Inhomogeneous Navier-Stokes equations in the half-space, with only bounded density

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Inhomogeneous Navier-Stokes equations in the half-space, with only bounded density

Raphaël Danchin, Ping Zhang

To cite this version:

Raphaël Danchin, Ping Zhang. Inhomogeneous Navier-Stokes equations in the half-space, with only

bounded density. Journal of Functional Analysis, Elsevier, 2014, 267, pp.2371-2436. �hal-00813552�

HALF-SPACE, WITH ONLY BOUNDED DENSITY

RAPHA¨ EL DANCHIN AND PING ZHANG

Abstract. In this paper, we establish the global existence of small solutions to the inhomo- geneous Navier-Stokes system in the half-space. The initial density only has to be bounded and close enough to a positive constant, and the initial velocity belongs to some critical Besov space. With a little bit more regularity for the initial velocity, those solutions are proved to be unique. In the last section of the paper, our results are partially extended to the bounded domain case.

Keywords: Inhomogeneous Navier-Stokes equations, Stokes system, critical Besov spaces, half-space, Lagrangian coordinates.

AMS Subject Classification (2000): 35Q30, 76D03 1. Introduction

We are concerned with the global well-posedness issue for the initial boundary value prob- lem pertaining to the following incompressible inhomogeneous Navier-Stokes equations:

(1.1)

 

 

 

 

 



∂ _t ρ + div(ρu) = 0 in R ₊ ×Ω,

∂ _t (ρu) + div(ρu ⊗ u) − µ∆u + ∇Π = 0 in R ₊ ×Ω,

div u = 0 in R ₊ ×Ω,

u = 0 on R ₊ ×∂Ω,

ρ| t=0 = ρ 0 , ρu| t=0 = ρ 0 u 0 in Ω,

where ρ = ρ(t, x) ∈ R ₊ , u = u(t, x) ∈ R ^d and Π = Π(t, x) ∈ R stand for the density, velocity field and pressure of the fluid, respectively, depending on the time variable t ∈ R ₊ and on the space variables x ∈ Ω. The positive real number µ stands for the viscosity coefficient.

We mainly consider the case where Ω is the half-space R ^d ₊ , except in the last section of the paper where it stands for a smooth bounded domain of R ^d (d ≥ 2).

The above system describes a fluid that is incompressible but has nonconstant density.

Basic examples are mixture of incompressible and non reactant flows, flows with complex structure (e.g. blood flow or model of rivers), fluids containing a melted substance, etc.

A number of recent works have been dedicated to the mathematical study of the above system. Global weak solutions with finite energy have been constructed by J. Simon in [19]

(see also the book by P.-L. Lions [17] for the variable viscosity case). In the case of smooth data with no vacuum, the existence of strong unique solutions goes back to the work of O.

Ladyzhenskaya and V. Solonnikov in [15]. More recently, the first author [8] established the well-posedness of the above system in the whole space R ^d in the so-called critical functional framework for small perturbations of some positive constant density. The basic idea is to use functional spaces that have the same scaling invariance as (1.1), namely

(1.2) (ρ, u, Π)(t, x) 7−→ (ρ, λu, λ ² Π)(λ ² t, λx), (ρ ₀ , u ₀ )(x) 7−→ (ρ ₀ , λu ₀ )(λx).

Date: April 11, 2013.

1

More precisely, in [8], global well-posedness was established assuming that kρ ₀ − 1k

B ˙

d

2,∞2

( R

)∩L

( R

) + µ ⁻¹ ku ₀ k

B ˙

d

2,12

( R

) ≪ 1.

Above ˙ B _p,r ^σ ( R ^d ) stands for a homogeneous Besov space on R ^d (see Definition 2.1 below).

This result was extended to more general Besov spaces by H. Abidi in [1], and H. Abidi and M. Paicu in [2], and to the half-space setting in [10]. The smallness assumption on the initial density was removed recently in [3, 4].

Given that in all those works the density has to be at least in the Besov space ˙ B

d

p,∞

, one cannot capture discontinuities across an hypersurface. In effect, the Besov regularity of the characteristic function of a smooth domain is only ˙ B

1

p,∞

. Therefore, those results do not apply to a mixture of two fluids with different densities.

Very recently, the first author and P. Mucha [11] noticed that it was possible to establish existence and uniqueness of a solution in the case of a small discontinuity, in a critical functional framework. More precisely, the global existence and uniqueness was established for any data (ρ 0 , u 0 ) such that for some p ∈ [1, 2d) and small enough constant c, we have

(1.3) kρ 0 − 1k

M( ˙ B

dp p,1

( R

))

+ µ ⁻¹ ku 0 k

B ˙

dp p,1

( R

)

≤ c.

Above, k · k

M( ˙ B

d p p,1

( R

))

is the multiplier norm associated to the Besov space ˙ B ⁻¹⁺

d p

p,1 ( R ^d ), which turns out to be finite for characteristic functions of C ¹ domains whenever p > d − 1. Therefore, initial densities with a discontinuity across an interface may be considered (although the jump has to be small owing to (1.3)). As observed later on in [12], large discontinuities may be considered if the initial velocity is smoother. In fact, therein, any initial density bounded and bounded away from 0 is admissible. Let us emphasize that in both works ([11] and [12]), using Lagrangian coordinates was the key to the proof of uniqueness.

A natural question is whether it is still possible to get existence and uniqueness in a critical functional framework where ρ ₀ is only bounded and bounded away from zero. As regards existence, a positive answer has been given recently by J. Huang, M. Paicu and the second author in [14], in the whole space setting, and uniqueness was obtained if assuming slightly more regularity for the velocity field. Let us emphasize that once again using Lagrangian coordinates is the key to uniqueness. Therefore, assumptions on the initial velocity have to ensure the velocity u to have gradient in L ¹ _loc ( R ₊ ; L ^∞ ( R ^d )) in order that Eulerian and Lagrangian formulations of the system are equivalent. While this property of the velocity field holds true if u 0 is in ˙ B ⁻¹⁺

d p

p,1 ( R ^d ), it fails if u 0 is only in ˙ B ⁻¹⁺

d

p,r

( R ^d ) for some r > 1. As a matter of fact, the question of uniqueness in a critical Besov framework for the velocity is open unless r = 1 (this latter case requires stronger assumptions on the density, as pointed out in [11]).

In the present work, we aim at extending the results of [14] to the half-space setting.

Because we shall consider only perturbations of the reference density 1, it is natural to set

a = 1/ρ − 1 so that System (1.1) translates into

(1.4)

 

 

 

 

 



∂ _t a + u · ∇a = 0 in R ₊ ×Ω,

∂ t u + u · ∇u + (1 + a)(∇Π − µ∆u) = 0 in R ₊ ×Ω,

div u = 0 in R ₊ ×Ω,

u = 0 on R ₊ ×∂Ω,

(a, u)| _t=0 = (a ₀ , u ₀ ) in Ω.

As in the whole space case considered in [14], the functional framework for solving (1.4) is motivated by classical maximal regularity estimates for the evolutionary Stokes system. In effect, the velocity field may be seen as the solution to the following system:

(1.5) ∂ _t u − µ∆u + ∇Π = −u · ∇u + a(µ∆u − ∇Π), div u = 0.

In the whole space case, we have for any 1 < p, r < ∞ and t > 0, k(∂ t u, µ∇ ² u, ∇Π)k _L

_(L

₎ ≤ C ³

µ ¹⁻

ku 0 k

B ˙

2r p,r

+ ku · ∇uk _L

_(L

₎ + ka(µ∆u − ∇Π)k _L

_(L

₎ ´ where we have used the notation kzk _L

t

(L

)

def = kzk _L

((0,t);L

( R

)) .

Given that ka(t)k _L

= ka ₀ k _L

for all time, it is clear that the last term may be absorbed by the l.h.s. if ka ₀ k _L

is small enough. Now, it is standard (see Corollary A.1 below) that in the simpler case where u just solves the heat equation with initial data u ₀ , having ∆u in L ^r (0, T ; L ^p ) is equivalent to u ₀ ∈ B ˙ ⁻¹⁺

d

p,r

provided −1 + ^d _p = 2 − ² _r · This implies that p and r have to be interrelated through p = _3r−2 ^dr , and thus ^d ₃ < p < d.

Here we aim at extending this simple idea to the half-space setting, or to C ² bounded domains.

In the half-space case, according to the above heuristics and because homogeneous Dirichlet boundary conditions are prescribed for the velocity, the natural solution space for (u, ∇Π) is

X _T ^{p,r def} = n

(u, ∇Π) with u ∈ C([0, T ]; ˙ B ²⁻

2

p,r

( R ^d ₊ )) and ∂ _t u, ∇ ² u, ∇Π ∈ L ^r (0, T ; L ^p ₊ ) o , where L ^p ₊ ^def = L ^p ( R ^d ₊ ) denotes the Lebesgue space over R ^d ₊ , and ˙ B ²⁻

2

p,r

( R ^d ₊ ) stands for the set of divergence free vector fields on R ^d ₊ with Besov regularity ˙ B _p,r ²⁻

( R ^d ₊ ) and null trace at the boundary (the exact meaning will be given in Definition 2.3 below).

We also introduce the following norm for all T > 0:

(1.6) k(u, ∇Π)k _X

T

def = µ ¹⁻

kuk

L

(0,T; ˙ B

( R

+

)) + k(∂ _t u, µ∇ ² u, ∇Π)k _L

(0,T ;L

) , and agree that X ^p,r and k · k _X

correspond to the above definition with T = +∞.

Before stating our main results, let us clarify what we mean by a weak solution to (1.4):

Definition 1.1. A global weak solution of (1.4) is any couple (a, u) satisfying:

• for any test function φ ∈ C _c ^∞ ([0, ∞) × Ω), there holds

(1.7)

Z ∞ 0

Z

Ω

a(∂ t φ + u · ∇φ) dx dt + Z

Ω

φ(0, ·)a 0 dx = 0, Z ∞

0

Z

Ω

φ div u dx dt = 0,

• for any vector valued function Φ = (Φ ¹ , · · · , Φ ^d ) ∈ C _c ^∞ ([0, ∞) × Ω) , one has (1.8)

Z ∞ 0

Z

Ω

n u · ∂ _t Φ + u ⊗ u : ∇Φ + (1 + a) ¡

µ∆u − ∇Π ¢ Φ o

dx dt + Z

Ω

u ₀ · Φ(0, ·) dx = 0.

Our main statement reads:

Theorem 1.1. Let a ₀ ∈ L ^∞ ₊ and u ₀ ∈ B ˙ ⁻¹⁺

d

p,r

( R ^d ₊ ) ∩ B ˙ ⁻¹⁺

d p

e

p,r ( R ^d ₊ ) with p ^def = _3r−2 ^dr ≤ p e ≤ _r−1 ^dr and r ∈ (1, ∞). There exist two positive constants c 0 = c 0 (r, d) and c 1 = c 1 (r, d) so that if

(1.9) ku ₀ k

B ˙

d p,r p

( R

+

)

≤ c ₀ µ and ka ₀ k _L

≤ c ₁

then (1.4) has a global solution (a, u, ∇Π) in the meaning of Definition 1.1 with Ω = R ^d

+ , satisfying ka(t)k _L

= ka ₀ k _L

for all t > 0 and u ∈ X ^p,r ∩ X ^{p,r e} . Moreover, there exist C ₁ = C ₁ (r, d) and C ₂ = C ₂ (r, d) so that

(1.10) k(u, ∇Π)k _X

+ µ ¹⁻

k∇uk

L

( R

;L

dr 2r−1

+

)

≤ C ₁ µ ¹⁻

ku ₀ k

B ˙

d p,r p

( R

+

)

and, if _α ^{1 def} = ¹ _{e p} − ^r−1 _dr ,

(1.11) k(u, ∇Π)k _X

+ µ ¹⁻

k∇uk _L

₍ R

;L

) ≤ C ₂ µ ¹⁻

ku ₀ k

B ˙

d p e

p,r

( R

)

. If in addition p > d e then ∇u ∈ L ^q ( R ₊ ; L ^∞ ₊ ) with q = ^{2 p e}

d+(

−1) e p and (1.12) µ

k∇uk _L

₍ R

;L

) ≤ C 2 ku 0 k

B ˙

dp e p,r

( R

+

)

, and uniqueness holds true.

Remark 1.1. In contrast with the whole space case in [14, 18], it is not clear that one may improve the isotropic smallness condition (1.9) to an anisotropic one allowing for arbitrarily large vertical velocity. The reason why is that even for solutions to the linear Stokes system in the half-space, the horizontal component of the velocity u ^h depends on both u ^h ₀ and u ^d ₀ (see the formula in Theorem 2.1 below).

Remark 1.2. We shall extend this statement to a more general critical (or almost critical) Besov setting, see Theorems 5.1 and 5.2 below. We could also deal with the local well- posedness of (1.4) with general velocity and small inhomogeneity. For a clear presentation, we skip the details here.

Let us briefly describe the plan of the rest of the paper. The next section is devoted to the linearized velocity equation of (1.4) in the half-space, that is the evolutionary Stokes system.

We first derive an explicit solution formula in the spirit of that of S. Ukai in [20], and then

deduce maximal regularity type estimates similar to those of the whole space. We consider

the general situation with prescribed (possibly nonzero) value for div u as it will be needed

when reformulating (1.4) in Lagrangian coordinates. The next two sections are devoted to

the proof of the existence part of Theorem 1.1, first under a stronger assumption on the

density, and next in the rough case corresponding to the hypotheses of the theorem. The

case of more general Besov spaces will be examined in Section 5. The proof of uniqueness

is postponed in Section 6. In the final section, we partially generalize Theorem 1.1 to the

bounded domain setting. Some technical lemmas related to maximal regularity and L ^p (L ^q )

estimates for the heat equation in the whole space (or Stokes system in bounded domains)

are presented in Appendix.

2. The evolutionary Stokes system in the half-space This section is devoted to the study of following system in the half-space:

(2.1)

 

 

 



∂ t u − µ∆u + ∇Π = f in R ₊ × R ^d ₊ ,

div u = g in R ₊ × R ^d

+ ,

u = 0 on R ₊ ×∂ R ^d ₊ ,

u| t=0 = u 0 on R ^d ₊ .

We shall first derive an explicit formula for the solution to this system, and next prove the key a priori estimates that are needed for getting the main results of our paper.

2.1. A solution formula. This part extends a prior work by S. Ukai [20] (see also [7]) to the case where there is a source term f in the velocity equation, and where the divergence constraint is nonhomogeneous. Let us recall that in [20], it was assumed that f = 0 and g = 0 (but u need not be zero at the boundary), and that in [7] nonzero f was considered (but still u is divergence free and f has trace zero). Furthermore, the gradient of the pressure was not computed therein, a computation that turns out to be essential for us as ∇Π appears in the right-hand side of the velocity equation (1.5).

Before writing out the formula, let us introduce a few notations. We denote ∆ = P d ℓ=1 ∂ _x ²

and ∆ _h = P d−1

ℓ=1 ∂ ² _x

, and define |D| ^±1 and |D _h | ^±1 to be the Fourier multipliers with symbol

|ξ _h | ^±1 = µ d−1 X

i=1

ξ _i ²

¶ ±

and |ξ| ^±1 = µ X d

i=1

ξ ² _i

¶ ±

, respectively.

The notations R _j and S _j stand for the Riesz transforms over R ^d and R ^d−1

h , namely R j def

= ∂ j |D| ⁻¹ for j = 1, · · · , d and S j def

= ∂ j |D _h | ⁻¹ for j = 1, · · · , d − 1.

We further set R _h ^def = (R ₁ , · · · , R _d−1 ) and S ^def = (S ₁ , · · · , S _d−1 ).

As in [20], for u = (u ^h , u ^d ) with u ^h = (u ¹ , · · · , u ^d−1 ), we define the operators V _d and V _h by

(2.2) V _d u ^def = −S · u ^h + u ^d and V _h u ^def = u ^h + Su ^d .

We shall see later on that both V _h u and V _d u satisfy a heat equation, this is the main motivation for considering those two quantities.

We also denote x = (x _h , x _d ) with x _h = (x ₁ , · · · , x _d−1 ). Let r be the restriction operator from R ^d to R ^d ₊ , that is rf ^def = f | R

+

, and e 0 (f ), e a (f ), e s (f ) be the extension operators given by

(2.3)

e ₀ (f) =

½ f for x _d ≥ 0,

0 for x _d < 0, e _a (f) =

½ f (x) for x _d ≥ 0,

−f (x _h , −x _d ) for x _d < 0, and e _s (f ) =

½ f (x) for x _d ≥ 0, f (x _h , −x _d ) for x _d < 0.

When solving (2.1), we shall repeatedly consider the following two equations:

(2.4) (∂ _d + |D _h |)w = |D _h |f in R ^d ₊ , and γw = 0 on ∂ R ^d ₊ . (2.5) (∂ _d + |D _h |)v = f in R ^d

+ , and γv = 0 on ∂ R ^d

+ .

Here and in what follows, γ stands for the trace operator on ∂ R ^d ₊ .

Finally we denote by H the harmonic extension operator from the hyperplane ∂ R ^d ₊ to the half-space R ^d ₊ . More precisely, for any given b : ∂ R ^d ₊ → R , we set Hb to be the unique solution going to zero at ∞ of

(2.6) ∆Hb = 0 in R ^d

+ , and γHb = b on ∂ R ^d

+ .

Introducing the Fourier transform F _h with respect to the horizontal component x _h , the function F _h (Hb) is explicitly given by the formula

(2.7) F _h (Hb)(ξ _h , x _d ) = e ^−|ξ

^|x

F _h b(ξ _h ), ξ _h ∈ R ^d−1 , x _d > 0, hence in particular

(2.8) (∂ _d + |D _h |)Hb = 0 in R ^d ₊ .

Lemma 2.1. For any smooth enough data f decaying at ∞, Equation (2.4) has a unique solution going to 0 at ∞, and Equation (2.5) has a unique solution with gradient going to 0 at infinity. Furthermore, denoting by U and P the solution operators for (2.4) and (2.5), one has

(2.9) U f = r R _h · S ¡

R _h · S e _a (f) + R _d e _s (f ) ¢

and P f (x _h , x _d ) = Z x

0

(I − U )f (x _h , y _d ) dy _d and the following identities are satisfied:

(1) ∇ _h U = U ∇ _h ; (2) ∂ _d U = (I − U )|D _h |;

(3) ∇ _h P = SU ; (4) ∂ _d P = I − U ; (5) ∆P = ∂ _d − |D _h |;

(6) [P, ∂ _d ] = −Hγ.

Proof. If w is a solution to (2.4) then it also satisfies the following Poisson equation:

½ −∆w = (|D _h | − ∂ _d )|D _h |f w| _x

₌₀ = 0,

the unique solution (decaying to 0 at infinity) of which is given by w = r(−∆) ⁻¹ e _a ¡

(|D _h | − ∂ _d )|D _h |f ¢ .

As e _a (|D _h | ² f) = |D _h | ² e _a (f) and e _a (∂ _d |D _h |f) = ∂ _d (e _s (|D _h |f )) = ∂ _d |D _h |e _s (f ), we get the formula for U f.

It is obvious that U commutes with ∇ _h . As regards commutation with ∂ _d , we notice that, by definition of U f,

∂ _d U f + |D _h |U f = |D _h |f, which yields (2).

It is clear that |D _h |P f satisfies (2.4), hence |D _h |P f = U f and ∇ _h P f = SU f. Similarly, the equation for P f yields

∂ _d P f = f − |D _h |P f = f − U f,

hence integrating with respect to the vertical variable gives the expression for P.

The next item is a direct consequence of the definition of P (just apply ∂ _d − |D _h | to the equation). Finally, we have by definition of P ∂ _d f,

(∂ _d + |D _h |)P ∂ _d f = ∂ _d f = (∂ _d + |D _h |)f − |D _h |f.

Hence, using (2.8),

(∂ _d + |D _h |)(P ∂ _d f − f + Hγf ) = −|D _h |f and (P ∂ _d f − f + Hγf )| _x

₌₀ = 0.

Therefore, using the definition of U, one may write

P ∂ _d f − f + Hγf = −U f.

Because ∂ _d P = I − U, it is easy to complete the proof of the last item. ¤ Remark 2.1. For functions vanishing at x _d = 0, operator U coincides with the expression

rR _h · S(R _h · S + R _d )e 0

that has been introduced in [20] and plays the role of the left-inverse of (Id +|D _h | ⁻¹ ∂ _d ) for functions of R ^d

+ vanishing at ∂ R ^d

+ . Our definition of operator U is slightly more general as it allows us to consider functions that do not vanish at x _d = 0.

The main result of this subsection reads:

Theorem 2.1. Given smooth and decaying data u 0 , f and g with g = div Q, the unique solution (u, ∇Π) of (2.1) is given by

(2.10)

u ^h =re ^µt∆ e _a (V _h u ₀ ) − µSP g − SU e ^µt∆ e _a (V _d u ₀ )

− SU Z t

0

e ^{µ(t−τ)∆} e _a ¡ e N f + Gk ¢ dτ + r

Z t 0

e ^{µ(t−τ)∆} e _a ( M f f + SGk ¢ dτ, u ^d =µP g + U e ^µt∆ e _a (V _d u ₀ ) + U

Z t 0

e ^{µ(t−τ)∆} e _a ¡ e N f + Gk ¢ dτ, and

(2.11)

∇ _h Π = r ¡

R _h · S(SR _d + R _h ) ¢¡

S · e _a (f ^h ) + e _s (f ^d ) ¢

+ S(U − I)Gk + µ(∇ _h − S∂ _d )g + S∂ _t ¡

S · U Q ^h + (I − U )Q ^d − HγQ ^d ¢

+ SU N f e + ¡

r(|D _h | − ∂ _d )∇ _h + SU ∆ ¢ µ

e ^µt∆ e _a (V _d u ₀ ) + Z t

0

e ^{µ(t−τ)∆} e _a ¡ e N f + Gk ¢ dτ

¶ ,

∂ _d Π = f ^d + µ(∂ _d − |D _h |)g − ∂ _t ¡

S · U Q ^h + (I − U)Q ^d − HγQ ^d ¢

− U ¡ e N f + Gk ¢

− ¡

r(|D _h | − ∂ _d )|D _h | + U ∆ ¢ µ

e ^µt∆ e a (V _d u 0 ) + Z t

0

e ^{µ(t−τ)∆} e a ¡ e N f + Gk ¢ dτ

¶ , where Gk, M f f and N f e are given by

(2.12)

Gk = −r ¡

R _d − R _h · S ¢¡

R _h · e _a (k ^h ) + R _d e _s (k ^d ) ¢

with k = ∂ _t Q − µ∇g, N f e = r ©£

1 + R ² _d − R _d R _h · S ¤

e _s (f ^d ) + R _d ² S · e _a (f ^h ) + R _d R _h · e _a (f ^h ) ª , M f f = rS £

R _d − R _h · S ¤¡

R _h · e a (f ^h ) + R _d e s (f ^d ) ¢

+ V _h f.

Proof. We shall essentially follow the arguments in [7, 20]. Note that setting

(2.13) u new (t, x) = µu old (µ ⁻¹ t, x), Π new (t, x) = Π old (µ ⁻¹ t, x), f new (t, x) = f old (µ ⁻¹ t, x) reduces the study to the case µ = 1, an assumption that we are going to make in the rest of the proof.

The basic idea is to reduce the study to that of the heat equation for the auxiliary functions V _h u and V _d u. As a first step, let us compute Π in terms of div f, g and of its trace at ∂ R ^d ₊ . Taking space divergence to (2.1) yields

( −∆Π = ∂ t g − ∆g − div f in R ^d ₊ ,

γΠ = b on ∂ R ^d ₊ Π → 0 as |x| → ∞, the solution of which is given by

Π = r(−∆) ⁻¹ e _a (div(k − f )) + Hb with k = ∂ _t Q − ∇g, which along with (2.8) implies that

(2.14) (∂ _d + |D _h |)Π = −r(∂ _d + |D _h |)(−∆) ⁻¹ e _a (div(f − k))

= M e _a (div(f − k)) with

(2.15) M h ^def = −r(∂ _d + |D _h |)(−∆) ⁻¹ h.

Let z ^def = (∂ _d + |D _h |)u ^d and N f ^def = (∂ _d + |D _h |)f ^d − ∂ _d M e _a (div f). We infer from (2.1) and (2.14) that

 



∂ _t z − ∆z = N f + ∂ _d M e _a (div k) in R ₊ × R ^d ₊ , (z − g)| _x

₌₀ = 0,

z| _t=0 = (∂ _d + |D _h |)u ^d ₀ . Note from the definition of M in (2.15) that

(2.16) ∂ _d M e _a (h) = e _a (h) − r|D _h |(∂ _d +|D _h |)(−∆) ⁻¹ £ e _a (h) ¤

. Therefore, because

(2.17) e _a (div k) = div _h e _a (k ^h ) + ∂ _d e _s (k ^d ) and

(2.18) M e _a (div k) = Gk,

we get

N f = |D _h |f ^d − r div _h e _a (f ^h ) + r|D _h |(∂ _d + |D _h |)(−∆) ⁻¹ £

div _h e _a (f ^h ) + ∂ _d e _s (f ^d ) ¤

= |D _h | N f. e

Now, using (2.16) with h = ∂ _t g − ∆g = div k and (2.17), we thus obtain

 



∂ _t (z − g) − ∆(z − g) = |D _h |¡ e N f + Gk ¢

in R ₊ × R ^d ₊ ,

(z − g)| x

=0 = 0 in R ₊ ×∂ R ^d ₊ ,

(z − g)| _t=0 = |D _h |V _d u ₀ in R ^d

+ .

Taking advantage of the solution formula for the heat equation in R ^d ₊ with homogeneous Dirichlet boundary conditions, we deduce that

(z − g)(t) = r|D _h | µ

e ^t∆ e _a (V _d u ₀ ) + Z t

0

e ^(t−τ)∆ e _a ¡ e N f + Gk ¢ dτ

¶ .

As z − g = (|D _h | + ∂ _d )(u ^d − P g) and u ^d − P g vanishes at x _d = 0, keeping in mind the definition of U, we get the second equality of (2.10).

To derive the solution formula for u ^h , we look at the equation satisfied by V _h u. Thanks to (2.1), (2.12) and (2.14), we get, observing

V _h f − S(|D _h | + ∂ _d )Π = V _h f − SM e _a (div(f − k)) = M f f + SGk,

that 





∂ _t V _h u − ∆V _h u = M f f + SGk in R ₊ × R ^d ₊ , V _h u| _x

₌₀ = 0 in R ₊ × R ^d ₊ , V _h u| _t=0 = V _h u ₀ in R ^d

+ , so that

(2.19) V _h u = re ^t∆ e _a (V _h u ₀ ) + r Z t

0

e ^(t−τ)∆ e _a ¡ f M f + SGk ¢ dτ.

Because u ^h = V _h u − Su ^d , combining the above identity and the second formula of (2.10), we obtain the solution formula for u ^h .

Let us finally derive (2.11). By virtue of (2.1) and (2.10), we may write

∂ _d Π = f ^d − (∂ _t − ∆)u ^d

= f ^d − (∂ _t − ∆)P g − (∂ _t − ∆)U µ

e ^t∆ e _a (V _d u ₀ ) + Z t

0

e ^(t−τ)∆ e _a ¡ e N f + Gk ¢ dτ

¶ . However, because

∆U h = r(∂ _d − |D _h |)|D _h |h, one may write

(∂ _t − ∆)U h = U(∂ _t − ∆)h + r(|D _h | − ∂ _d )|D _h |h + U ∆h.

Note also that, by virtue of Lemma 2.1,

∂ t P g = ∂ t (P div _h Q ^h + P ∂ _d Q ^d ) = ∂ t (S · U Q ^h + (I − U )Q ^d − HγQ ^d ) and that

∆P g = (∂ _d − |D _h |)g, which ensures that

(2.20) ∂ _d Π = f ^d − ∂ _t (S · U Q ^h + (I − U )Q ^d − HγQ ^d ) + (∂ _d − |D _h |)g − U ¡ e N f + Gk ¢

− ¡

r(|D _h | − ∂ _d )|D _h | + U ∆ ¢ µ

e ^t∆ e _a (V _d u ₀ ) + Z t

0

e ^(t−τ)∆ e _a ( N f e + Gk) ¢ dτ

¶

· On the other hand, by virtue of (2.14), one has

|D _h |Π = −∂ _d Π + M e _a (div(f − k)),

and

M e _a (div f ) = −(∂ _d + |D _h |)(−∆) ⁻¹ ¡

div _h e _a (f ^h ) + ∂ _d e _s (f ^d ) ¢

= e _s (f ^d ) − (∂ _d + |D _h |)(−∆) ⁻¹ ¡

div _h e _a (f ^h ) + |D _h |e _s (f ^d ) ¢ ,

which together with (2.20) gives rise to (2.11). This completes the proof of Theorem 2.1. ¤ 2.2. A priori estimates. Let us first briefly recall the definition of homogeneous Besov spaces in R ^d . Let χ : R ^d → [0, 1] be a smooth nonincreasing radial function supported in B(0, 1) and such that χ ≡ 1 on B(0, 1/2), and let

ϕ(ξ) ^def = χ(ξ/2) − χ(ξ).

The homogeneous Littlewood-Paley decomposition of any tempered distribution u on R ^d is defined by

∆ ˙ _k u ^def = ϕ(2 ^−k D)u = F ⁻¹ ¡

ϕ(2 ^−k ·)F u ¢

, k ∈ Z where F stands for the Fourier transform on R ^d .

Definition 2.1. For any s ∈ R and (p, r) ∈ [1, +∞] ² , the homogeneous Besov space B ˙ _p,r ^s ( R ^d ) stands for the set of tempered distributions f so that

kf k _{B ˙}

( R

)

def = ° °2 ^sk k ∆ ˙ _k fk _L

( R

)

° °

ℓ

( Z ) < ∞ and for all smooth compactly supported function θ over R ^d , we have

(2.21) lim

λ→+∞ θ(λD)f = 0 in L ^∞ ( R ^d ).

Remark 2.2. Condition (2.21) means that functions in homogeneous Besov spaces are re- quired to have some decay at infinity (see [5] for more details). In particular, we have

(2.22) f = X

k∈ Z

∆ ˙ _k f in S ^′ ( R ^d )

whenever f satisfies (2.21). In this paper, we will only consider exponents s < d/p so that for f with finite B ˙ _p,r ^s ( R ^d ) semi-norm, (2.21) and (2.22) are equivalent.

The homogeneous Besov spaces on the half-space are defined by restriction:

Definition 2.2. For any s ∈ R , and (p, r) ∈ [1, +∞] ² we denote by B ˙ _p,r ^s ( R ^d ₊ ) the set of distributions u on R ^d ₊ admitting some extension u e ∈ B ˙ _p,r ^s ( R ^d ) on R ^d . Then we set

kuk _{B ˙}

( R

+

)

def = inf k uk e _{B ˙}

( R

)

where the infimum is taken on all the extensions of u in B ˙ _p,r ^s ( R ^d ).

We also need to introduce some spaces of divergence free vector fields vanishing at the boundary ∂ R ^d ₊ . We proceed as follows:

Definition 2.3. For 1 < p < ∞ and 0 < s < 2, we denote by B ˙ _p,r ^s ( R ^d

+ ) the completion of the set of divergence free vector fields with coefficients in W ^2,p ( R ^d ₊ ) ∩ W ₀ ^1,p ( R ^d ₊ ) (where W ₀ ^1,p ( R ^d

+ ) stands for the subspace of W ^1,p ( R ^d

+ ) functions with null trace at ∂ R ^d

+ ) for the norm k · k B ˙

( R

+

) .

It is classical (see e.g. [10]) that spaces ( ˙ B _p,r ^s ( R ^d ₊ )) ^d (with the divergence free condition)

and ˙ B _p,r ^s ( R ^d ₊ ) coincide whenever 1 < p, r < ∞ and 0 < s < 1/p.

The following result extends Lemma 3.2 of [20] to the context of Besov spaces.

Lemma 2.2. Operators R _h , R _d and S map L ^p ( R ^d ) in itself for any 1 < p < ∞, and, with no restriction on s, p, r, we have

kRzk B ˙

( R

) ≤ Ckzk B ˙

( R

) for R ∈ {R _h , R _d , S}.

Operators V _h , V _d , U, G, M f and N e map L ^p ₊ in itself if 1 < p < ∞, and B ˙ ^s _p,r ( R ^d ₊ ) in itself if 1 < p, r < ∞ and 0 < s < 2.

Proof. The result in the Lebesgue spaces just follows from the fact that all those operators are combinations of Riesz transforms so that Calderon-Zygmund theorem applies. The result in homogeneous Besov spaces stems from the fact that the Riesz operators are Fourier multipliers of degree 0, hence map any homogeneous Besov space in itself. ¤

We are now ready to establish a first family of a priori estimates for System (2.1).

Proposition 2.1. Let 1 < p, r < ∞ and the data u ₀ , f, g fulfill the following hypotheses:

• u ₀ ∈ B ˙ _p,r ²⁻

( R ^d

+ );

• f ∈ L ^r ( R ₊ ; L ^p ₊ );

• ∇g ∈ L ^r ( R ₊ ; L ^p ₊ ), g = div Q with ∂ _t Q ∈ L ^r ( R ₊ ; L ^p ₊ ) and the following compatibility conditions are fulfilled ¹

(2.23) γu ^d ₀ = 0, g| t=0 = 0, and ∂ t (γQ ^d ) = 0.

Then System (2.1) has a unique solution (u, ∇Π) with u ∈ C _b ( R ₊ ; ˙ B ²⁻

2

p,r

( R ^d ₊ )) and ∂ _t u, ∇ ² u, ∇Π ∈ L ^r ( R ₊ ; L ^p ₊ ), with also ∇u ∈ L ^q ( R ₊ ; L ^m ₊ ) whenever q ∈ [r, ∞) and m ∈ [p, ∞] satisfy

0 ≤ 1 − 2 r + 2

q ≤ d

p and d

m = d

p − 1 + 2 r − 2

q · Furthermore, the following inequality is fulfilled for all t > 0:

(2.24) µ ¹⁻

kuk

L

( ˙ B

2r p,r

( R

+

)) + µ ¹⁻

⁺

k∇uk _L

t

(L

) + k(∂ t u, µ∇ ² u, ∇Π)k _L

t

(L

)

≤ C ³

µ ¹⁻

ku ₀ k

B ˙

( R

+

) + k(f, µ∇g, ∂ _t Q)k _L

t

(L

)

´ ·

Finally, if g ≡ 0 then we have u ∈ C( R ₊ ; ˙ B _p,r ²⁻

( R ^d ₊ )).

Remark 2.3. As regards the bounds for ∇u, we shall often use the following two cases:

• p = _3r−2 ^dr , q = 2r and m = _2r−1 ^dr ,

• p > d, q = ^2p

d+(

−1)p and m = +∞.

Proof. We concentrate on the proof of the estimates in L ^r (0, T ; L ^p ₊ ) for ∇ ² u and ∇Π. Indeed, once the pressure has been determined, u may be seen as a solution of the heat equation with source term in L ^r ( R ₊ ; L ^p ₊ ), the solution of which is given by

(2.25) u(t) = r

µ

e ^µt∆ e _a (u ₀ ) + Z t

0

e ^{µ(t−τ)∆} e _a (f − ∇Π) dτ

¶

·

1 The condition on u

is ensured by the fact that u

∈ B ˙

( R

). As for Q, it stems from the fact that

div Q is quite smooth.

Therefore combining Corollary A.1, Lemma A.1 and Lemma 2.2 allows to bound u(t) in B ˙ _p,r ²⁻

( R ^d

+ ) in terms of the data and of the norm of ∇Π in L ^r (0, t; L ^p ₊ ). In addition, because any function in L ^r ( R ₊ ; L ^p ₊ ) may be approximated by smooth functions compactly supported in R ₊ × R ^d ₊ , and because u ₀ may be approximated by functions in W ^2,p ( R ^d ₊ ) ∩ W ₀ ^1,p ( R ^d ₊ ), the above formula guarantees that u is continuous in time, with values in ˙ B ²⁻

2

p,r

( R ^d ₊ ) (or in B ˙ ²⁻

2

p,r

( R ^d ₊ ) if div u ≡ 0).

In what follows, we assume that µ = 1, which is not restrictive owing to the change of variables (2.13). Of course, when proving estimates, one may consider separately the three cases where only one element of the triplet (u ₀ , f, g) is nonzero, a consequence of the fact that (2.1) is linear.

Step 1. Case u ₀ ≡ 0 and f ≡ 0. Then the formula for u ^d given by Theorem 2.1 reduces to u ^d = P g + U

Z t 0

e ^(t−τ)∆ e _a (Gk) dτ, and using the algebraic relations provided by Lemma 2.1 thus yields

∇ ² _h u ^d = SU ∇ h g + U Z t

0

e ^(t−τ)∆ ∇ ² _h e a (Gk) dτ,

∇ _h ∂ _d u ^d = (I − U )∇ _h g + (I − U ) Z t

0

e ^(t−τ)∆ ∇ _h |D _h |e a (Gk) dτ,

∂ _d ² u ^d = (∂ _d + (U −I )|D _h |)g + Z t

0

e ^(t−τ)∆ ∂ _d |D _h |e a (Gk) dτ + (I −U ) Z t

0

e ^(t−τ)∆ ∆ _h e a (Gk) dτ.

The important fact is that all the terms corresponding initially to P g may be written A∇g where A stands for some 0-th order operator for which Lemma 2.2 applies. A similar ob- servation holds for the terms with the time integral so that applying Lemma A.1 eventually yields

(2.26) k∇ ² u ^d k _L

T

(L

) ≤ C ¡

k∇gk _L

T

(L

) + kGkk _L

T

(L

)

¢ .

At this point, we use Lemma 2.2 to bound the right-hand side by k(∇g, ∂ _t Q)k _L

T

(L

) , and we thus get

(2.27) k∇ ² u ^d k _L

T

(L

) ≤ Ck(∇g, ∂ t Q)k _L

T

(L

) . It is clear that ∇ ² u ^h also satisfies (2.27): indeed (2.10) gives

(2.28) ∇ ² u ^h = r

Z t 0

e ^(t−τ)∆ ∇ ² Se a (Gk) dτ − S∇ ² u ^d .

Let us now concentrate on the pressure. Keeping in mind (2.20) and (2.23), we may write

∂ _d Π = (∂ _d − |D _h |)g − S · U ∂ _t Q ^h − (I − U )∂ _t Q ^d − U Gk

− ¡

r(|D _h | − ∂ _d )|D _h | + U ∆ ¢ Z ^t

0

e ^(t−τ)∆ e _a (Gk) dτ.

Therefore, combining Lemmas 2.2 and A.1 gives

(2.29) k∂ _d Πk _L

T

(L

) ≤ Ck(∂ _t Q, ∇g)k _L

T

(L

) . Finally, because

(2.30) ∇ _h Π = −SGk − S∂ _d Π,

it is clear that ∇ _h Π also satisfies (2.29).

Step 2. Case f ≡ 0 and g ≡ 0. With no loss of generality, one may assume that u ₀ ∈ W ₀ ^1,p ( R ^d ₊ ) ∩ W ^2,p ( R ^d ₊ ). From Theorem 2.1 and Lemma 2.1, we readily get

∇ ² _h u ^d = U e ^t∆ ∇ ² _h e a (V _d (u 0 )),

∇ _h ∂ _d u ^d = (I − U )e ^t∆ ∇ _h |D _h |e a (V _d (u 0 )),

∂ _d ² u ^d = re ^t∆ ∂ _d |D _h |e a (V _d (u 0 )) + r(I − U )e ^t∆ e a (∆ _h V _d (u 0 )).

Therefore, combining Corollary A.1 and Lemma 2.2, k∇ ² u ^d k _L

T

(L

) ≤ C ³

ke a (∇ ² _h V _d (u 0 ))k

B ˙

2r p,r

( R

)

+ke _a (∇ _h |D _h |V _d (u ₀ ))k

B ˙

2r

p,r

( R

) + ke _s (∂ _d |D _h |V _d (u ₀ ))k

B ˙

2r p,r

( R

)

´

≤ Cku 0 k

B ˙

2r p,r

( R

+

) .

Note that in order to bound the last term, we used the fact that because V _d u ₀ is null at the boundary, we have

|D _h |∂ _d e _a (V _d (u ₀ )) = e _s (∂ _d |D _h |V _d (u ₀ )) ∈ B ˙ ⁻

2

p,r

( R ^d ).

Owing to (2.28), ∇ ² u ^h satisfies the same inequality. Finally,

∂ _d Π = − ¡

r(|D _h | − ∂ _d )|D _h | + U ∆ ¢

e ^t∆ e _a (V _d u ₀ ) hence, according to Lemma 2.2 and Corollary A.1,

k∂ _d Πk _L

T

(L

) ≤ Ck∇ ² e ^t∆ e _a (V _d u ₀ )k _L

T

(L

)

≤ Cku 0 k

B ˙

2r p,r

( R

+

) . Of course, (2.30) implies that ∇ _h Π has the same bound.

Step 3. Case u ₀ ≡ 0 and g ≡ 0 . As in the previous steps, owing to

∇ ² u ^h = r Z t

0

e ^(t−τ)∆ ∇ ² e _a ( M f f ) dτ − S∇ ² u ^d ,

∇ _h Π = SGf − S∂ _d Π,

it suffices to bound ∇ ² u ^d and ∂ _d Π. The formulae for the second spatial derivatives of u ^d now read

∇ ² _h u ^d = U Z t

0

e ^(t−τ)∆ ∇ ² _h e _a ( N f e ) dτ,

∇ _h ∂ _d u ^d = (I − U) Z t

0

e ^(t−τ)∆ ∇ _h |D _h |e _a ( N f e ) dτ,

∂ _d ² u ^d = r Z t

0

e ^(t−τ)∆ ∂ _d |D _h |e _a ( N f e ) dτ + (I − U ) Z t

0

e ^(t−τ)∆ ∆ _h e _a ( N f e ) dτ.

Therefore applying Lemmas 2.2 and A.1, k∇ ² u ^d k _L

T

(L

) ≤ Cke _a ( N f e )k _L

T

(L

)

≤ Ckf k _L

T

(L

) .

For the pressure, we have

∂ _d Π − f ^d = −U N f e − ¡

r(|D _h | − ∂ _d )|D _h | + U ∆ ¢ Z ^t

0

e ^(t−τ)∆ e a ( N f e ) dτ, therefore, using once again Lemmas 2.2 and A.1, we obtain

k∂ _d Π − f ^d k _L

T

(L

) ≤ C ¡

kU N f e k _L

T

(L

) + ke _a ( N f e )k _L

T

(L

)

¢

≤ Ckfk _L

T

(L

) .

Step 4. Estimates for ∇u . The starting point is the following classical Gagliardo-Nirenberg inequality on R ^d :

(2.31) k∇zk _L

( R

) ≤ Ckzk ^1−θ

B ˙

( R

)

k∇ ² zk ^θ _L

_{( R}

)

with θ ∈ (0, 1], m ≥ p, 0 ≤ 1 − ² _r + ^2θ _r ≤ ^d _p and _m ^d = ^d _p − 1 + ² _r − ^2θ _r ·

If θ ∈ (0, 1) then this inequality may be easily proved by decomposing u into low and high frequencies by means of an homogeneous Littlewood-Paley decomposition (see e.g. [5] Chap.

2 for the proof of similar inequalities). The case θ = 1 corresponds to the classical Sobolev inequality. We omit the proof as it is standard.

We claim that this inequality extends to the half-space setting if considering functions u ∈ W ₀ ^1,p ( R ^d ₊ ) ∩ W ^2,p ( R ^d ₊ ). In effect, we observe that for such functions we have the following identities:

∇ _h (e _a u) = e _a (∇ _h u) and ∂ _d (e _a u) = e _s (∂ _d u).

As ∇ h (e a u) also has null trace at ∂ R ^d ₊ , one can thus write

∇ ² _h (e _a u) = e _a (∇ ² _h u) and ∂ _d ∇ _h (e _a u) = e _s (∂ _d ∇ _h u).

Even though ∂ _d (e _a u) need not be zero at ∂ R ^d ₊ , it is symmetric with respect to the vertical variable, whence

∂ _d ² (e _a u) = e _a (∂ _d ² u).

Applying (2.31) to z = e _a u, and the above relations for the second order derivatives, we thus gather

k∇uk L

≤ k∇(e a u)k _L

( R

) ,

≤ Cke _a uk ^1−θ

B ˙

2r p,r

( R

)

k∇ ² (e _a u)k ^θ

L

( R

) ,

≤ Ckuk ^1−θ

B ˙

( R

)

k∇ ² uk ^θ _L

.

Hence, taking the L ^q norm with respect to time of both sides (with q = r/θ ), we discover that

k∇uk _L

T

(L

) ≤ Ckuk ^1−θ

L

( ˙ B

2r p,r

( R

+

))

k∇ ² uk ^θ _L

(L

) .

Bounding the right-hand side according to the previous steps leads to the desired estimate of k∇uk _L

T

(L

) in (2.24). ¤

In order to solve System (1.4) for more general data, it will be suitable to extend the above

estimates to the case where the index of regularity of u ₀ is not related to r. This motivates

the following statement:

Proposition 2.2. Let 1 < p, r < ∞ and 0 < s < 2. Let (u ₀ , f, g) satisfy the compatibility conditions of Proposition 2.1, and be such that

u ₀ ∈ B ˙ ^s _p,r ( R ^d ₊ ), t ^α (f, ∇g, ∂ _t Q) ∈ L ^r ( R ₊ ; L ^p ₊ ) with α ^def = 1 − s 2 − 1

r .

Then System (2.1) has a unique solution (u, ∇Π) with t ^α (∂ t u, ∇ ² u, ∇Π) ∈ L ^r ( R ₊ ; L ^p ₊ ) and for all T > 0,

(2.32) kt ^α (∂ _t u, ∇ ² u, ∇Π)k _L

T

(L

) ≤ C ³ ku ₀ k _{B ˙}

p,r

( R

) + kt ^α (f, ∇g, ∂ _t Q)k _L

T

(L

)

´ ·

Furthermore, the following properties hold true:

(1) For any couple (p ₂ , r ₂ ) so that s < 1 + d

p − d

p ₂ < 2 + 2 r ₂ − 2

r , p 2 ≥ p, r 2 ≥ r we have t ^β ∇u ∈ L ^r

( R ₊ ; L ^p ₊

) with β = ¹ ₂ − _2p ^d

+ _2p ^d − ₂ ^s − _r ¹

, and

kt ^β ∇uk _L

T

(L

) ≤ C ³ ku ₀ k

B ˙

dp+ d p2 p2,r2

( R

+

)

+ kt ^α (f, ∇g, ∂ _t Q)k _L

T

(L

)

´ ·

(2) For any couple (p ₃ , r ₃ ) so that s < d

p − d p 3

< 2 + 2 r 3

− 2

r , p ₃ ≥ p, r ₃ ≥ r, we have t ^γ u ∈ L ^r

( R ₊ ; L ^p ₊

) with γ = − _2p ^d

3

+ _2p ^d − ^s ₂ − _r ¹

3

, and kt ^γ uk _L

T

(L

) ≤ C ³ ku ₀ k

B ˙

dp+ d p3 p3,r3

( R

+

)

+ kt ^α (f, ∇g, ∂ _t Q)k _L

T

(L

)

´ ·

Proof. Let us first assume that only g is nonzero. Then we start with the formula t ^α ∇ ² _h u ^d = SU t ^α ∇ _h g + U t ^α

Z t 0

e ^(t−τ)∆ ∇ ² _h e _a (Gk) dτ.

From Lemmas 2.2 and A.2, we immediately infer that, if αr ^′ < 1, kt ^α ∇ ² _h u ^d k _L

t

(L

) ≤ C ¡

kt ^α ∇ _h gk _L

t

(L

) + kt ^α e _a (Gk)k _L

t

(L

( R

))

¢ , whence

(2.33) kt ^α ∇ ² _h u ^d k _L

t

(L

) ≤ Ckt ^α (∇g, ∂ _t Q)k _L

t

(L

) . Similarly, as

t ^α ∇ _h ∂ _d u ^d = (I − U )t ^α ∇ _h g + (I − U )t ^α Z t

0

e ^(t−τ)∆ ∇ _h |D _h |e _a (Gk) dτ, and

t ^α ∂ _d ² u ^d = (∂ _d + (U − I )|D _h |)t ^α g + t ^α Z t

0

e ^(t−τ)∆ ∂ _d |D _h |e _a (Gk) dτ +(I − U )t ^α

Z t 0

e ^(t−τ)∆ ∆ _h e _a (Gk) dτ, it is clear that kt ^α ∇ ² u ^d k _L

t

(L

) is bounded by the right-hand side of (2.33). Because t ^α ∇ ² u ^h = r t ^α

Z t 0

e ^(t−τ)∆ ∇ ² Se _a (Gk) dτ − t ^α S∇ ² u ^d ,

the same inequality holds true for t ^α ∇ ² u ^h .

In order to bound the pressure, we use the fact that

t ^α ∂ _d Π = t ^α (∂ _d − |D _h |)g − t ^α S · U ∂ _t Q ^h − (I − U )t ^α ∂ _t Q ^d − t ^α U Gk

− ¡

r(|D _h | − ∂ _d )|D _h | + U ∆ ¢ t ^α

Z t 0

e ^(t−τ)∆ e _a (Gk) dτ.

Note that the terms in the right-hand side may be handled by means of Lemmas 2.2 and A.2, exactly as we did for t ^α ∇ ² _h u ^d . Therefore, we have

kt ^α ∂ _d Πk _L

t

(L

) ≤ Ckt ^α (∇g, ∂ _t Q)k _L

t

(L

) . Owing to (2.30), it is clear that t ^α ∇ _h Π satisfies the same inequality.

Let us now consider the case f ≡ 0, g ≡ 0 and u 0 ∈ W ₀ ^1,p ( R ^d ₊ ) ∩ W ^2,p ( R ^d ₊ ) (with no loss of generality). As usual, because one may go from u ^d to u ^h through

u ^h = re ^t∆ e _a (V _h u ₀ ) − Su ^d , we concentrate on t ^β ∇ ² u ^d . We start with the formula

t ^α ∇ ² _h u ^d (t) = U t ^α e ^t∆ e _a (∇ ² _h V _d u ₀ ) = U t ^α e ^t∆ ∇ ² _h V _d e _a (u ₀ ), which, in view of Lemmas 2.2 and A.5 ensures that

kt ^α ∇ ² _h u ^d k _L

t

(L

) ≤ CkV _d e _a (u ₀ )k _{B ˙}

p,r

( R

) ≤ Cku ₀ k _{B ˙}

( R

+

) with α = 1 − s 2 − 1

r · Similarly, we have

t ^α ∇ _h ∂ _d u ^d = (I − U )t ^α e ^t∆ ∇ _h |D _h |e _a (V _d u ₀ ), hence

(2.34) kt ^α ∇ _h ∂ _d u ^d k _L

t

(L

) ≤ Cku ₀ k B ˙

( R

+

) . Finally, t ^α ∂ _d ² u ^d satisfies

t ^α ∂ _d ² u ^d = rt ^α e ^t∆ e s (∂ _d |D _h |V _d u 0 ) + (I − U )t ^α e ^t∆ e a (∆ _h V _d u 0 )

= rt ^α e ^t∆ ∂ _d |D _h |e a (V _d u 0 ) + (I − U )t ^α e ^t∆ e a (∆ _h V _d u 0 )

because e _s (∂ _d V _d u ₀ ) = ∂ _d e _a (V _d u ₀ ) owing to the fact that V _d u ₀ vanishes on ∂ R ^d ₊ . Hence t ^α ∂ ² _d u ^d satisfies (2.34), too, and we conclude that

(2.35) kt ^α ∇ ² u ^d k _L

t

(L

) ≤ Cku 0 k B ˙

( R

+

) . Bounding ∇Π is strictly analogous.

In order to prove the estimate for t ^α ∇ ² u in the case g ≡ 0 and u ₀ ≡ 0, we use the fact that

t ^α ∇ ² _h u ^d = U t ^α Z t

0

e ^(t−τ)∆ ∇ ² _h e _a ( N f e ) dτ, t ^α ∇ _h ∂ _d u ^d = (I −U ) t ^α Z t

0

e ^(t−τ)∆ ∇ _h |D _h |e _a ( N f e ) dτ and t ^α ∂ _d ² u ^d = r

Z t 0

e ^(t−τ)∆ ∂ _d |D _h |e _a ( N f e ) dτ + (I − U ) t ^α Z t

0

e ^{(t−τ )∆} ∆ _h e _a ( N f e ) dτ.

Then combining Lemmas 2.2 and A.2 readily gives kt ^α ∇ ² u ^d k _L

t

(L

) ≤ Ckt ^α f k _L

t

(L

) .

Bounding t ^α ∇ ² u _h and t ^α ∇Π works the same.

Let us finally go to the proof of estimates for t ^β ∇u and t ^γ u. By virtue of (2.25) and of the definition of B (see the appendix), we have

t ^β ∇u(t) = r ³

t ^β ∇e ^t∆ e _a (u ₀ ) + t ^β Be _a (f − ∇Π) ´

· Therefore, applying Lemmas A.3 and A.5 yields

kt ^β ∇uk _L

t

(L

) ≤ C ³

ke _a (u ₀ )k _{B ˙}

p2,r2

( R

) + kt ^α e _a (f − ∇Π)k _L

t

(L

( R

))

´ , whenever p ₂ ≥ p, r ₂ ≥ r, s ₂ = _p ^d

2

− ^d _p + s, β = α + d

2 µ 1

p − 1 p ₂

¶

− 1 2 + 1

r − 1 r ₂ = 1

2 − s 2

2 − 1 r ₂ and

s < 1 + d p − d

p 2 < 2 + 2 r 2 − 2

r ·

Combining with the fact that e _a is continuous on functions of ˙ B _p ^s

_,r

( R ^d

+ ) with null trace at the boundary, and with (2.32), we get

kt ^β ∇uk _L

t

(L

) ≤ C ³

ku ₀ k _{B ˙}

( R

+

) + kt ^α (f, ∇g, ∂ _t Q)k _L

t

(L

)

´ ·

Finally, in order to bound t ^γ u, we use the formula t ^γ u(t) = r ³

t ^γ e ^t∆ e _a (u ₀ ) + t ^γ Ce _a (f − ∇Π) ´

· Applying Lemmas A.4 and A.5 yields

kt ^γ uk _L

t

(L

) ≤ C ³

ke a (u 0 )k B ˙

( R

) + kt ^α e a (f − ∇Π)k _L

t

(L

( R

)) , with p ₃ , ≥ p, r ₃ ≥ r, s ₃ = _p ^d

3

− ^d _p + s, γ = α + d

2 µ 1

p − 1 p ₃

¶

− 1 + 1 r − 1

r ₃ = − s ₃ 2 − 1

r ₃ and

s < d p − d

p ₃ < 2 + 2 r ₃ − 2

r ·

Combining with the fact that e _a is continuous on functions of ˙ B _p ^s

_,r

( R ^d ₊ ) with null trace, and with (2.32), we get

kt ^γ uk _L

t

(L

) ≤ C ³

ku 0 k B ˙

( R

+

) + kt ^γ (f, ∇g, ∂ t Q)k _L

t

(L

)

´ .

This completes the proof of the proposition. ¤

3. Existence of smooth solutions

As a first step for proving Theorem 1.1, we here establish the global existence of strong

solutions for (1.4) in the case of a globally Lipschitz bounded density. As for the velocity, we

assume that it has slightly sub-critical regularity. Here is our statement (recall that the space

X ^p,r has been defined just above (1.6)):