Gevrey regularity for a system coupling the Navier-Stokes system with a beam : the non-flat case

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Gevrey regularity for a system coupling the

Navier-Stokes system with a beam : the non-flat case

Mehdi Badra, Takéo Takahashi

To cite this version:

Mehdi Badra, Takéo Takahashi. Gevrey regularity for a system coupling the Navier-Stokes system

with a beam : the non-flat case. 2019. �hal-02303258�

Gevrey regularity for a system coupling the Navier-Stokes system with a beam: the non-flat case

Mehdi Badra ¹ and Tak´ eo Takahashi ²

1 Institut de Math´ ematiques de Toulouse ; UMR5219; Universit´ e de Toulouse ; CNRS ; UPS IMT, F-31062 Toulouse Cedex 9, France

2 Universit´ e de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France October 2, 2019

Abstract

We consider a bi-dimensional viscous incompressible fluid in interaction with a beam located at its boundary. We show the existence of strong solutions for this fluid-structure interaction system, extending a previous result [3] where we supposed that the initial deformation of the beam was small. The main point of the proof consists in the study of the linearized system and in particular in proving that the corresponding semigroup is of Gevrey class.

Keywords: fluid-structure, Navier-Stokes system, Gevrey class semigroups 2010 Mathematics Subject Classification. 76D03, 76D05, 35Q74, 76D27

Contents

1 Introduction 2

2 Change of variables and linearization 5

2.1 The system written in a fixed domain . . . . 5 2.2 The linear system . . . . 6

3 Definition and properties of some operators 8

4 Commutator estimates 12

4.1 The system written in a domain with a flat boundary . . . . 12 4.2 Commutator estimate . . . . 14

5 Estimation of V e

_λ⁻¹

16

6 Proof of Theorem 2.4 20

6.1 Estimation of V

_λ⁻¹

. . . . 21 6.2 Proof of Theorem 2.4 . . . . 22

7 Proof of the local in time existence for (1.3) 24

A Formula for the change of variables 25

F

η

Γ

₋₁

− − − −

Γ

_η

0 L

Figure 1: Our geometry

1 Introduction

This work is devoted to the mathematical analysis of a fluid-structure interaction system where the fluid is modeled by the Navier-Stokes system whereas the structure is a beam situated at a part of the fluid domain.

We consider here the bi-dimensional case in space, that is the fluid domain is a subset of R

²

whereas the beam domain is an interval. Another important assumption for our analysis is to assume periodic boundary conditions in the direction orthogonal to the beam deformation. To be more precise, let L > 0 be the length of the beam and let us set

I

^def

= R /L Z . (1.1)

For any deformation η : I → (−1, ∞), we also consider the corresponding fluid domain F

η

def

= {(x

1

, x

2

) ∈ I × R ; x

2

∈ (0, 1 + η(x

1

))} . (1.2) The boundary of F

η

can be splitted into a “deformable” part

Γ

η

def

= {(s, 1 + η(s)), s ∈ I} , and a “fixed” part

Γ

−1

def

= I × {0}.

We recall the geometry in Figure 1.

Let us denote by v and p the velocity and the pressure of the fluid. Then, the system modeling the interaction between the viscous incompressible fluid and the beam is



 

 



 

 



∂

t

v + (v · ∇)v − div T (v, p) = 0, t > 0, x ∈ F

η(t)

, div v = 0, t > 0, x ∈ F

η(t)

,

v(t, s, 1 + η(t, s)) = (∂

t

η)(t, s)e

2

, t > 0, s ∈ I, v = 0 t > 0, x ∈ Γ

−1

,

∂

tt

η + α

1

∂

ssss

η − α

2

∂

ss

η = −e H

η

(v, p), t > 0, s ∈ I,

(1.3)

with the initial conditions

η(0, ·) = η

⁰1

, ∂

t

η(0, ·) = η

⁰2

and v(0, ·) = v

⁰

in F

_η0

1

. (1.4)

The two first equations correspond to the Navier-Stokes system, whereas the last equation is the beam equation. We have considered the no-slip boundary conditions (third and forth equations). The canonical basis of R

²

is denoted by (e

1

, e

2

) and we have also used the following notations:

T (v, p)

^def

= 2νD(v) − pI

2

, D(v) = 1

2 (∇v + (∇v)

^∗

) , (1.5)

H e

η

(v, p)

^def

= n

(1 + |∂

s

η|

²

)

^1/2

[ T (v, p)n] (t, s, 1 + η(t, s)) · e

2

o

. (1.6)

We assume that the constants satisfy

ν > 0 (viscosity), α

1

> 0, α

2

> 0.

Finally, the vector fields n is the unit exterior normal to F

η(t)

: n = −e

2

on Γ

−1

and on Γ

η(t)

, n(t, x

1

, x

2

) = 1

p 1 + |∂

s

η(t, x

1

)|

²

−∂

s

η(t, x

1

) 1

. (1.7)

An important remark is that a solution to (1.3) satisfies d

dt Z

L

0

η(t, s) ds = 0.

By assuming that the mean value of η

₁⁰

is zero, this leads to Z

L

0

η(t, s) ds = 0 (t > 0). (1.8)

We denote by M the orthogonal projection from L

²

(I) onto L

²0

(I) where L

²0

(I)

^def

=

f ∈ L

²

(I) ; Z

L

0

f(s) ds = 0

. (1.9)

Taking the projection of the last equation of (1.3) on L

²0

(I) gives

∂

tt

η + A

1

η = − H

η

(v, p), t > 0, s ∈ I, (1.10) where

H

η

(v, p)

^def

= M H e

η

(v, p), (1.11)

and where A

1

is the operator for the structure defined by

H

S^def

= L

²₀

(I), D(A

1

)

^def

= H

⁴

(I) ∩ L

²₀

(I), (1.12) A

1

: D(A

1

) → H

S

, η 7→ α

1

∂

ssss

η − α

2

∂

ss

η. (1.13) One can check that for any θ > 0,

D(A

^θ1

) = H

^4θ

(I) ∩ L

²0

(I). (1.14)

The projection of the last equation of (1.3) on L

²0

(I)

^⊥

allows us to determine the constant for the pressure (see [3] for more details): at the contrary to the classical Navier-Stokes system without structure, here the pressure is not determined up to a constant.

The classical Lebesgue and Sobolev spaces are denoted by L

^α

, H

^k

and we use the notation C

⁰

for the space of continuous maps and C

b⁰

for the space of continuous and bounded maps. We use the bold notation for the spaces of vector fields: L

^α

= (L

^α

)

²

, H

^k

= (H

^k

)

²

etc. Since the fluid domain is moving, we introduce spaces of the form H

¹

(0, T ; L

^q

(F

η

)), L

²

(0, T ; H

^k

(F

η

)), etc. with T 6 ∞. If η(t, ·) > −1 (t ∈ (0, T )), then

v ∈ H

¹

(0, T ; L

^q

(F

η

)) if y 7→ v(t, y

1

, y

2

(1 + η(t, y

1

)) ∈ H

¹

(0, T ; L

^q

(F

0

)) and similarly, for the other spaces. We also write

H

0^α

(I)

^def

= H

^α

(I) ∩ L

²0

(I) (α > 0).

Finally, we use C as a generic positive constant that does not depend on the other terms of the inequality.

The value of the constant C may change from one appearance to another.

Let us write our hypotheses for the initial conditions: there exists ε > 0 such that

η

1⁰

∈ W

^7,∞

(I) ∩ L

²0

(I), η

⁰2

∈ H

₀^1+ε

(I) η

⁰1

> −1 in I, (1.15) v

⁰

∈ H

¹

(F

_η0

1

), (1.16)

with

div v

⁰

= 0 in F

_η0

1

, v

⁰

(s, 1 + η

⁰1

(s)) = η

2⁰

(s)e

2

s ∈ I, v

⁰

= 0 on Γ

−1

. (1.17)

Our main result on (1.3) is the existence and uniqueness of strong solutions for small times:

Theorem 1.1. For any [v

⁰

, η

⁰₁

, η

⁰₂

] satisfying (1.15)–(1.17), there exist T > 0 and a strong solution (η, v, p) of (1.3) with

η(t, ·) > −1 t ∈ [0, T ], (1.18)

v ∈ L

²

(0, T ; H

²

(F

η

) ∩ C

⁰

([0, T ]; H

¹

(F

η

)) ∩ H

¹

(0, T ; L

²

(F

η

)), p ∈ L

²

(0, T ; H

¹

(F

η

)), (1.19) η ∈ L

²

(0, T ; H

₀^7/2

(I)) ∩ C

⁰

([0, T ]; H

₀^5/2

(I)) ∩ H

¹

(0, T ; H

₀^3/2

(I)),

∂

t

η ∈ L

²

(0, T ; H

₀^3/2

(I)) ∩ C

⁰

([0, T ]; H

₀^1/2

(I)) ∩ H

¹

(0, T ; (H

₀^1/2

(I))

⁰

),

(1.20) the first four equations of (1.3) are satisfied almost everywhere or in the trace sense and (1.10) holds in L

²

(0, T ; H

₀^1/2

(I)

⁰

).

This solution is unique locally: if (η

^(∗)

, v

^(∗)

, p

^(∗)

) is another solution with the same regularity, there exists T

^∗

> 0 such that

(η

^(∗)

, v

^(∗)

, p

^(∗)

) = (η, v, p) on [0, T

^∗

].

In order to prove the above result, a first standard step consists in rewriting the Navier-Stokes system in the fixed spatial domain

F

^def

= F

_η0

1

, (1.21)

by using a change of variables. Then, one of the main ingredients to obtain Theorem 1.1 is a result on a linear system associated with (1.3):



 



 



∂

t

w − div T (w, q) = F, t > 0, y ∈ F, div w = 0 t > 0, y ∈ F,

w(t, s, 1 + η

⁰1

(t, s)) = (∂

t

η)(t, s)e

2

t > 0, s ∈ I,

∂

tt

η + A

1

η = − H

η⁰₁

(w, q) + G, t > 0, s ∈ I,

(1.22)

with the initial conditions

w(0, ·) = w

⁰

, η(0, ·) = ζ

1⁰

, ∂

t

η(0, ·) = ζ

2⁰

. (1.23) For this system, we have the following result

Theorem 1.2. Assume

η

₁⁰

∈ W

^7,∞

(I), η

₁⁰

> −1 in I.

Suppose F ∈ L

²

(0, ∞; L

²

(F)) and G ∈ L

²

(0, ∞; D(A

^1/8₁

)), ε > 0,

ζ

₁⁰

∈ H

₀^3+ε

(I), ζ

₂⁰

∈ H

₀^1+ε

(I), w

⁰

∈ H

¹

(F ), (1.24) div w

⁰

= 0 in F , w

⁰

(s, 1 + ζ

₁⁰

(s)) = ζ

₂⁰

(s)e

2

s ∈ I, w

⁰

= 0 on Γ

−1

. (1.25) Then (1.22)-(1.23) admits a unique solution

w ∈ L

²

(0, ∞; H

²

(F)) ∩ C

b⁰

([0, ∞); H

¹

(F)) ∩ H

¹

(0, ∞; L

²

(F)), q ∈ L

²

(0, ∞; H

¹

(F )/ R ), (1.26) η ∈ L

²

(0, ∞; D(A

^7/8₁

)) ∩ C

b⁰

([0, ∞); D(A

^5/8₁

)) ∩ H

¹

(0, ∞; D(A

^3/8₁

)), (1.27) and

∂

t

η ∈ L

²

(0, ∞; D(A

^3/8₁

)) ∩ C

b⁰

([0, ∞); D(A

^1/8₁

)) ∩ H

¹

(0, ∞; D(A

^1/8₁

)

⁰

). (1.28) Moreover, there exists C

0

> 0 such that

kwk

_L2(0,∞;H²(F))∩C⁰

b([0,∞);H¹(F))∩H¹(0,∞;L²(F))

+ kqk

L²(0,∞;H¹(F)/R)

+ kηk

L²(0,∞;D(A^7/8₁ ))∩C⁰

b([0,∞);D(A^5/8₁ ))∩H¹(0,∞;D(A^3/8₁ ))

+ k∂

t

ηk

L²(0,∞;D(A^3/8₁ ))∩C⁰_b([0,∞);D(A^1/8₁ ))∩H¹(0,∞;D(A^1/8₁ )⁰)

6 C

0

kw

⁰

k

H¹(F)

+ kζ

1⁰

k

_D(A3/4+ε

1 )

+ kζ

2⁰

k

_D(A1/4+ε

1 )

+ kF k

_L2(0,∞;L²(F))

+ kGk

L²(0,∞;D(A^1/8₁ ))

. (1.29)

In [3], we obtained Theorem 1.2 only in the case η

⁰1

= 0 so that the result on (1.3) was reduced to the

case of small initial deformations. Here we are no longer restricted to this hypothesis. As in [3], the proof

of Theorem 1.2 relies on resolvent estimates and results on semigroup of Gevrey class. More precisely, it is

a consequence of Theorem 2.4.

Remark 1.3. As explained above, the main novelty here is to remove the restriction of smallness of η

⁰₁

that was needed in [3]. Our method to obtain the result for the linear system is based on commutator estimates (see Section 4). The main drawback of such approach is that we need a more regular initial deformation (W

^7,∞

instead of H

^3+ε

). Even without this condition, we have as in our previous result a loss of regularity for (η, ∂

t

η): the continuity of (η, ∂

t

η) lies in H

^5/2

(0, L) × H

^1/2

(0, L) but we need to impose that at initial time, it belongs to W

^7,∞

(0, L) × H

^1+ε

(0, L) for some ε > 0. This is due to this model that couples two dynamical systems of different nature and in particular the linear system (1.22) couples the Stokes system and the beam equation and the corresponding semigroup is not analytic but only of Gevrey class as stated in Theorem 1.2.

With an appropriate damping on the beam equation, we can recover an analytic semigroup. More precisely, in the original model proposed in [11] (for the blood flow in a vessel), the beam equation in (1.3) is replaced by

∂

tt

η + α

1

∂

ssss

η − α

2

∂

ss

η − δ∂

tss

η = −e H

η

(v, p), (1.30) with δ > 0.

Several works analyze such a model: [6] (existence of weak solutions), [4], [10] and [8] (existence of strong solutions), [12] (stabilization of strong solutions), [2] (stabilization of weak solutions around a stationary state). In all these works, the damping term −δ∂

tss

η is crucial. Few works have tackled the case without damping: the existence of weak solutions is proved in [7]. In [9], the existence of local strong solutions is obtained for a structure described by either a wave equation (α

1

= δ = 0 and α

2

> 0 in (1.30)) or a beam equation with inertia of rotation (α

1

> 0, α

2

= δ = 0 and with an additional term −∂

ttss

η in (1.30)).

Finally, in our previous work [3] we proved the existence and uniqueness of strong solutions in the case of an undamped beam equation but for small initial deformations.

The outline of the article is as follows: in Section 2, we construct and use a change of variables to write system (1.3) in a cylindrical domain and then linearize it. Section 3 is devoted to the introduction of several useful operators together with their properties. In order to prove Theorem 1.2 we need to estimate commutators appearing due the fact that our initial domain F is not flat. Such estimates allows us to deduce resolvent estimates in Section 6 by estimating the inverse of the operator V

λ

(see (6.6)). At first, we first estimate an approximation of V

_λ⁻¹

in Section 5. Finally, in Section 7 we recall the idea of the proof of Theorem 1.1 based on Theorem 1.2, by using a fixed point argument.

2 Change of variables and linearization

2.1 The system written in a fixed domain

In this section, we defined and use a standard change of variables to rewrite system (1.3) in a cylindrical domain. We set

X

_η1,η²

: F

_η1

→ F

_η2

, (y

1

, y

2

) 7→

y

1

, y

2

1 + η

²

(y

1

) 1 + η

¹

(y

1

)

, (2.1)

whose inverse is X

_η2,η¹

. In our case, we consider X(t, ·)

^def

= X

_η0

1,η(t)

: (y

1

, y

2

) 7→

y

1

, y

2

1 + η(t, y

1

) 1 + η

⁰₁

(y

1

)

, (2.2)

Y (t, ·)

^def

= X (t, ·)

⁻¹

= X

_η(t),η0

1

: (x

1

, x

2

) 7→

x

1

, x

2

1 + η

⁰1

(x

1

) 1 + η(t, x

1

)

, (2.3)

so that X (t, ·) transforms F = F

η⁰

onto F

η(t)

. Then, we write

a

^def

= Cof(∇Y )

^∗

, b

^def

= Cof(∇X )

^∗

, (2.4) w(t, y)

^def

= b(t, y)v(t, X(t, y)) and q(t, y)

^def

= p(t, X(t, y)), (2.5) so that

v(t, x) = a(t, x)w(t, Y (t, x)) and p(t, x) = q(t, Y (t, x)). (2.6)

After some calculation (see for instance [3]), system (1.3), (1.4) rewrites,



 

 



 

 



∂

t

w − div T (w, q) = F b (ξ, w, q) in (0, ∞) × F, div w = 0 in (0, ∞) × F,

w(t, s, 1) = (∂

t

η)(t, s)e

2

t > 0, s ∈ I, w = 0 t > 0, y ∈ Γ

−1

,

∂

tt

η + A

1

η = − H

η⁰₁

(w, q) + G b

_η0

1

(ξ, w), t > 0,

(2.7)

with the initial conditions

η(0, ·) = η

1⁰

, ∂

t

η(0, ·) = η

2⁰

and w(0, y) = w

⁰

(y)

^def

= b(0, y)v

⁰

(X(0, y)) (y ∈ F), (2.8) where we have the following definitions:

F b

α

(η, w, q)

^def

= ν X

i,j,k

b

αi

∂

²

a

ik

∂x

²_j

(X)w

k

+ 2ν X

i,j,k,`

b

αi

∂a

ik

∂x

j

(X) ∂w

k

∂y

`

∂Y

`

∂x

j

(X)

+ ν X

j,`,m

∂

²

w

α

∂y

`

∂y

m

∂Y

`

∂x

j

(X ) ∂Y

m

∂x

j

(X) − δ

`,j

δ

m,j

+ ν X

j,`

∂w

α

∂y

`

∂

²

Y

`

∂x

²_j

(X )

− X

k,i

∂q

∂y

k

det(∇X) ∂Y

α

∂x

i

(X) ∂Y

k

∂x

i

(X ) − δ

α,i

δ

k,i

− X

i,j,k,m

b

αi

∂a

ik

∂x

j

(X )a

jm

(X )w

k

w

m

− 1

det(∇X ) [(w · ∇)w]

_α

− [b(∂

t

a)(X)w]

_α

− [(∇w)(∂

t

Y )(X )]

_α

, (2.9)

G b

_η0

1

(η, w)(t, s) = νM (

2 X

k,`

δ

2,k

δ

2,`

− a

2k

(X) ∂Y

`

∂x

2

(X) ∂w

k

∂y

`

+ ∂

s

(η − η

1⁰

) ∂w

2

∂y

1

+ ∂w

1

∂y

2

+ ∂

s

η



 X

k,`

a

2k

(X) ∂Y

`

∂x

1

(X ) − δ

2,k

δ

1,`

∂w

k

∂y

`

+ X

k,`

a

1k

(X) ∂Y

`

∂x

2

(X) − δ

1,k

δ

2,`

∂w

k

∂y

`





+ X

k

∂

s

η ∂a

2k

∂x

1

(X) + ∂a

1k

∂x

2

(X )

− 2 ∂a

2k

∂x

2

(X )

w

k

)

(t, s, 1 + η

1⁰

(s)). (2.10) Moreover, we recall that

H

η⁰₁

(w, q)(t, s) = M n

(1 + |∂

s

η

⁰1

|

²

)

^1/2

[ T (v, p)n] (t, s, 1 + η

⁰1

(s)) · e

2

o

. (2.11)

2.2 The linear system

From the previous section, and in particular from system (2.7)-(2.8), we are led to consider the linear system (1.22)-(1.23) written in the fixed domain F (defined by (1.21)). We introduce the notation

C

^+def

= {λ ∈ C ; Re(λ) > 0} . (2.12)

C

⁺α def

=

λ ∈ C

⁺

; |λ| > α . (2.13)

Let us consider the following functional spaces V

^θ

(F)

^def

= n

f ∈ H

^θ

(F) ; div f = 0 o

, (2.14)

V

^θn

(F)

^def

= n

f ∈ H

^θ

(F) ; div f = 0, f · n = 0 on ∂F o

(θ ∈ [0, 1/2)), (2.15) V

^θn

(F)

^def

=

n

f ∈ H

^θ

(F) ; div f = 0, f = 0 on ∂F o

(θ ∈ (1/2, 1]), (2.16)

V

^θ

(∂F)

^def

=

f ∈ H

^θ

(∂F ) ; Z

∂F

f · n dγ = 0

(θ > 0). (2.17)

We introduce the operator Λ : L

²

(I) → L

²

(∂F) defined by

(Λη)(y) = (M η(s)) e

2

if y = (s, 1 + η

⁰₁

(s)) ∈ Γ

_η0 1

, (Λη)(y) = 0 if y ∈ Γ

−1

.

(2.18) The adjoint Λ

^∗

: L

²

(∂F ) → L

²

(I) of Λ is given by

(Λ

^∗

v)(s) = M

(1 + |∂

s

η

⁰₁

(s)|

²

)

^1/2

v(s, 1 + η

₁⁰

(s)) · e

2

. (2.19)

Since η

1⁰

∈ W

^7,∞

(I), then for any θ ∈ [0, 4],

Λ(H

^θ

(I)) ⊂ V

^θ

(∂F) (2.20)

and

Λ

^∗

(H

^θ

(∂F)) ⊂ D(A

^θ/4₁

). (2.21)

In particular

kΛηk

_Hθ(∂F)

> c(θ)kA

^θ/4₁

ηk

H_S

(η ∈ D(A

^θ/4₁

)). (2.22) We can also define the Stokes operator

D( A )

^def

= V

¹_n

(F) ∩ H

²

(F), A

^def

= P ∆ : D( A ) → V

⁰_n

(F), (2.23) where P : L

²

(F ) → V

n⁰

(F ) is the Leray projection operator.

We consider the space L

²

(F) × D(A

^1/2₁

) × H

S

equipped with the scalar product:

Dh

w

⁽¹⁾

, η

₁⁽¹⁾

, η

₂⁽¹⁾

i , h

w

⁽²⁾

, η

₁⁽²⁾

, η

⁽²⁾₂

iE

= Z

F

w

⁽¹⁾

· w

⁽²⁾

dy +

A

^1/2₁

η

₁⁽¹⁾

, A

^1/2₁

η

₁⁽²⁾

H_S

+

η

₂⁽¹⁾

, η

⁽²⁾₂

H_S

, and we introduce the following spaces:

H

^def

= n

[w, η

1

, η

2

] ∈ L

²

(F) × D(A

^1/2₁

) × H

S

; w · n = (Λη

2

) · n on ∂F, div w = 0 in F o

, (2.24)

V

^def

= n

[w, η

1

, η

2

] ∈ H

¹

(F) × D(A

^3/4₁

) × D(A

^1/4₁

) ; w = Λη

2

on ∂F, div w = 0 in F o .

We denote by P

0

the orthogonal projection from L

²

(F) × D(A

^1/2₁

) × H

S

onto H. We have the following regularity result on P

0

(see [3]):

Lemma 2.1. For any θ ∈ [0, 1],

P

0

∈ L(H

^θ

(F ) × D(A

^1/2+θ/4₁

) × D(A

^θ/4₁

)), (2.25) and

P

0

∈ L(L

²

(F) × D(A

^3/8₁

) × D(A

^1/8₁

)

⁰

). (2.26) We now define the linear operator A

0

: D(A

0

) ⊂ H → H:

D(A

0

)

^def

= V ∩ h

H

²

(F) × D(A

1

) × D(A

^1/2₁

) i

, (2.27)

and for

w, η

1

, η

2

∈ D(A

0

), we set

A e

0



 w η

1

η

2





def

=











∆w η

2

−A

1

η

1

− Λ

^∗

(2D(w)n)











(2.28)

and

A

0

def

= P

0

A e

0

. (2.29)

By using the above operators, we can rewrite the linear system (1.22), as follows d

dt



 w

η

∂

t

η



 = A

0



 w η

∂

t

η



 + P

0



 F 0 G



 ,



 w η

∂

t

η



 (0) =



 w

⁰

η

⁰1

η

⁰2



 . (2.30)

We also recall the following result (see [2, Proposition 3.4, Proposition 3.5 and Remark 3.6]).

Proposition 2.2. The operator A

0

defined by (2.27)–(2.29) has compact resolvents, it is the infinitesimal generator of a strongly continuous semigroup of contractions on H and it is exponentially stable on H.

We have also the following result (see [2, Proposition 3.8]).

Proposition 2.3. For θ ∈ [0, 1], the following equalities hold D((−A

0

)

^θ

) =

h

H

^2θ

(F) × D(A

^1/2+θ/2₁

) × D(A

^θ/2₁

) i

∩ H if θ ∈ (0, 1/4) , (2.31)

D((−A

0

)

^θ

) = n

[w, η

1

, η

2

] ∈ h

H

^2θ

(F) × D(A

^1/2+θ/2₁

) × D(A

^θ/2₁

) i

∩ H ; w = Λη

2

on ∂F o

if θ ∈ (1/4, 1]. (2.32) One of the main goals of this article is to show the following result:

Theorem 2.4. There exists C > 0 such that for all λ ∈ C

⁺

|λ|

^1/2

(λI − A

0

)

⁻¹

_L(H)

6 C. (2.33)

Moreover, there exists a constant C > 0 such that for all λ ∈ C

⁺

(λI − A

0

)

⁻¹

z

H²(F)×D(A^7/8₁ )×D(A^3/8₁ )

+ |λ|

(λI − A

0

)

⁻¹

z

L²(F)×D(A^3/8₁ )×D(A^1/8₁ )⁰

6 C kzk

L²(F)×D(A^5/8₁ )×D(A^1/8₁ )

z ∈ H ∩

L

²

(F) × D(A

^5/8₁

) × D(A

^1/8₁

)

. (2.34) Using the above theorem and Theorem 5.1 in [3], we deduce Theorem 1.2.

3 Definition and properties of some operators

This section is devoted to the introduction of several operators that are used to prove the resolvent estimates in Theorem 2.4. In this section we assume η

⁰₁

∈ W

^4,∞

(I). It implies in particular that the domain F is of class C

^3,1

.

For all λ ∈ C

⁺

, we define the solution (w

η

, q

η

) (that depends on λ) of







λw

η

− div T (w

η

, q

η

) = 0 in F, div w

η

= 0 in F, w

η

= Λη on ∂F,

(3.1) where Λ is defined by (2.18). The above problem is well-posed (see, for instance, [3, Proposition 4.4]) and if we define the operators

W

λ

η

^def

= w

η

, Q

λ

η

^def

= q

η

, (3.2)

since F is of class C

^3,1

, we have

W

λ

∈ L(D(A

^7/8₁

), H

⁴

(F )) ∩ L(D(A

^1/8₁

), H

¹

(F )) ∩ L(D(A

^1/8₁

)

⁰

, L

²

(F)) (3.3) and

Q

λ

∈ L(D(A

^3/8₁

), H

¹

(F)/ R ). (3.4) We also define the operator

L

λ

∈ L(D(A

^3/8₁

), D(A

^1/8₁

))

by

L

λ

η

^def

= Λ

^∗

T (w

η

, q

η

)n

|∂F

. (3.5)

We decompose L

λ

with the operators

K

λ

∈ L(D(A

^1/8₁

)

⁰

, D(A

^1/8₁

)), G

λ

∈ L(D(A

^1/8₁

), D(A

^1/8₁

)

⁰

) ∩ L(D(A

^3/8₁

), D(A

^1/8₁

)) defined by

hK

λ

η, ζi

_D(A1/8

1 ),D(A^1/8₁ )⁰ def

=

Z

F

w

η

· w

ζ

dy (3.6)

and

hG

λ

η, ζi

_D(A1/8

1 )⁰,D(A^1/8₁ ) def

= 2ν

Z

F

Dw

η

: Dw

ζ

dy = 2ν Z

L

0

Λ

^∗

((Dw

η

)n) ζ ds − ν Z

F

∆w

η

· w

ζ

dy.

(The second relation holds if η ∈ D(A

^3/8₁

)).

The operators K

λ

and G

λ

are related to the operator L

λ

defined by (3.5): multiplying (3.1) by w

ζ

and integrating by part, we deduce that

L

λ

= λK

λ

+ G

λ

. (3.7)

We recall the following result (see Proposition 3.1 in [3]):

Proposition 3.1. The operators K

λ

and G

λ

defined above are positive and self-adjoint. Moreover there exist 0 < ρ

1

< ρ

2

such that for any λ such that Re λ > 0, we have

ρ

1

kA

^1/8₁

ηk

²H_S

6 hG

λ

η, ηi

_D(A1/8

1 )⁰,D(A^1/8₁ )

6 ρ

2

kA

^1/8₁

ηk

²H_S

+ |λ|kA

^−1/8₁

ηk

²H_S

(η ∈ D(A

^1/8₁

)), (3.8) 0 6 hK

λ

η, ηi

D(A^1/8₁ ),D(A^1/8₁ )⁰

6 ρ

2

kA

^−1/8₁

ηk

²H_S

(η ∈ D(A

^1/8₁

)

⁰

). (3.9) Note that we have

K

λ

η = −Λ

^∗

{ T (ϕ

η

, π

η

)n|

∂F

} (3.10) where







λϕ

η

− div T (ϕ

η

, π

η

) = W

λ

η in F, div ϕ

η

= 0 in F, ϕ

η

= 0 on ∂F ,

(3.11) and where W

λ

is defined by (3.2).

Next, we define an important operator in what follows:

V

λ

= λ

²

I + λL

λ

+ A

1

= λ

²

(I + K

λ

) + λG

λ

+ A

1

, (3.12) and an “approximation”:

V e

λ

def

= λ

²

(I + K

λ

) + 2ρλA

^1/4₁

+ A

1

, (3.13) where ρ > 0 is a constant to be fixed later.

Let us consider







λ b v − div T ( v, b p) = b f in F, div b v = 0 in F, b v = 0 on ∂F.

(3.14) Proposition 3.2. Let γ ∈ [0, 1/4), θ ∈ [γ, 1].

1. There exists C > 0 such that for any f ∈ H

^2γ

(F ) and for any λ ∈ C

⁺0

, the solution ( b v, p) b of (3.14) satisfies

k vk b

_H2θ(F)

6 C|λ|

^θ−γ−1

kfk

_H2γ(F)

. (3.15) 2. There exists C > 0 such that for any f ∈ H

^2θ

(F) and for any λ ∈ C

⁺0

, the solution ( b v, p) b of (3.14)

satisfies

k b vk

_H2+2θ(F)

+ k b pk

_H1+2θ(F)

6 C

|λ|

^θ−γ

kfk

H^2γ(F)

+ kfk

_H2θ(F)

. (3.16)

Proof. Using that the Stokes operator A (defined by (2.23)) is the infinitesimal generator of an analytic semigroup and that C

⁺0

⊂ ρ( A ), we have the existence of a constant C such that

k(− A )

^α

(λI − A )

⁻¹

gk

_L2(F)

6 C|λ|

^α−1

kgk

_L2(F)

(g ∈ V

⁰n

(F), λ ∈ C

⁺0

, α ∈ [0, 1]).

Using that for γ ∈ [0, 1/4), P ∈ L(H

^2γ

(F), D((− A )

^γ

)) (see [1, Section 2.1]), we have k(−A )

^γ

P fk

L²(F)

6 Ckfk

H^2γ(F)

(γ ∈ [0, 1/4), f ∈ H

^2γ

(F)).

Gathering the two above estimates with the fact that D((− A )

^θ

) ⊂ H

^2θ

(F), we can deduce k vk b

_H2θ(F)

6 C

(− A )

^θ

(λI − A )

⁻¹

P f

L²(F)

6 C|λ|

^θ−γ−1

kfk

_H2γ(F)

. For the second estimate, we use the following classical estimate for Stokes system:

k b vk

_H2+2θ(F)

+ k b pk

_H1+2θ(F)

6 C |λ|k vk b

_H2θ(F)

+ kfk

_H2θ(F)

, and we combine it with (3.15).

Using the above proposition, we can define the following operator T

λ

∈ L(L

²

(F ), D(A

^1/8₁

)), T

λ

f

^def

= −Λ

^∗

T ( b v, p)n b

|∂F

. (3.17) We have in particular that the norm of T

λ

in L(L

²

(F ), D(A

^1/8₁

)) is independent of λ.

Proposition 3.3. For θ ∈ [0, 1], ε ∈ (0, 1/4) and λ ∈ C

⁺0

, the operators W

λ

and Q

λ

defined by (3.2) satisfy kW

λ

ηk

_H2θ(F)

6 CkA

^θ/2−1/8₁

ηk

H_S

(θ < 1/4), (3.18) kW

λ

ηk

_H2θ(F)

6 C

kA

^θ/2−1/8₁

ηk

H_S

+ |λ|

^θ

kA

^−1/8₁

ηk

H_S

(θ > 1/4), (3.19) kW

λ

ηk

_H2θ(F)

6 C

kA

^θ/2−1/8₁

ηk

H_S

+ |λ|

^θ−1/4+ε

kηk

H_S

, θ > 1/4 − ε, (3.20) kW

λ

ηk

_H2+2θ(F)

+ kQ

λ

ηk

_H1+2θ(F)

6 C

kA

^θ/2+3/8₁

ηk

H_S

+ |λ|

^1+θ

kA

^−1/8₁

ηk

H_S

, (3.21)

kW

λ

ηk

_H2+2θ(F)

+ kQ

λ

ηk

_H1+2θ(F)

6 C

kA

^θ/2+3/8₁

ηk

H_S

+ |λ|

^3/4+ε+θ

kηk

H_S

. (3.22)

Proof. We write

W

λ

η = W

0

η + z

η

, Q

λ

η = Q

0

η + ζ

η

,

with 





λz

η

− div T (z

η

, ζ

η

) = −λW

0

η in F, div z

η

= 0 in F,

z

η

= 0 on ∂F.

(3.23) Using (3.3), there exists a positive constant C such that

kW

0

ηk

_H2θ(F)

6 CkA

^θ/2−1/8₁

ηk

H_S

(θ ∈ [0, 2], η ∈ D(A

^θ/2−1/8₁

)). (3.24) Combining the above relation with (3.15) we deduce the following relations:

kz

η

k

_H2θ(F)

6 C|λ|

^θ

kA

^−1/8₁

ηk

H_S

,

kz

η

k

_H2θ(F)

6 CkA

^θ/2−1/8₁

ηk

H_S

if θ ∈ [0, 1/4), kz

η

k

_H2θ(F)

6 C|λ|

^θ−1/4+ε

kηk

H_S

(θ > 1/4 − ε).

Then (3.18), (3.19) and (3.20) follow by combining the above inequalities with (3.24).

From (3.16) we deduce

kz

η

k

_H2+2θ(F)

+ kζ

η

k

_H1+2θ(F)

6 C

|λ|

^3/4+ε+θ

kW

0

ηk

_H1/2−2ε(F)

+ |λ|kW

0

ηk

_H2θ(F)

, (3.25)

and

kz

η

k

_H2+2θ(F)

+ kζ

η

k

_H1+2θ(F)

6 C

|λ|

^1+θ

kW

0

ηk

L²(F)

+ |λ|kW

0

ηk

_H2θ(F)

. (3.26)

Moreover, from (3.24), we have

|λ|kW

0

ηk

_H2θ(F)

6 C

|λ|

^1+θ

kW

0

ηk

_L2(F)

_1+θ¹

kW

0

ηk

_H2+2θ(F)

_1+θ^θ

6 C

kW

0

ηk

_H2+2θ(F)

+ |λ|

^1+θ

kW

0

ηk

L²(F)

6 C

kA

^θ/2+3/8₁

ηk

H_S

+ |λ|

^1+θ

kA

^−1/8₁

ηk

H_S

,

and

|λ|kW

0

ηk

_H2θ(F)

6 C

|λ|

^{1+θ−1/4+ε}

kW

0

ηk

_H1/2−2ε(F)

_{1+θ−1/4+ε}¹

kW

0

ηk

_H2+2θ(F)

_{1+θ−1/4+ε}^θ−1/4+ε

6 C

kW

0

ηk

_H2+2θ(F)

+ |λ|

^3/4+ε+θ

kW

0

ηk

_H1/2−2ε(F)

6 C

kA

^θ/2+3/8₁

ηk

H_S

+ |λ|

^3/4+ε+θ

kηk

H_S

.

Combining the above estimates with (3.25) and (3.26), we deduce (3.21) and (3.22).

Proposition 3.4. Let θ ∈ [1/4, 1/2) and λ ∈ C

⁺0

, then

kA

^θ/2₁

K

λ

ηk

H_S

6 CkA

^θ/2−1/4₁

ηk

H_S

. (3.27) Let ε ∈ (0, 1/4), λ ∈ C

⁺0

and θ ∈ [1/2, 2], then

kA

^θ/2₁

K

λ

ηk

H_S

6 C

kA

^θ/2−1/4₁

ηk

H_S

+ |λ|

^θ−1/2+ε

kηk

H_S

. (3.28)

Proof. From (3.10) and properties on the trace operator and on Λ

^∗

, kA

^θ/2₁

K

λ

ηk

H_S

6 C

kϕ

η

k

_H3/2+2θ(F)

+ kπ

η

k

_H1/2+2θ(F)

. (3.29)

Using (3.11), (3.16) and (3.18), we deduce from the above estimate that

kA

^θ/2₁

K

λ

ηk

H_S

6 CkW

λ

ηk

_H2θ−1/2(F)

6 CkA

^θ/2−1/4₁

ηk

H_S

if θ ∈ [1/4, 1/2).

On the other hand, if θ ∈ [1/2, 2], using (3.11), (3.16), (3.18) and (3.20), we deduce from (3.29) that kA

^θ/2₁

K

λ

ηk

H_S

6 C

|λ|

^θ−1/2+ε

kW

λ

ηk

_H1/2−2ε(F)

+ kW

λ

ηk

_H2θ−1/2(F)

6 C

|λ|

^θ−1/2+ε

kηk

H_S

+ kA

^θ/2−1/4₁

ηk

H_S

.

Proposition 3.5. Assume α > 0. Let θ ∈ (−1/2, 1/2), then

kA

^θ/2₁

(I + K

λ

)

⁻¹

ηk

H_S

6 CkA

^θ/2₁

ηk

H_S

(λ ∈ C

⁺

). (3.30) Let ε > 0 and θ ∈ [1/2, 2], then

kA

^θ/2₁

(I + K

λ

)

⁻¹

ηk

H_S

6 C

kA

^θ/2₁

ηk

H_S

+ |λ|

^θ−1/2+ε

kηk

H_S

(λ ∈ C

⁺α

). (3.31) Proof. We write

ζ = (I + K

λ

)

⁻¹

η, ζ = η − K

λ

ζ.

First, using the positivity of K

λ

stated in (3.9), we find

kζk

H_S

6 kηk

H_S

. (3.32)

Moreover, we have the relation

kA

^θ/2₁

ζk

H_S

6 kA

^θ/2₁

ηk

H_S

+ kA

^θ/2₁

K

λ

ζk

H_S

. (3.33)

Assume first θ ∈ [1/4, 1/2). Then combining (3.27) and (3.32) with (3.33), we deduce (3.30). Interpolating (3.30) and (3.32), we also deduce that (3.30) holds for θ ∈ [0, 1/2). To prove (3.30) for θ ∈ (−1/2, 0) we use a duality argument. First, for θ ∈ (0, 1/2) (3.30) can be rewritten as

kA

^θ/2₁

(I + K

λ

)

⁻¹

A

^−θ/2₁

k

L(H_S)

< +∞.

Thus, noticing that (A

^θ/2₁

(I + K

λ

)

⁻¹

A

^−θ/2₁

)

^∗

= A

^−θ/2₁

(I + K

λ

)

⁻¹

A

^θ/2₁

we also have kA

^−θ/2₁

(I + K

λ

)

⁻¹

A

^θ/2₁

k

L(H_S)

< +∞,

which is equivalent to (3.30) for θ ∈ (−1/2, 0).

Next, we assume θ ∈ [1/2, 2] and ε > 0. We deduce from (3.33) and (3.28) that kA

^θ/2₁

ζk

H_S

6 kA

^θ/2₁

ηk

H_S

+ C

kA

^θ/2−1/4₁

ζk

H_S

+ |λ|

^θ−1/2+ε

kζk

H_S

. (3.34)

If θ ∈ [1/2, 1), then θ − 1/2 ∈ [0, 1/2) and we can use (3.30) and (3.32) in the above relation to deduce (3.31).

If θ ∈ [1, 3/2), then θ − 1/2 ∈ [1/2, 1) and we can use (3.31) and (3.32) in (3.34) to deduce kA

^θ/2₁

ζk

H_S

6 kA

^θ/2₁

ηk

H_S

+ C

kA

^θ/2−1/4₁

ηk

H_S

+ |λ|

^θ−1+ε

kζk

H_S

+ |λ|

^θ−1/2+ε

kζk

H_S

.

Since |λ| > α we have |λ|

^θ−1+ε

6 α

^−1/2

|λ|

^θ−1/2+ε

and it yields (3.31). We can then repeat the same argument for θ ∈ [3/2, 2) and θ = 2.

4 Commutator estimates

The aim of this section is to show the following result:

Lemma 4.1. Assume η

1⁰

∈ W

^7,∞

(I). For ε ∈ (0, 1/4), there exists a constant C > 0 such that for any λ ∈ C

⁺0

,

k[A

^3/8₁

, K

λ

]ηk

H_S

6 C(|λ|

⁻¹

kA

^1/2+ε

ηk

H_S

+ |λ|

^ε

kηk

H_S

).

Here we have denoted by [A, B] the commutator of A and B: [A, B] = AB − BA.

4.1 The system written in a domain with a flat boundary

We transform the systems (3.1) and (3.11) written in F = F

_η0

1

into systems written in the domain F

0

= I × (0, 1).

We use the change of variables

X e : F

0

→ F, (y

1

, y

2

) 7→ y

1

, y

2

(1 + η

⁰₁

(y

1

)) ,

Y e : F → F

0

, (x

1

, x

2

) 7→

x

1

, x

2

1 + η

⁰₁

(x

1

)

. We write

e a

^def

= Cof(∇e Y )

^∗

, e b

^def

= Cof(∇ X e )

^∗

. We set

w(y) e

^def

= e b(y)w( X e (y)) and q(y) e

^def

= q( X(y)), e (4.1) so that

w(x) = e a(x) w( e Y e (x)) and q(x) = q( e Y e (x)). (4.2) We set

[ L w] e

_α^def

= X

i,j,k

e b

αi

∂

²

e a

ik

∂x

²_j

( X) e w e

k

+ 2 X

i,j,k,`

e b

αi

∂ e a

ik

∂x

j

( X e ) ∂ w e

k

∂y

`

∂ Y e

`

∂x

j

( X e )

+ X

j,`,m

∂

²

w e

α

∂y

`

∂y

m

∂ Y e

`

∂x

j

( X) e ∂ Y e

m

∂x

j

( X e ) + X

j,`

∂ w e

α

∂y

`

∂

²

Y e

`

∂x

²_j

( X e ), (4.3)

[ Ge q]

_α^def

= det(∇ X) e X

k,i

∂ e q

∂y

k

∂ Y e

α

∂x

i

( X) e ∂ Y e

k

∂x

i

( X). e (4.4)

Then some calculation yields h

e b∆w( X e ) i

α

= [ L w] e

_α

, h

e b∇q( X) e i

α

= [ Ge q]

_α

. (4.5)

We recall the derivation of (4.5) in Appendix A.

We now consider systems (3.1) and (3.11) and using the change of variables (4.1), we introduce the new states

w e

η

def

= e b(w

η

◦ X e ), q e

η

def

= q

η

◦ X, e ϕ e

η

def

= e b(ϕ

η

◦ X), e π e

η

def

= π

η

◦ X. e

From the above relations, systems (3.1) and (3.11) are transformed into the following systems







λ w e

η

− ν L w e

η

+ Ge q

η

= 0 in F

0

, div w e

η

= 0 in F

0

, w e

η

= Λ

0

η on ∂F

0

(4.6)

and 





λ ϕ e

η

− ν Le ϕ

η

+ Ge π

η

= w e

η

in F

0

, div ϕ e

η

= 0 in F

0

,

ϕ e

η

= 0 on ∂ F

0

.

(4.7) Here, we have also transformed the operator Λ defined by (2.18) into the operator Λ

0

: L

²

(I) → L

²

(∂F

0

) defined by

( (Λ

0

η)(s, 1) = (M η(s)) e

2

(Λ

0

η)(s, 0) = 0 , s ∈ I. (4.8)

From (3.10) and (2.19), we have the following formula K

λ

η = −M

νD(ϕ

η

)(s, 1 + η

⁰₁

(s))

−∂

s

η

⁰₁

(s) 1

· e

2

− π

η

(s, 1 + η

⁰₁

(s))

. (4.9)

Thus

K

λ

η = −M (ν Df ϕ

η

(s, 1) − e π

η

(s, 1)) , (4.10) with

Df ϕ

η

= 1 2 (−∂

s

η

1⁰

)



 X

k

∂ e a

2k

∂x

1

( X)( e ϕ f

η

)

k

+ X

k,`

e a

2k

( X e ) ∂( ϕ f

η

)

k

∂y

`

∂ Y e

`

∂x

1

( X e ) + X

k,`

e a

1k

( X e ) ∂( ϕ f

η

)

k

∂y

`

∂ Y e

`

∂x

2

( X e )





+ X

k

∂ e a

2k

∂x

2

( X)( e ϕ f

η

)

k

+ X

k,`

e a

2k

( X e ) ∂( ϕ f

η

)

k

∂y

`

∂ Y e

`

∂x

2

( X). e (4.11) In what follows, we write the above operators by splitting the derivatives with respect to y

1

and y

2

. More precisely, we introduce the set O

^β_α

(β > α) of operators of the form

f 7→ X

i6α

c

^(β−i)_i

∂

1ⁱ

f, (4.12)

where c

^(k)_i

are functions of the form

c

^(k)_i

= c b

i

(η

₁⁰

(y

1

), . . . , ∂

_s^k

η

₁⁰

(y

1

), y

2

), (4.13) with c b

i

a smooth function and k ∈ N . These operators are thus depending on y

2

but it can be seen as a parameter.

For instance, using (A.1)–(A.8), we deduce

D f = D

⁽¹⁾

f + D

⁽⁰⁾

∂

2

f, L f = L

⁽²⁾

f + L

⁽¹⁾

∂

2

f + L

⁽⁰⁾

∂

²2

f, G f = G

⁽¹⁾

f + G

⁽⁰⁾

∂

2

f, (4.14)

where L

⁽²⁾

∈ O

³₂

, D

⁽¹⁾

, L

⁽¹⁾

, G

⁽¹⁾

∈ O

²₁

, D

⁽⁰⁾

, L

⁽⁰⁾

, G

⁽⁰⁾

∈ O

¹₀

.

4.2 Commutator estimate

First we show the following result:

Proposition 4.2. Assume B ∈ O

α^β

and η

⁰1

∈ W

^4+β,∞

(I). For any θ ∈ (0, 1) and for any s > 0, if s > 4θ + α − 1, then there exists C > 0 such that

[A

^θ1

M, B ]f

L²(I)

6 Ckfk

H^s(I)

(f ∈ H

^s

(I)). (4.15) Proof. We write

B f = X

i6α

c

^(β−i)_i

∂

1ⁱ

f.

Then we recall the following formula (see for instance [13, p. 98]):

A

^θ1

= 1 Γ(θ)Γ(1 − θ)

Z

∞ 0

t

^θ−1

A

1

(tI + A

1

)

⁻¹

dt.

In particular,

[A

^θ1

M, B ] = A

^θ1

M B − B A

^θ1

M = A

^θ1

M B M − M B A

^θ1

M − (I − M) B A

^θ1

M + A

^θ1

M B (I − M ). (4.16) By using the identity A

1

(tI + A

1

)

⁻¹

= I − t(tI + A

1

)

⁻¹

we deduce that

A

1

(tI + A

1

)

⁻¹

M B M − M B A

1

(tI + A

1

)

⁻¹

M = −t(tI + A

1

)

⁻¹

M B M + tM B (tI + A

1

)

⁻¹

M

= t(tI +A

1

)

⁻¹

(−M B (tI+A

1

)+(tI+A

1

)M B )(tI+A

1

)

⁻¹

M = t(tI +A

1

)

⁻¹

(−M B A

1

+A

1

M B )(tI+A

1

)

⁻¹

M and the first two terms in (4.16) give

A

^θ₁

M B M − M B A

^θ₁

M = 1 Γ(θ)Γ(1 − θ)

Z

∞ 0

t

^θ

(tI + A

1

)

⁻¹

M [A

1

M, B ](tI + A

1

)

⁻¹

M dt.

Moreover,

[A

1

M, B ]f = α

1

∂

1⁴

− α

2

∂

1²

M X

i6α

c

^(β−i)_i

∂

1ⁱ

f − X

i6α

c

^(β−i)_i

∂

1ⁱ

α

1

∂

1⁴

− α

2

∂

²1

M f

= α

1

∂

1⁴

− α

2

∂

²1

X

i6α

c

^(β−i)_i

∂

ⁱ1

f − X

i6α

c

^(β−i)_i

∂

1ⁱ

α

1

∂

1⁴

− α

2

∂

²1

f

and thus

[A

1

M, B ] ∈ O

^β+4_α+3

. (4.17)

We deduce that for f ∈ H

^s

(I),

k(A

^θ1

M B M − M B A

^θ₁

M )f k

L²(I)

6 1 Γ(θ)Γ(1 − θ)

Z

∞ 0

t

^θ

k(tI + A

1

)

⁻¹

M [A

1

M, B ](tI + A

1

)

⁻¹

M fk

L²(I)

dt 6 C

Z

∞ 0

t

^θ−1

k[A

1

M, B ](tI + A

1

)

⁻¹

M fk

L²(I)

dt 6 C Z

∞

0

t

^θ−1

k(tI + A

1

)

⁻¹

M fk

H^α+3(I)

dt 6 C

Z

1 0

t

^θ−1

kA

1

(tI + A

1

)

⁻¹

A

α−1 4

1

M fk

_L2(I)

dt + Z

∞

1

t

^θ−1

kA

α+3−s 4

1

(tI + A

1

)

⁻¹

A

s 4

1

M fk

_L2(I)

dt

6 C Z

1

0

t

^θ−1

kM fk

H^α−1(I)

dt + Z

∞

1

t

^θ−1

1

t

1−(α+3−s)/4

kM fk

H^s(I)

dt

6 CkM fk

H^s(I)

. (4.18) For the third term in (4.16), we write

(I − M ) B A

^θ1

M f = 1 L

Z

L 0

X

i6α

c

^(β−i)_i

∂

1ⁱ

A

^θ1

M f

! ds = 1

L Z

L

0

X

i6α

(−1)

ⁱ

A

^θ1

M ∂

1ⁱ

c

^(β−i)_i

! f ds

1 L

Z

L 0

A

^θ−1₁

X

i6α

(−1)

ⁱ

M (α

1

∂

₁ⁱ⁺⁴

− α

2

∂

₁ⁱ⁺²

)c

^(β−i)_i

!

f ds (4.19)

that yields

k(I − M ) B A

^θ1

M fk

_L2(I)

6 C X

i6α

kc

^(β−i)_i

k

_Hi+4(I)

!

kfk

_L2(I)

. (4.20)

For the last term in (4.16), we simply write k(A

^θ1

M B )(I − M )fk

L²(I)

=

(A

^θ₁

M c

^(β)₀

) 1 L

Z

L 0

f ds

L²(I)

6 Ckc

^(β)₀

k

_H4θ(I)

kfk

L²(I)

.

Combining this, (4.20) and (4.18), we deduce the result.

From Proposition 4.2 and (4.14), we deduce in particular that if η

⁰₁

∈ W

^7,∞

(I) then for all ε > 0 there exists C = C(ε) > 0 such that

[A

^3/8₁

M, D ]f

L²(I)

6 C kf k

_H3/2+ε(I)

(4.21)

and

[A

^3/8₁

M, L ]f

L²(F₀)

6 C kfk

_H5/2+ε(F₀)

,

[A

^3/8₁

M, G ]f

L²(F₀)

6 C kfk

_H3/2+ε(F₀)

. (4.22) We are in position to prove Lemma 4.1:

Proof of Lemma 4.1. We recall that K

λ

is given by (3.10), (3.11), (3.2) and (3.1). After the change of variables, we have formulas (4.10), (4.11) and (4.14) and thus

[A

^3/8₁

, K

λ

]η = M h

ν D , A

^3/8₁

M i

ϕ e

η

(s, 1) − M ν De ϕ(s, 1) + M π(s, e 1) (4.23) where

ϕ e = A

^3/8₁

M ϕ e

η

− ϕ e

A^3/8₁ η

and e π = A

^3/8₁

M π e

η

− e π

A^3/8₁ η

. (4.24)

From (4.6) and (4.7), we have







λ ϕ e − ν Le ϕ + Ge π = w e + [A

^3/8₁

M, ν L ] ϕ e

η

− [A

^3/8₁

M, G ] e π

η

in F

0

, div ϕ e = 0 in F

0

,

ϕ e = 0 on ∂F

0

(4.25) where

w e = A

^3/8₁

M w e

η

− w e

A^3/8₁ η

and q e = A

^3/8₁

M e q

η

− e q

A^3/8₁ η

satisfy







λ w e − ν L w e + Ge q = [A

^3/8₁

M, ν L ] w e

η

− [A

^3/8₁

M, G ] q e

η

in F

0

, div w e = 0 in F

0

,

w e = 0 on ∂F

0

.

(4.26) From (3.15), (3.16) and using the change of variables in Section 4.1, we deduce

k wk e

_L2(F₀)

6 C|λ|

⁻¹

[A

^3/8₁

M, ν L ] w e

η

− [A

^3/8₁

M, G ] q e

η

L²(F₀)

(4.27)

and

k ϕk e

_H2(F₀)

+ k πk e

_H1(F₀)

6 C

e w + [A

^3/8₁

M, ν L ] ϕ e

η

− [A

^3/8₁

M, G ] e π

η

L²(F₀)

. (4.28) From (4.27), (4.22), (3.22) and using the change of variables in Section 4.1, we deduce

k wk e

_L2(F₀)

6 C|λ|

⁻¹

k w e

η

k

_H5/2+ε(F₀)

+ k q e

η

k

_H3/2+ε(F₀)

6 C

|λ|

⁻¹

kA

^1/2+ε/2₁

ηk

H_S

+ |λ|

^ε

kηk

H_S

. (4.29) Using (4.28) and (4.22), we find

k ϕk e

H²(F₀)

+ k e πk

H¹(F₀)

6 C

k wk e

L²(F₀)

+ k ϕ e

η

k

_H5/2+ε(F₀)

+ k e π

η

k

_H3/2+ε(F₀)

. (4.30)

From (3.16), (3.18), (3.20) and using the change of variables in Section 4.1, we deduce k ϕ e

η

k

_H5/2+ε(F₀)

+ k e π

η

k

_H3/2+ε(F₀)

6 C

|λ|

^ε

k w e

η

k

_H1/2−ε(F₀)

+ k w e

η

k

_H1/2+ε(F₀)

6 C

|λ|

^ε

kηk

H_S

+ kA

^ε/2₁

ηk

H_S

. (4.31)

Combining the above equation with (4.29) and (4.30), we deduce k ϕk e

H²(F₀)

+ k e πk

H¹(F₀)

6 C

|λ|

⁻¹

kA

^1/2+ε/2₁

ηk

H_S

+ |λ|

^ε

kηk

H_S

+ kA

^ε/2₁

ηk

H_S

. (4.32)

By using the above estimate, (4.23), (4.31), (4.21), and trace estimates,

[A

^3/8₁

, K

λ

]η

H_S

6

M h

D , A

^3/8₁

M i ϕ e

η

(·, 1)

H_S

+ kM De ϕ(·, 1)k

_H

S

+ kM e π(·, 1)k

_H

S

6 C k ϕ e

η

k

_H2+ε(F₀)

+ k ϕk e

_H2(F₀)

+ k πk e

_H1(F₀)

6 C

|λ|

⁻¹

kA

^1/2+ε/2₁

ηk

H_S

+ |λ|

^ε

kηk

H_S

+ kA

^ε/2₁

ηk

H_S

. The conclusion follows from

kA

^ε/2₁

ηk

H_S

6 (|λ|

^ε

kηk

H_S

)

^1+ε1

|λ|

⁻¹

kA

^1/2+ε/2₁

ηk

H_S

_1+ε^ε

6 C

|λ|

^ε

kηk

H_S

+ |λ|

⁻¹

kA

^1/2+ε/2₁

ηk

H_S

.

5 Estimation of V e _λ ⁻¹

The aim of this section is to estimate V e

_λ⁻¹

where V e

λ

is defined by (3.13). We recall that the notation C

⁺α

is introduced in (2.13).

Theorem 5.1. There exists α > 0 such that for all λ ∈ C

⁺α

the operator V e

λ

: D(A

1

) → H

S

is an isomorphism and for θ ∈ [0, 1] the following estimates hold

sup

λ∈C⁺α

|λ|

^3/2−2θ

kA

^θ1

V e

_λ⁻¹

k

L(H_S)

< +∞. (5.1)

Moreover,

sup

λ∈C⁺α

|λ|

^3/2−2θ

kA

^θ1

V e

_λ^∗−1

k

L(H_S)

< +∞. (5.2)

Proof. Note that it is sufficient to consider the cases θ = 0 and θ = 1, the other cases are obtained by interpolation.

Let us consider λ ∈ C

⁺α

and η ∈ D(A

1

). Then from (3.13)

V e

λ

η λ

²

2

H_S

=

(I + K

λ

)η + 2ρA

^1/4₁

η λ + A

1

η

λ

²

2

H_S

. (5.3)

Using (3.9) and Proposition 3.4, we deduce that

kηk

H_S

6 k(I + K

λ

)

^1/2

ηk

H_S

6 Ckηk

H_S

. (5.4) Thus, combining (5.3) and (5.4), we deduce

V e

λ

η λ

²

2

H_S

>

(I + K

λ

)

^1/2

η + 2ρ (I + K

λ

)

^−1/2

A

^1/4₁

η

λ + (I + K

λ

)

^−1/2

A

1

η λ

²

2

H_S

=

(I + K

λ

)

^−1/2

η + K

λ

η + A

1

η λ

²

2

H_S

+ 4ρ

²

(I + K

λ

)

^−1/2

A

^1/4₁

η λ

2

H_S

+ 4ρ Re 1

λ

A

^1/8₁

η

2

H_S

+ 4ρ Re 1

|λ|

²

λ D

(I + K

λ

)

⁻¹

A

1

η, A

^1/4₁

η E

H_S

. (5.5) We can write

D

(I + K

λ

)

⁻¹

A

1

η, A

^1/4₁

η E

H_S

= k(I + K

λ

)

^−1/2

A

^5/8₁

ηk

²H_S

+ Dh

(I + K

λ

)

⁻¹

, A

^3/8₁

i

A

^5/8₁

η, A

^1/4₁

η E

H_S

= k(I + K

λ

)

^−1/2

A

^5/8₁

ηk

²H_S

+ D

(I + K

λ

)

⁻¹

A

^5/8₁

η, h

A

^3/8₁

, K

λ

i

(I + K

λ

)

⁻¹

A

^1/4₁

η E

H_S

. (5.6)