)kBσ p,∞dt. Proof:
The proof follows the same line as in proposition 4.6. Remark 9 In the sequel, we will use the theorem 4.6 and the proposition 4.7 when p1 = ∞. Indeed we will have u in eL1(B
N p2+1
p2,∞), then according proposition 2.1, ∇u is in
e
L1(B∞,∞0 )so we obtain the following control: Z t 0 sup l≥0 k∇Slu(t0, ·)kL∞ l + 1 dt 0 . kuk e L1(B N p2+1 p2,∞) .
So it will allow to get estimates on the density with a small loss of regularity on a small time interval.
5
Proof of the existence for theorem 1.1
We use a standard scheme:
1. We smooth out the data and get a sequence of smooth solutions (an, un)n∈N to (1.4) on a bounded interval [0, Tn] which may depend on n.
2. We exhibit a positive lower bound T for Tn, and prove uniform estimates in the space: ET = eCT(B N p1+ ε 2 p1,∞ ) × eCT(B N p2−1 p2,r ) ∩ eL 1(BNp+1 p2,r )) N . for the smooth solution (an, un).
3. We use compactness to prove that the sequence converges, up to extraction, to a solution of (1.4).
Construction of approximate solutions We smooth on the data as follows:
an0 = Sna0, u0n= Snu0 and fn= Snf.
Note that we have:
∀l ∈ Z, k∆lan0kLp1 ≤ k∆la0kLp1 and kan0k B N p1+ε 0 p,∞ ≤ ka0k B N p1+ε 0 p,∞ ,
and similar properties for un
0 and fn, a fact which will be used repeatedly during the
next steps. Now, according [2], one can solve (1.4) with the smooth data (an0, un0, fn). We get a solution (an, un) on a non trivial time interval [0, Tn] such that:
an∈ eC([0, Tn), HN +ε), un∈ C([0, Tn), HN) ∩ eL1Tn(H N +2)
and ∇Πn∈ eL1Tn(HN). (5.32)
Uniform bounds
Let Tnbe the lifespan of (an, un), that is the supremum of all T > 0 such that (1.4) with
initial data (an0, un0) has a solution which satisfies (5.32). Let T be in (0, Tn), we aim at
getting uniform estimates in ET for T small enough. For that, we need to introduce the
solution (unL, ∇ΠnL) to the nonstationary Stokes system:
(L) ∂tunL− µ∆unL+ ∇ΠnL= fn, divunL= 0, (unL)t=0= un0.
Now, the vectorfields ˜un= un− un
Land ∇Πn= ∇ΠnL+ ∇ ˜Πnsatisfy the parabolic system:
∂tu˜n− µ(1 + an)∆˜un+ (1 + an)∇ ˜Πn= H(an, un, ∇Πn), div˜un= 0, ˜ un(0) = 0, (5.33)
which has been studied in proposition 3.5 with:
H(an, un, ∇Πn) = an(µ∆unL− ∇ΠnL) − un· ∇un. Define m ∈ Z by: m = inf{p ∈ Z/ 2¯νX l≥q 2l N p1k∆la0kLp1 ≤ c¯ν} (5.34)
where c is small enough positive constant (depending only N ) to be fixed hereafter. Let: ¯b = 1 + sup x∈RN a0(x), A0 = 1 + 2ka0k B N p1+ε 0 p1,∞ , U0 = ku0k B N p2−1 p2,r + kf k e L1(B N p2−1 p2,r ) ,
and eU0 = 2CU0+ 4C ¯νA0 (where C stands for a large enough constant depending only
N which will be determined when applying proposition 2.2, 3.5 and 4.6 in the following computations.) We assume that the following inequalities are fulfilled for some η > 0, α > 0: (H1) kan− Smank e L∞ T(B N p1 p1,∞)∩L∞ ≤ cν ¯ν−1, (H2) C ¯ν2T kank2 e L∞ T(B N p1 p1,∞)∩L∞ ≤ 2−2mν, (H3) 1 2b ≤ 1 + a n(t, x) ≤ 2¯ b for all (t, x) ∈ [0, T ] × RN, (H4) kank e L∞T(B N p1+ ε2 p1,∞ )∩L∞ ≤ A0, (H5) kunLk e L1(B N p2+1 p2,r ) ≤ η, (H6) kue nk e L∞(B N p2−1 p2,r ) + νkue nk e L1(B N p2+1 p2,r ) ≤ eU0η, (H7) k∇ΠnLk e L1(B N p2−1 p2,r ) ≤ η, (H8) k∇ eΠnk e L1(B N p2−1 p2,r ) ≤ eΠ0η,
Remark that since:
1 + Sman= 1 + an+ (Sman− an),
assumptions (H1) and (H3) insure that:
inf
(t,x)∈[0,T ]×RN(1 + Sma
n)(t, x) ≥ 1
4b, (5.35)
provided c has been chosen small enough (note that νν¯ ≤ ¯b).
We are going to prove that under suitable assumptions on T and η (to be specified below) if condition (H1) to (H8) are satisfied, then they are actually satisfied with strict
inequalities. Since all those conditions depend continuously on the time variable and are strictly satisfied initially, a basic boobstrap argument insures that (H1) to (H8) are
indeed satisfied for a small Tn0 ≤ Tn. First we shall assume that η satisfies:
so that denoting: ( ˜Vn)p 2,1−1r(t) = Z t 0 ( ˜Vn)0p 2,1−1r(s)ds and (V n L)p2,1−1r(t) = Z t 0 (VLn)0p 2,1−1r(s)ds, with: ( ˜Vn)0p 2,1−1r(s) = sup l≥0 2l N p2k∇Slu˜n(s)kLp2 (l + 1)1−1r ds and (VLn)0p 2,1−1r(s) = sup l≥0 2l N p2k∇SlunL(s)kLp2 (l + 1)1−1r ds,
We recall now that according proposition 2.4: ( ˜Vn)p 2,1−1r(t) ≤ Ck˜u nk e L1 t(B N p2+1 p2,r ) and (VLn)p 2,1−1r(t) ≤ Cku n Lk e L1 t(B N p2+1 p2,r ) ,
we have, according to (H5) and (H6):
e C(e2)1−r k˜unk e L1 t(B N p2+1 p2,r ) +kun Lk e L1 t(B N p2+1 p2,r ) < 2. (5.37) In order to bound an in eL∞T (B N p1+ ε 2
p1,∞ ), we apply theorem 4.5 and get:
kank e L∞ T(B N p1+ ε2 p1,∞ ) < eC(ε2) 1−r k˜unk e L1 t(B N p2 p2,r) + kunLk e L1 t(B N p2 p2,r) ka0k B N p1+ε p1,∞ . (5.38)
Moreover as we know that kankL∞ ≤ ka0kL∞, (H4) is satisfied with a strict inequality.
Next, applying classical proposition on heat equation ( see [11]) yields: kunLk e L∞ T(B N p2−1 p2,r ) ≤ U0, (5.39) κνkunLk e L1 T(B N p2+1 p2,r ) ≤ X l∈Z 2lr( N p2−1)(1 − e−κν22lT)r(k∆lu0kLrp2 + k∆lf krL1(R+,Lp2)) 1 r. (5.40) Hence taking T such that:
X l∈Z 2lr( N p2−1)(1 − e−κν22lT)r(k∆lu0kr L2+ k∆lf krL1(R+,L2)) 1r < κην, (5.41)
insures that (H5) is strictly verified.
Since (H1), (H2) and (5.35) are satisfied, proposition 3.5 may be applied with α = ε2.
We get: keunk e L∞ T(B N p2−1 p2,r ) + νkeunk e L1 T(B N p2+1 p2,r ) ≤ CeCZn m(T ) ku0k B N p2−1 p2,r + AκT,n× kan(∆un L− ∇ΠnL)k e L1(B N p2−1 p2,r ) + kun· ∇unk e L1(B N p2−1 p2,r ) + AT,nkuk e L1 T(B N p2+1−α 0 p2,r ) .
with Zmn(T ) = 2mε¯ν2ν−1RT 0 ka nk2 B N p1 p1,∞∩L∞
dτ. Next using Bony’s decomposition and divun= 0, one can write:
div(un⊗ un) = T∂jv(u n)j+ T
(un)j∂jun+ ∂jR(un, (un)j),
with the summation convention over repeated indices.
Hence combining proposition 1.4.1 and 1.4.2 in [11] with the fact that eLρT(B
N p2− 1 2 p2,r ) ,→ e LρT(B− 1 2 ∞,∞) for ρ = 43 or ρ = 4, we get: kdiv(un⊗ un)k e L1 T(B N p2−1 p2,r ) ≤ Ckunk e L 4 3 T(B N p2+ 12 p2,r ) kunk e L4 T(B N p2− 12 p2,r ) .
By taking advantage of proposition 3.5, 2.2, 2.1 and Young’inequality, we end up with:
kuenk e L∞T(B N p2−1 p2,r ) + νkuenk e L1 T(B N p2+1 p2,r ) ≤ eCZmn(T ) ku0k B N p2−1 p2,r + kank e L∞T(B N p1+ ε2 p1,∞ ∩L∞) + 1κ × CkunLk e L1 T(B N p2+1 p2,r ) (¯νkank L∞ T(B N p1 p1,∞∩L∞) + kunLk e L∞ T(B N p2−1 p2,r ) ) + k∇ΠnLk e L1 T(B N p2−1 p2,r ) × kank L∞ T(B N p1 p1,∞∩L∞) + (kank e L∞ T(B N p1+ ε2 p1,∞ ∩L∞) + 1)Tε2k e unk e L1 T(B N p2+1 p2,r ) .
with C = C(N ) and Cg = (N, g, b, ¯b). Now, using assumptions (H2), (H4), (H5), (H6)
and (H7), and inserting (5.37) in the previous inequality and choosing T enough small
gives: kuenk e L∞T(B N p2−1 p2,r ) + keunk e L1 T(B N p2+1 p2,r ) ≤ 2C(¯νA0+ U0)η + 2CgT A0,
hence (H6) is satisfied with a strict inequality provided:
CgT < C ¯νη. (5.42)
To show that (H7) and (H8) are strictly verified on (0, T 0
n), we proceed similarly as for
(H5) and (H6). We now have to check whether (H1) is satisfied with strict inequality.
For that we apply proposition 4.6 which yields for all m ∈ Z, sup l≥m 2l N p1k∆lankL∞ T(Lp1)≤ C sup l≥m 2l N p1k∆la0kLp1) k˜unk e L1 t(B N p2 p2,r) + kunLk e L1 t(B N p2 p2,r) (5.43)
Using (5.36) and (H5), (H6), we thus get:
kan− Smank L∞ T(B N p1 p1,∞)∩L∞ ≤ sup l≥m 2l N p1k∆la0kLp1 + C log 2(1 + ka0kB N p1 p1,∞∩L∞ )(1 + ν−1Le0)η.
Hence (H1) is strictly satisfied provided that η further satisfies:
C log 2(1 + ka0kBp1N p1,∞∩L∞ )(1 + ν−1Le0)η < cν 2¯ν. (5.44) So H1 is strictly verified.
away and that a0 ∈ L∞.
Next, according to (H4) condition (H2) is satisfied provide:
T < 2
−2mν
C ¯ν2A2 0
(5.45)
One can now conclude that if T < Tn has been choosen so that conditions (5.41), (5.42) and (5.45) are satisfied (with η verifying (5.36) and (5.44), and m defined in (5.34) and n ≥ m then (an, un, Πn) satisfies (H1) to (H8), thus is bounded independently of n on
[0, T ].
We still have to state that Tnmay be bounded by below by the supremum ¯T of all times T such that (5.41), (5.42) and (5.45) are satisfied. This is actually a consequence of the uniform bounds we have just obtained, and of a theorem of blow-up of R. Danchin in [14]. Indeed, by combining all these informations, one can prove that if Tn < ¯T then (an, un, ∇Πn) is actually in: e L∞Tn(B N p1+ε 0 p1,∞ ∩ B N 2+1 2,1 ) × eL ∞ Tn(B N p2−1 p2,r ∩ B N 2 2,1) ∩ eL1Tn(B N p2+1 p2,r ∩ B N 2+2 2,1 ) N × eL1Tn(B N p2−1 p2,r ∩ B N 2 2,1).
hence may be continued beyond ¯T (see the remark on the lifespan in [14] where a control of ∇u in eL1(B∞,∞0 ) is required). We thus have Tn≥ ¯T .
Compactness arguments
We now have to prove that (an, un)n∈N tends (up to a subsequence) to some function (a, u) which belongs to ET and satisfies (1.4). The proof is based on Ascoli’s theorem
and compact embedding for Besov spaces. As similar arguments have been employed in [12] or [13], we only give the outlines of the proof.
• Convergence of (an) n∈N:
We use the fact thatean= an− an
0 satisfies:
∂tea
n= −un· ∇an.
Since (un)n∈N is uniformly bounded in eL1T(B N p2+1 p2,r ) ∩ eL ∞ T (B N p2−1 p2,r ), it is, by
interpo-lation, also bounded in eLr
0 T(B N p2−1+ 2 r0 p2,r ) for any r 0 ∈ [1, +∞]. By taking r = 2 and using the standard product laws in Besov spaces, we thus easily gather that (∂tea
n) is uniformly bounded in eL2T(B N p1−1 p1,∞). k∂tea nk e L2 T(B N p1−1 p1,∞) . kunk e L2 T(B N p2 p2,r) k∇ank e L∞ T(B N p1−1 p1,∞) . Hence (ean) n∈N is bounded in eL∞T (B N p1−1 p1,∞ ∩ B N p1 p1,∞) and equicontinuous on [0, T ] with values in B N p1−1
p1,∞. Since the embedding B N p1−1 p1,∞ ∩ B N p1 p1,∞ ,→ B N p1−1 p1,∞ is (locally)
compact, and (an0)n∈N tends to a0 in B N p1
p1,∞, we conclude that (a n)
extraction) to some distribution a. Given that (an) n∈N is bounded in eL∞T (B N p1+ ε 2 p1,r ), we actually have a ∈ eL∞T (B N p1+ ε 2 p1,r ). • Convergence of (un L)n∈N:
From the definition of unL and classical proposition on Stokes equation, it is clear that (unL)n∈N and (∇ΠnL)n∈N tend to solution uLand ∇ΠL to:
∂tuL− µ∆ul+ ∇ΠL= f, uL(0) = u0 in eL∞T (B N p2−1 p2,r ) ∩ eL1T(B N p2+1 p2,r ) for (unL)n∈N and eL1T(B N p2−1 p2,r ) for (∇ΠnL)n∈N. • Convergence of (uen)n∈N: We use the fact that:
∂tue
n= −un L· ∇eu
n−
e
un· ∇un+ (1 + an)∆uen+ an∆unL− unL· ∇unL− ∇ eΠn.
As (an)n∈Nis uniformly bounded in eL∞T (B N p1 p1,∞) and (u n) n∈N is uniformly bounded in eL∞T (B N p2−1 p2,r ) ∩ eL 4 3(B N p2+ 1 2
p2,r ), it is easy to see that the last term of the right-hand
side is uniformly bounded in eL∞T (B
N p2−1
p2,r ) and that the other terms are uniformly
bounded in eL43(B N p2− 3 2 p2,r ). Hence (uen)n∈Nis bounded in eL∞T (B N p2−1
p2,r ) and equicontinuous on [0, T ] with values in
B N p2−1 p2,r +B N p2− 3 2
p2,r . This enables to conclude that (ue
n)
n∈Nconverges (up to extraction)
to some functionu ∈ ee L∞T (B N p2−1 p2,r ) ∩ eL 1 T(B N p2+1 p2,r ).
We proceed similarly for (ΠnL)n∈N and ( eΠn)n∈N. By interpolating with the bounds pro-vided by the previous step, one obtains better results of convergence so that one can pass to the limit in the mass equation and in (5.33). Finally by setting u =u + ue L and Π = eΠ + ΠL, we conclude that (a, u, Π) satisfies (1.4).
In order to prove continuity in time for a it suffices to make use of proposition 4.5. In-deed, a0 is in B N p1+ε p1,∞ ∩ L ∞, and having a ∈ eL∞ T (B N p1+ ε 2 p1,∞ ) ∩ L ∞ and u ∈ eL1 T(B N p2+1 p2,r ) insure
that ∂ta + u · ∇a belongs to eL1T(B N p1
p1,∞). Similarly, continuity for u may be proved by
using that u0 ∈ B N p2
p2,r and that (∂tu − µ∆u) ∈ eL 1 T(B
N p2−1 p2,r ).
6
The proof of the uniqueness
6.1 Uniqueness when 1 ≤ p2 < 2N , N2 < p11 +p12 and N ≥ 3.
In this section, we focus on the case N ≥ 3 and postpone the analysis of the other cases (which turns out to be critical) to the next section. Throughout the proof, we assume that we are given two solutions (a1, u1) and (a2, u2) of (1.4) which belongs to:
e C([0, T ]; B N p1+ ε 2 p1,∞ ) ∩ L ∞ × e C([0, T ]; B N p2−1 p2,r ) ∩ eL 1(0, T ; B N p2+1 p2,r ) N .
Let δa = a2− a1, δu = u2− u1 and δΠ = Π2− Π1. The system for (δa, δu) reads:
(
∂tδa + u2· ∇δa = −δu · ∇a1,
∂tδu − (1 + a2)(µ∆δu − ∇δΠ) = F (ai, ui, Πi).
(6.46)
with:
F (ai, ui, Πi) = u1· ∇δu + δu · ∇u2+ δa(µ∆u1− ∇Π1).
The function δa may be estimated by taking advantage of proposition 4.5 with s =
N p1 − 1 +
ε
2. We get for all t ∈ [0, T ],
kδa(t)k B N p1−1 p1,∞ ≤ Ckδu · ∇a1k e L1T(B N p1−1+ ε2 p1,∞ ) exp C εr−1(Vp1,1−1r(t)) r,
We have then by proposition 2.2 and 2.4: kδa(t)k B N p1−1 p1,∞ ≤ Ckδuk e L1 T(B N p2 p2,r) ka1k e L∞ T(B N p1+ ε2 p1,∞ ) exp C εr−1(ku 2k e L1 T(B N p2+1 p2,r ) )r, (6.47)
For bounding δu, we aim at applying proposition 3.5 to the second equation of (6.46). So let us fix an integer m such that:
1 + inf (t,x)∈[0,T ]×RNSma 2≥ b 2 and ka 2− S ma2k L∞ T(B N p1 p1,∞) ≤ cν ¯ ν. (6.48) Now applying proposition 3.5 with s = Np
2− 2 insures that for all time t ∈ [0, T ], we have:
kuk e L∞ T(B N p2−2 p2,r ) + κνkuk e L1 T(B N p2 p2,r) + k∇δΠk e L1 T(B N p2−2 p2,r ) ≤ eCZm(T )× kPF (ai, ui, Πi)k e L1 T(B N p2−2 p2,r ) + (ν(p2− 1) p2 )ATkuk e L1 T(B N p2−α 0 p,r ) . (6.49) with Zm(t) = 2mµ2ν−1 Rt 0ka(τ )k2 B N p1 p1,∞∩L∞ dτ .
Hence, applying proposition 2.2, corollary 1 and the fact that divδu = 0, we get as exemple: kδu · ∇u1k e L1 T(B N p2−2 p2,r ) . ku1k 1 2 e L∞ T(B N p2−1 p2,r ) ku1k 1 2 e L1 T(B N p2+1 p2,r ) kδuk e L∞ T(B N p2−2 p2,r ) + kδuk e L∞ T(B N p2 p2,r) .
By the fact that N2 < p1
1 + 1 p2, N ≥ 3 and 1 p2 ≤ 1 N + 1 p1 imply that: kδa(µ∆u1− ∇δΠ)k e L1 T(B N p2−2 p2,r ) . kδak e L∞ T(B N p1−1 p1,∞) k∆u1k e L1 T(B N p2−1 p1,r ) + k∇Π1k e L1 T(B N p2−1 p2,r ) . (6.50)
Now let choose T1 enough small to controll in (6.49) ATkuk e L1 T(B N p2−α 0 p,r ) and in (6.50) k∆u1k e L1 T(B N p2−1 p1,r ) + k∇Π1k e L1 T(B N p2−1 p2,r )
and by the fact that ka1k e L∞ T(B N p1+ ε2 p1,∞ ) ≤ c with c small, we obtain finally:
with: FT = eCT([0, T ], B N p1 p1,∞) × eL 1 T(B N p2 p2,r) ∩ eL ∞ T (B N p2−2 p2,r ) ×Le1T(B N p2−2 p2,r ).
We obtain so (δa, δu, ∇δΠ) = 0 on [0, T1] for T1 enough small. By connectivity we obtain
that (δa, δu, δ∇Π) = 0 on [0, T ]. This conclude this case.
6.2 Uniqueness when:N2 = p1 1 +
1
p2 or p2 = 2N or N = 2.
The above proof fails in dimension two. One of the reasons why is that the product of functions does not map B
N p1−1 p1,∞ × B N p2−1 p2,r in B N p2−2
p2,r but only in the larger space B −1 2,∞. This
induces us to bound δa in eL∞T (B
N p1−1 p1,∞) and δu in eL ∞ T (B N p2−2 p2,∞) ∩ eL 1 T(B N p2 p2,∞). In fact, it
is enough to study only the case N2 = p1
1 + 1
p2. Indeed the other cases deduct from this
case. If p2 = 2N then p1 = 2N3 as p11 ≤ N1 +p12 and N2 ≤ p11 +p12. So it is a particular
case of N2 = p1
1 + 1
p2. For N = 2, we begin with p2= 4 and p1= 4
3 and by embedding we
get the result for 1 ≤ p1≤ 43, 1 ≤ p2≤ 4 and for 1 ≤ p1 ≤ 4, 1 ≤ p2≤ 43.
Moreover in your case, it exists two possibilities, one when 1 < p2< 2 and when p2 ≥ 2.
The first case is resolved by embedding so we have just to treat the case p2 ≥ 2. We
want show that (δa, δu, ∇δΠ) ∈ GT where:
GT = eCT([0, T ], B N p1−1+ ε 2 p1,∞ ) × eL 1 T(B N p2 p2,∞) ∩ eL ∞ T (B N p2−2 p2,∞) ×Le1T(B N p2−2 p2,∞).
In fact we proceed exactly as in the previous proof but we benefit that a is in eCT([0, T ], B N p1−1+
ε 2 p1,∞ )
which give sense to the product δ∆u. It conclude the proof of uniqueness.
6.3 Proof of theorem 1.2
The proof follow strictly the same lines than the proof of theorem 1.1 except that we profits of the fact that in the proposition 3.5 in the case of p2 = 2 we do not need of
conditions of smallness on the initial density a0.
7
Proof of theorem 1.3 and 1.4
In this case we have a control on u only in eL1(B
N p2+1
p2,∞), that is why to control the density
in this case we need to use the proposition 4.6. The rest of the proofs is similar to proof 1.1.
8
Appendix
8.1 Elliptic estimates
This section is devoted to the study of the elliptic equation:
div(b∇Π) = divF. (8.51)
with b = 1 + a.
Proposition 8.8 Let 0 < α < 1, (p, r) ∈ [1, +∞]2 and σ ∈ R satisfy α ≤ σ ≤ α +pN1. Then the operator Hb : F → ∇Π is a linear bounded operator in Bp,rσ and the following
estimate holds true:
bk∇ΠkBσ p,r . A |σ| min(1,α)kQF k Bσ p,r, (8.52) with if α 6= 1: A = 1 + b−1k∇bk B N p1+α−1 p1,r . Proof:
Let us first rewrite (8.51) as follows:
div(bm∇Π) = divF − Em, (8.53)
with Em= div((Id − Sm)a∇Π).
Apply ∆q to (8.53) we get:
div(bm∇∆qΠ) = divFq− ∆q(Em) + Rq, (8.54)
with Rq = div(bm∇Πq) − ∆qdiv(bm∇Π). Multiplying (8.54) by ∆qΠ|∆qΠ|p−2 and
inte-grate, we gather: Z RN bm|∇Πq|2|Πq|p−2dx + Z RN bm|∇|Πq|2|pdx ≤ (kdivFqkLp+ kRqkLp)kΠqkp−1 Lp + Z RN |(Id − Sm)a| |∇Πq|2|Πq|p−2dx + Z RN |(Id − Sm)a| |∇|Πq|2|pdx, (8.55)
Assuming that m has been choose so large as to satisfy: ka − Smak L∞T(B N p1+α p1,∞ )∩L∞ ≤ b 2, an by using lemma A5 in [16]: 22qkΠqkLp . 2qkFqkLp+ kRqkLp.
By multipliyng by 2q(s−1) and by integrating on lr we get:
bk∇ΠkBs
p,r . kQf kBsp,r + kRqkBp,rs .
The commutator may be bounded thanks to lemma 1 with 0 < α < 1, σ = s − 1. We have get: bk∇ΠkBs p,r . kQF kBsp,r + k∇Smak B N p1+α p1,r k∇ΠkBs−α p,r .
Therefore complex interpolation entails: k∇ΠkBs−α p,r ≤ k∇Πk s−α s Bs p,rk∇Πk α s B0 p,∞.
Note that, owing to Bersntein inequality, we have: k∇Smak B N p1+α p1,r . 2m+αkak B N p1 p1,r . We have then: bk∇ΠkBs p,r . kQf kBp,rs + 2 m+αkak B N p1 p1,∞ k∇Πk s−α s Bs p,rk∇Πk α s B0 p,∞.
8.2 Commutator estimates
This section is devoted to the proof of commutator estimates which have been used in section 2 and 3. They are based on paradifferentiel calculus, a tool introduced by J.-M. Bony in [6]. The basic idea of paradifferential calculus is that any product of two distributions u and v can be formally decomposed into:
uv = Tuv + Tvu + R(u, v) = Tuv + T 0 vu
where the paraproduct operator is defined by Tuv =PqSq−1u∆qv, the remainder
oper-ator R, by R(u, v) =P
q∆qu(∆q−1v + ∆qv + ∆q+1v) and T 0
vu = Tvu + R(u, v).
Inequality (3.23) is a consequence of the following lemma:
Lemma 1 Let p1 ∈ [1, +∞], p ∈ [1, +∞], α ∈ (1 − Np, 1[, k ∈ {1, · · · , N } and Rq =
∆q(a∂kw) − ∂k(a∆qw). There exists c = c(α, N, σ) such that:
2qσk eRqkLp ≤ Ccqkak B N p1+α p1,r kwkBσ+1−α p,r (8.56) whenever −pN 1 < σ ≤ N p1 + α and where cq∈ L r.
In the limit case σ = −pN
1, we have for some constant C = C(α, N ):
2−q N p1k eRqkLp ≤ Ckak B α+ N p1 p,1 kwk B − N p1+1−α p,∞ . (8.57) Proof
The proof is based on Bony’s decomposition which enables us to split Rq into:
Rq= ∂k[∆q, Ta]w | {z } R1 q − ∆qT∂kaw | {z } R2 q + ∆qT∂kwa | {z } R3 q + ∆qR(∂kw, a) | {z } R4 q − ∂kT∆0qwa | {z } R5 q .
By using the fact that:
R1q = q+4 X q0=q−4 ∂k[∆q, Sq0−1a]∆ q0w,
Using the definition of the operator ∆q leads to:
[∆q, Sq0 −1a]∆q0w(x) = − Z h(y) Sq0 −1a(x) − Sq0−1a(x − 2 −qy)∆ q0w(x − 2 −qy)dy. and: [∆q, Sq0 −1a]∆q0w(x) ≤ k∇Sq0−1akL∞2−q Z
2qN|h(2qu)||2qu||∆q0w|(x − u)du,
≤ 2qNk∇Sq0 −1akL∞| h(2q·)| · | ∗ ∆ q0w|(x). So we get: k[∆q, Sq0 −1a]∆q0wkLp≤ k∇Sq0−1akL∞k∆q0wkLp
we readily get under the hypothesis that α < 1, 2qσkR1qkLp . q+4 X q0=q−4 2qσk∇Sq0 −1akL∞k∆ q0wkLp. (8.58) We have then: 2qσkR1qkLp . cqk∇ak B∞,∞α−1kwkBp,rσ+1−α. (8.59)
In the case α = 1, we get:
2qσkR1qkLp . cqk∇akB0
∞,1kwkBp,rσ+1−α. (8.60)
For bounding R2q, standard continuity results for the paraproduct insure that if α < 1, R2q satisfies that: 2qσkR2qkLp ≤ cqk∇ak B∞,∞α−1kwkBp,rσ+1−α. and if α = 1 2qσkR2qkLp≤ cqk∇ak Bα−1∞,1kwkBp,rσ+1−α.
Standard continuity results for the paraproduct insure that R3q satisfies: 2qσkR3qkLp . cqk∇wk Bσ−α− N∞,∞ p kak B N p +α p,r . (8.61) provided σ − α − Np < 0. If σ − α − Np = 0 then: 2qσkRq3kLp. cqk∇wkB0 ∞,1kak B N p +α p,r . (8.62)
Next, standard continuity result for the remainder insure that under the hypothesis σ > −Np, we have: 2qσkR4qkLp . cqk∇wk Bσ−αp,r kak B N p +α p,∞ . (8.63)
For bounding R5q we use the decomposition: R5q= X
q0≥q−3
∂k(Sq0+2∆qw∆q0a),
which leads (after a suitable use of Bernstein and H¨older inequalities) to: 2qσkR5qkLp . X q0≥q−3 2qσ2q 0 k∆q0akL∞kS q0+2∆qwkLp . X q0≥q−2 2(q−q 0 )(α+Np−1) 2q(σ+1−α)k∆qwkLp2q 0 (Np+α)k∆ q0akLp.
Hence, since α + Np − 1 > 0, we have:
2qσkR5qkLp . cqk∇wk Bσ+1−αp,r kak B N p +α p,∞ .
Combining this latter inequality with (8.59), (8.61) and (8.63), and using the embedding B N p p,r,→ B r−Np
∞,∞ for r = Np + α − 1, σ − α completes the proof of (8.56).
The proof of (8.57) is almost the same: for bounding R1
q, Rq2, R3q and R5q, it is just
a matter of changing P
q into supq. We proceed similarly for R4q.
References
[1] H. ABIDI. ´Equation de Navier-Stokes avec densit´e et viscosit´e variables dans l’espace critique. Th`ese de l’Universit´e Paris VI.
[2] H. ABIDI and M. PAICU. ´Equation de Navier-Stokes avec densit´e et viscosit´e vari-ables dans l’espace critique. Annales de l’institut Fourier, 57 no. 3 (2007), p. 883-917. [3] S. ANTONTSEV, A. KAZHIKOV and V. MONAKOV. Boundary value problems in mechanics of nonhomogeneous fluids. Translated from the Russian. Studies in mathe-matics and its applications, 22. North-Holland publishing co. Amsterdam, 1990. [4] H. BAHOURI and J.-Y. CHEMIN. ´Equations d’ondes quasilin´eaires et estimation de
Strichartz, Amer. J. Mathematics 121, (1999), 1337-1377.
[5] H. BAHOURI, J.-Y. CHEMIN and R. DANCHIN. Fourier analysis and nonlinear partial differential equations, to appear in Springer.
[6] J.-M. BONY. Calcul symbolique et propagation des singularit´es pour les ´equations aux d´eriv´ees partielles non lin´eaires, Annales Scientifiques de l’´ecole Normale Sup´erieure 14, (1981), 209-246.
[7] M. CANNONE, Y. MEYER and F. PLANCHON. Solutions auto-similaires des ´equations de Navier-Stokes. S´eminaire sur les ´equations aux d´eriv´ees partielles, 1993-1994, exp. No12 pp. ´Ecole polytech, palaiseau, 1994.
[8] J-Y CHEMIN. Fluides parfaits incompressibles. Ast´erisque, 230, 1995.
[9] J.-Y. CHEMIN. Th´eor`emes d’unicit´e pour le syst`eme de Navier-Stokes tridimension-nel, J. d’Analyse Math., 77, (1999), 25-50.
[10] J.-Y. CHEMIN and N. LERNER. Flot de champs de vecteurs non lipschitziens et ´equations de Navier-Stokes, J. Differential Equations., 121, (1995), 247-286.
[11] R. DANCHIN. Fourier analysis method for PDE’s, Preprint Novembre 2005. [12] R. DANCHIN. Local and global well-posedness results for flows of inhomogeneous
viscous fluids. Advances in differential equations, 9 (2004), 353-386.
[13] R. DANCHIN. Density-dependent incompressible viscous fluids in critical spaces. Proc. Roy. Soc. Edinburgh Sect. A, 133(6), 1311-1334, 2003.
[14] R. DANCHIN. The inviscid limit for density-dependent incompressible fluids. Ann. Fac. Sci. Toulouse Math., S´er. 6, 15 no. 4 (2006), p. 637-688.
[15] R. DANCHIN. Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density. Communications in Partial Differential Equations,32:9,1373 1397, 2007.
[16] R. DANCHIN. Local Theory in critical Spaces for Compressible Viscous and Heat-Conductive Gases,[ Communication in Partial Differential Equations. 26 (2001), 7-8, 1183-1233]. Commun. Partial differential equations. 27, (2002), 11-12, 2531-2532. [17] R. DANCHIN. Global Existence in Critical Spaces for Flows of Compressible Viscous
and Heat-Conductive Gases, Arch. Rational Mech. Anal., 160, (2001), 1-39.
[18] R. DANCHIN. On the uniqueness in critical spaces for compressible Navier-Stokes equations. NoDEA Nonlinear Differentiel Equations Appl, 12(1), 111-128, 2005. [19] B. DESJARDINS. Linear transport equations with initial values in Sobolev spaces
and application to the Navier-Stokes equations, Differential and Integral Equations, 10 (1997), 587-598.
[20] B. DESJARDINS. Global existence results for the incompressible density-dependent Navier-Stokes equations in the whole space. Differential and integral equations, 10 (1997), 587-598.
[21] B. DESJARDINS. Regularity results for two-dimensional flows of multiphase viscous fluids. Arch. Rational mech. Anal., 137, (1997), 135-158.
[22] E. FERNANDEZ-CARA and F. GUILLEN. The existence of nonhomogeneous vis-cous and incompressible flow in unbounded domains. Comm. in Partial Differential Equation, 17, (1992), 1253-1265.
[23] H. FUJITA and T. KATO. On the Navier-Stokes initial value problem I, Archive for Rational Mechanics and Analysis, 16, (1964), 269-315.
[24] S. ITOH and A. TANI. Solvability of nonstationnary problems for nonhomogeneous incompressible fluids and the convergence with vanishing viscosity, Tokyo Journal of Mathematics, 22, (1999), 17-42.
[25] V. KAZHIKOV. Resolution of boundary value problems for nonhomogeneous viscous fluids. Dokl. Akad. Nauh, 216, (1974), 1008-1010.
[26] H. KOCH and D. TATARU. Well-posedness for the Navier-Stokes equations, Ad-vances in Math. 157, 920010, 22-35.
[27] O. LADYZHENSKAYA and V. SOLONNIKOV. The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids. J. Soviet Math. 9, (1978), 697-749.
[28] P-. L. LIONS, Mathematical topics in fluid dynamics. Vol 1 incompressible models, Oxford university press 1996.
