Gevrey regularity for a system coupling the Navier-Stokes system with a beam equation

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Gevrey regularity for a system coupling the Navier-Stokes system with a beam equation

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 equation. SIAM Journal on Mathematical Analysis, Society for Industrial and Applied

Mathematics, In press, 51 (6), pp.4776-4814. �10.1137/18M1196212�. �hal-02160011�

Gevrey regularity for a system coupling the Navier-Stokes system with a beam equation

Mehdi Badra ¹ and Tak´ eo Takahashi ²

1 Institut de Math´ ematiques de Toulouse, UMR5219; Universit´ e de Toulouse, CNRS;

UPS, F-31062 Toulouse Cedex 9, France

2 Universit´ e de Lorraine, CNRS, Inria, IECL, F-54000 Nancy December 11, 2018

Abstract

We analyse a bi-dimensional fluid-structure interaction system composed by a viscous incompressible fluid and a beam located at the boundary of the fluid domain. Our main result is the existence and uniqueness of strong solutions for the corresponding coupled system. The proof is based on a the study of the linearized system and a fixed point procedure. In particular, we show that the linearized system can be written with a Gevrey class semigroup. The main novelty with respect to previous results is that we do not consider any approximation in the beam equation.

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

Contents

1 Introduction 2

2 The system written in a fixed domain 5

3 The linear system 8

4 Gevrey type resolvent estimates for A

0

10

4.1 Preliminaries . . . . 12 4.2 Estimation of V

⁻¹

(λ) . . . . 16 4.3 Proof of Theorem 4.1 . . . . 19

5 Regularity result for the linear system 20

5.1 Regularity results for Gevrey linear systems . . . . 20 5.2 Proof of Theorem 5.1 . . . . 22

6 Fixed Points 23

6.1 Local in time existence . . . . 24 6.2 Uniqueness . . . . 28 6.3 Small data . . . . 28

A A technical result 29

1 Introduction

In this article, we are interested in the interaction of a Navier-Stokes fluid and of a beam. More precisely, we consider a planar domain F(t) where evolves a viscous incompressible fluid modeled by the classical Navier- Stokes system. The fluid domain is a transformation of an infinite horizontal strip through the displacement of a beam located at its upper side. More precisely, we write for any η : R → (−1, ∞)

F

#

(η)

^def

=

(x

1

, x

2

) ∈ R

²

, x

1

∈ R , x

2

∈ (0, 1 + η(x

1

)) , (1.1) Γ

#

(η)

^def

= {(s, 1 + η(s)), s ∈ R } , Γ

fix,#

def

= R × {0}. (1.2) Then the fluid domain writes F(t)

^def

= F

#

(η(t)) where η(t, s) (t > 0, s ∈ R ) is the vertical displacement of the beam. If we denote by v and p the velocity and the pressure of the fluid, our system is the following



 

 

 

 



 

 

 

 



∂

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 ∈ R , v = 0 t > 0, x ∈ Γ

fix,#

,

v Le

1

− periodic t > 0,

∂

tt

η + α

1

∂

ssss

η − α

2

∂

ss

η = −e H

^η

(v, p), t > 0, s ∈ R , η L − periodic t > 0,

(1.3)

with the initial conditions

η(0) = η

1⁰

, ∂

t

η(0) = η

⁰2

and v(0, ·) = v

⁰

. (1.4) We focus in this article in the case of a periodic solution in the direction e

1

, where (e

1

, e

2

) is the canonical basis of R

²

. We have also used the following notation:

T (v, p)

^def

= 2νD(v) − pI

2

, D(v) = 1

2 (∇v + (∇v)

^∗

) , (1.5)

H e

η

(v, p)(t, s)

^def

= n

(1 + |∂

s

η(t, s)|

²

)

^1/2

[ T (v, p)n] (t, s, 1 + η(t, s)) · e

2

o

. (1.6)

Above, the constant ν > 0 is the viscosity and the vector fields n is the unit exterior normal to F

#

(η(t)) and in particular, 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)

The constants α

1

and α

2

are assumed to satisfy

α

1

> 0, α

2

> 0. (1.8)

Due the spatial periodicity, we also use the following notation (see Figure 1): for any η : [0, L] → (−1, ∞), F(η)

^def

=

(x

1

, x

2

) ∈ R

²

, x

1

∈ (0, L), x

2

∈ (0, 1 + η(x

1

)) , (1.9) Γ(η)

^def

= {(s, 1 + η(s)), s ∈ (0, L)} , Γ

fix

def

= (0, L) × {0}. (1.10) A formal calculation on system (1.3) shows that

d dt

Z

L 0

η(t, s) ds = 0.

For simplicity, and without loss of generality, we assume that the mean value of η

⁰1

is zero and thus Z

L

0

η(t, s) ds = 0 (t > 0). (1.11)

This leads us to consider the following spaces

L

^p_#

(0, L)

^def

= {f ∈ L

^p_loc

( R ) ; f(· + L) = f} (p ∈ [1, ∞]), (1.12)

F(η)

Γ

fix

Γ

0

Γ

L

Γ(η)

0 L

Figure 1: Our geometry

L

^p_#,0

(0, L)

^def

=

f ∈ L

^p_#

(0, L) ; Z

L

0

f(s) ds = 0

(p ∈ [1, ∞]), (1.13)

H

_#,0^α

(0, L)

^def

= H

_loc^α

( R ) ∩ L

²_#,0

(0, L) (α > 0), (1.14) and the orthogonal projection

M : L

²_#

(0, L) → L

²_#,0

(0, L).

We also define the operator for the structure:

H

S

def

= L

²#,0

(0, L), D(A

1

)

^def

= H

#,0⁴

(0, L), (1.15) A

1

: D(A

1

) → H

S

, η 7→ α

1

∂

ssss

η − α

2

∂

ss

η. (1.16) One can check that for any α ∈ [0, 1],

D(A

^α1

) = H

#,0^4α

(0, L). (1.17)

Note that from relation (1.11), the pressure is not determined up to a constant as in the usual Navier-Stokes system. Indeed, from the beam equation in (1.3) and relation (1.11) we deduce

Z

L 0

p(t, s, η(t, s)) − 2ν n

(1 + |∂

s

η|

²

)

^1/2

[D(v)n] (t, s, 1 + η(t, s)) · e

2

o ds = 0

that can determine the constant for the pressure. In what follows, we only write the projection of the beam equation in H

S

:

∂

tt

η + A

1

η = − H

^η

(v, p), t > 0, (1.18) where

H

η

(v, p)

^def

= M H e

η

(v, p). (1.19)

In what follows, we need spaces of the form H

¹

(0, T ; L

^α

(F(η))) and 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

^α

(F (η))) if y 7→ v(t, y

1

, y

2

(1+η(t, y

1

)) is in H

¹

(0, T ; L

^α

(F(0))) and similarly, v ∈ L

²

(0, T ; H

^k

(F(η))) if y 7→ v(t, y

1

, y

2

(1 + η(t, y

1

))) is in L

²

(0, T ; H

^k

(F(0))). We have used the following notation: L

^α

for the Lebesgue spaces, H

^k

for the Sobolev spaces and C

b

for the set of the 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.

As explained above, our aim is to show the existence and uniqueness of strong solutions for system (1.3).

We mean by a strong solution of system (1.3) a strong solution for the fluid equations: that is

v ∈ L

²

(0, T ; H

²

(F (η)) ∩ C

b

([0, T ]; H

¹

(F(η))) ∩ H

¹

(0, T ; L

²

(F(η))), p ∈ L

²

(0, T ; H

¹

(F (η))) (1.20) and the first four equations of (1.3) are satisfied almost everywhere or in the trace sense and a solution for the structure equation with the regularity

η ∈ L

²

(0, T ; H

_#,0^7/2

(0, L)) ∩ C

b

([0, T ]; H

_#,0^5/2

(0, L)) ∩ H

¹

(0, T ; H

_#,0^3/2

(0, L)),

∂

t

η ∈ L

²

(0, T ; H

_#,0^3/2

(0, L)) ∩ C

b

([0, T ]; H

_#,0^1/2

(0, L)) ∩ H

¹

(0, T ; (H

_#,0^1/2

(0, L))

⁰

),

(1.21)

and (1.18) holds in L

²

(0, T ; H

_#,0^1/2

(0, L)

⁰

).

Remark 1.1. From the third equation of (1.3), we see that the regularity of v and of ∂

t

η are related by trace theorems. The regularity (1.21) of ∂

t

η that we consider is thus quite natural when looking at the regularity (1.20) of v.

We are now in position to state our main results. We assume there exists ε > 0 such that

η

⁰₁

∈ H

_#,0^3+ε

(0, L), η

⁰₂

∈ H

_#,0^1+ε

(0, L) (1.22) with

η

₁⁰

> −1 in R . (1.23)

and

v

⁰

∈ H

¹

(F(η

1⁰

)), (1.24)

with

div v

⁰

= 0 in F

#

(η

1⁰

), (1.25)

v

⁰

Le

1

− periodic, (1.26)

v

⁰

(s, 1 + η

₁⁰

(s)) = η

⁰₂

(s)e

2

s ∈ R , (1.27)

v

⁰

= 0 on Γ

fix,#

. (1.28)

We first give the existence and uniqueness of strong solutions for small times.

Theorem 1.2. Assume [v

⁰

, η

₁⁰

, η

⁰₂

] satisfies (1.22)–(1.28). Then there exist T > 0 and C

0

> 0 such that if

kη

₁⁰

k

_H2(0,L)

6 C

0

(1.29)

there exists a strong solution (η, v, p) of (1.3) with

η(t, ·) > −1 t ∈ [0, T ]. (1.30)

This solution is unique locally: if (η

^(∗)

, v

^(∗)

, p

^(∗)

) is another solution, there exists T

^∗

> 0 such that (η

^(∗)

, v

^(∗)

, p

^(∗)

) = (η, v, p) on [0, T

^∗

].

We can also obtain the following result

Theorem 1.3. There exists C

0

> 0 such that for any [v

⁰

, η

₁⁰

, η

₂⁰

] satisfying (1.22)–(1.28) and [v

⁰

, η

⁰₁

, η

⁰₂

]

H¹(F(η₁⁰))×H^3+ε

#,0(0,L)×H^1+ε

#,0(0,L)

6 C

0

, there exists a solution (η, v, p) of (1.3) with

η(t, ·) > −1 t ∈ [0, ∞). (1.31)

Let us give some comments about our main results. Theorem 1.2 states a local in time existence of strong solutions whereas Theorem 1.3 gives the global existence of strong solutions for small initial conditions. We do not recover the global in time existence of strong solutions as for the standard Navier-Stokes equations.

Moreover, we see that there is a loss of regularity for (η, ∂

t

η): we have the continuity of (η, ∂

t

η) in H

_#,0^5/2

(0, L)×

H

_#,0^1/2

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

_#,0^3+ε

(0, L) × H

_#,0^1+ε

(0, L) for some ε > 0.

Finally, the uniqueness holds only locally in time. All the above points are due to the coupling of the Navier-Stokes equations with the beam equation which modify the nature of the Navier-Stokes system.

More precisely, the linearized system (3.23) that we consider in Section 3 is composed by a Stokes system and a beam equation and the corresponding semigroup is not analytic but only of Gevrey class (see Section 5). This is stated in Theorem 5.1 and is a part of our main results.

Another important remark is that in this work we have focused on a particular geometry: our linear

result, Theorem 5.1, is proved in the case where the fluid domain is a rectangle. This explains why we need

the smallness conditions (1.29) in Theorem 1.2. This hypothesis on the geometry implies the commutativity

of some operators (see Proposition 4.6) and this simplifies the resolvent estimates in Section 4. The result

should hold true in a general geometry, but the corresponding proof should be more involved. This will be

the subject of a forthcoming paper.

The model presented above, mainly system (1.3) was proposed in [16] for a model a blood flow in a vessel.

It is important to remark that in their model, the beam equation is damped by a term of the form −δ∂

tss

η.

More precisely in (1.3), the beam equation is replaced by

∂

tt

η + α

1

∂

ssss

η − α

2

∂

ss

η − δ∂

tss

η = −e H

^η

(v, p). (1.32) Several authors have studied this model: [5] (existence of weak solutions), [2], [14] and [10] (existence of strong solutions), [17] (stabilization of strong solutions), [1] (stabilization of weak solutions around a stationary state). In all these works, the damping term plays an important role. In particular, with this term, the linearized system is parabolic that is the underlying semigroup is analytic.

Up to our knowledge, there exists only one result in the case without damping, that is (1.32) for δ = 0:

the existence of weak solutions is obtained in [9] (by passing to the limit δ → 0). Note that recently in [11], the authors show the existence of local strong solutions for a structure described by either a wave equation (α

1

= δ = 0 and α

2

> 0 in (1.32)) or a beam equation with inertia of rotation (α

1

> 0, α

2

= δ = 0 and with an additional term −∂

ttss

η in (1.32)).

In particular, they do not treat our case (α

1

> 0, α

2

> 0 and δ = 0 in (1.32)) and therefore, our work gives the first results of existence and uniqueness of strong solutions in the case of an undamped beam equation.

Moreover, we develop a new general approach for the analysis of fluid-structure interaction systems based on Gevrey class semigroups. More precisely, our idea consists in linearizing the problem and in showing that the linearized system (that is (3.23) in our case) is of Gevrey class (see [19] for this notion). We then derive some regularity properties for our linear system and perform a fixed point argument to deduce the well-posedness of system (1.3). Several works considered Gevrey class semigroups, but this is usually done for elastic structures: [6], [7], [18], [8], [22], etc.

The outline of the article is as follows: in Section 2, we perform a change of variables to write system (1.3) in a cylindrical domain. The rest of the paper concerns the resulting system. Section 3 presents the linear system associated with this nonlinear system, and we show in Section 4 the Gevrey class of this system by estimating the resolvent of the corresponding operator. In Section 5, we prove some regularity properties of the linear system. Part of this section is general for any Gevrey class systems. Finally, Section 6 is devoted to the proof of the main results, Theorem 1.2 and Theorem 1.3, by using a fixed point argument.

Notation

We complete here some notation that we use all along the paper. We denote by L(X

1

, X

2

) the space of the bounded linear operators from X

1

to X

2

. We also set for T ∈ (0, ∞]

W (0, T ; X

1

, X

2

)

^def

=

w ∈ L

²

(0, T ; X

1

); dw

dt ∈ L

²

(0, T ; X

2

)

. We recall (see [3, Rem. 4.1 p. 156 and Prop. 4.3 p. 159]) the following embediing

W (0, ∞; X

1

, X

2

) , → C

b

([0, ∞), [X

1

, X

2

]

1/2

), (1.33) where [X

1

, X

2

]

θ

denotes the complex interpolation method.

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.

2 The system written in a fixed domain

We transform the system (1.3) written in the non cylindrical domain [

t>0

{t} × F(η(t)) into a system written in the domain

(0, ∞) × F, where

F

^def

= F(0) = (0, L) × (0, 1). (2.1)

We also define F

#

def

= R × (0, 1), Γ

^def

= Γ(0) = (0, L) × {1}, Γ

#

def

= Γ

#

(0) = R × {1}, (2.2) and

Γ

b

def

= Γ ∪ Γ

fix

, Γ

b,#

def

= Γ

#

∪ Γ

fix,#

= ∂F

#

, (2.3)

where Γ

fix

, Γ

fix,#

are defined in (1.2) and (1.10).

Using the particular geometry of the problem, one can consider the general changes of variables X

_η1,η²

: F(η

¹

) → F(η

²

), (y

1

, y

2

) 7→

y

1

, y

2

1 + η

²

(y

1

) 1 + η

¹

(y

1

)

, (2.4)

whose inverse is X

_η2,η¹

. Our change of variables is thus defined by

X (t, ·)

^def

= X

0,η(t)

: (y

1

, y

2

) 7→ (y

1

, y

2

(1 + η(t, y

1

))) , (2.5) Y (t, ·)

^def

= X (t, ·)

⁻¹

= X

η(t),0

: (x

1

, x

2

) 7→

x

1

, x

2

1 1 + η(t, x

1

)

. (2.6)

We write

a

^def

= Cof(∇Y )

^∗

, b

^def

= Cof(∇X )

^∗

. (2.7) We set

w(t, y)

^def

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

^def

= p(t, X(t, y)), (2.8) so that

v(t, x) = a(t, x)w(t, Y (t, x)) and p(t, x) = q(t, Y (t, x)). (2.9) Remark 2.1. Note that we use the cofactor matrix Cof(∇X ) of ∇X in (2.8). Such a change of variables allows us to keep the divergence free condition and the structure of the boundary conditions (see [13], [4], [1, Lemma 2.3]).

Then some calculation yields [b∆v(X)]

_α

= X

i,j,k

b

αi

∂

²

a

ik

∂x

²_j

(X)w

k

+ 2 X

i,j,k,`

b

αi

∂a

ik

∂x

j

(X ) ∂w

k

∂y

`

∂Y

`

∂x

j

(X )

+ X

j,`,m

∂

²

w

α

∂y

`

∂y

m

∂Y

`

∂x

j

(X) ∂Y

m

∂x

j

(X ) + X

j,`

∂w

α

∂y

`

∂

²

Y

`

∂x

²_j

(X), (2.10) [b [(v · ∇)v] (X )]

_α

= X

i,j,k,m

b

αi

∂a

ik

∂x

j

(X )a

j,m

(X)w

k

w

m

+ 1

det(∇X ) [(w · ∇)w]

_α

, (2.11) [b∇p(X)]

_α

= det(∇X) X

k,i

∂q

∂y

k

∂Y

α

∂x

i

(X) ∂Y

k

∂x

i

(X ), (2.12)

b∂

t

v(X ) = b(∂

t

a)(X )w + ∂

t

w + (∇w)(∂

t

Y )(X). (2.13) For the other equations of (1.3), we need to use in a more precise way (2.5) and (2.6). We have the following formulas

∇X (t, y

1

, y

2

) =

1 0 y

2

∂

s

η 1 + η

, b(t, y

1

, y

2

) =

1 + η 0

−y

2

∂

s

η 1

, (2.14)

∇Y (t, x

1

, x

2

) =





1 0

−x

2

∂

s

η (1 + η)

²

1 1 + η



 , a(t, x

1

, x

2

) =





 1

1 + η 0

x

2

∂

s

η (1 + η)

²

1





 . (2.15)

Thus the boundary condition of (1.3) on Γ

#

(η) rewrites

w(s, 1) = ∂

t

η(t, s)e

2

t > 0, s ∈ R .

We also deduce from (1.5), (1.6), (1.7), (1.19) and (2.5) that H

η

(v, p)(t, s) = M

[ν(∇v + (∇v)

^∗

) − pI

2

] ◦ X(t, s, 1)

−∂

s

η(t, s) 1

· e

2

= −M q(t, s, 1) + M ("

ν X

k

∂a

ik

∂x

j

(X)w

k

+ ν X

k,`

a

ik

(X ) ∂w

k

∂y

`

∂Y

`

∂x

j

(X ) + ν X

k

∂a

jk

∂x

i

(X)w

k

+ ν X

k,`

a

jk

(X) ∂w

k

∂y

`

∂Y

`

∂x

i

(X )

#

i,j

(t, s, 1)

−∂

s

η(t, s) 1

· e

2

)

. (2.16) Moreover, we recall that

H

0

(w, q)(t, s) = M {[ T (w, q)e

2

] (t, s, 1) · e

2

} . (2.17) In particular,

G(η, w)(t, s) b

^def

= H

0

(w, q)(t, s) − H

η

(v, p)(t, s)

= νM (

2 X

k,`

δ

2,k

δ

2,`

− a

2k

(X ) ∂Y

`

∂x

2

(X) ∂w

k

∂y

`

+ ∂

s

η



 X

k,`

a

2k

(X ) ∂w

k

∂y

`

∂Y

`

∂x

1

(X) + X

k,`

a

1k

(X) ∂w

k

∂y

`

∂Y

`

∂x

2

(X)





+ X

k

∂

s

η ∂a

2k

∂x

1

(X ) + ∂a

1k

∂x

2

(X )

− 2 ∂a

2k

∂x

2

(X )

w

k

)

. (2.18) We also define

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

, (2.19) Then system (1.3), (1.4) rewrites,



 

 

 



 

 

 



∂

t

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

#

, div w = 0 in (0, ∞) × F

#

,

w(t, s, 1) = (∂

t

η)(t, s)e

2

t > 0, s ∈ R , w = 0 t > 0, y ∈ Γ

fix,#

, w Le

1

− periodic t > 0,

∂

tt

η + A

1

η = − H

0

(w, q) + G(η, w), b t > 0,

(2.20)

with the initial conditions

η(0) = η

₁⁰

, ∂

t

η(0) = η

⁰₂

and w(0, y) = w

0

(y)

^def

= b(0, y)v

0

(X(0, y)), y ∈ F. (2.21) Remark 2.2. We can notice that in the reference geometry we use here (where n = e

2

on Γ),

H

0

(w, q)(t, s) = −M q(t, s, 1).

This can be done by standard calculation. Nevertheless, this particular form of H

⁰

is never used in what

follows.

Using (2.14) and (2.15), we can precise some terms of the above nonlinearities:

a(X ) =





 1 1 + η 0 y

2

∂

s

η 1 + η 1





 , ∇Y (X ) =





1 0

−y

2

∂

s

η 1 + η

1 1 + η



 , (2.22)

∇Y (X) − I

2

=





0 0

−y

2

∂

s

η 1 + η

−η 1 + η



 , det(∇X) = 1 + η, (2.23)

∂a

∂x

1

(X ) =









−∂

s

η

(1 + η)

²

0

y

2

∂

ss

η(1 + η) − 2(∂

s

η)

²

(1 + η)

²

0







 , ∂a

∂x

2

(X ) =





0 0

∂

s

η (1 + η)

²

0



 , (2.24)

∂

²

a

∂x

1

∂x

2

(X) =





0 0

∂

ss

η(1 + η) − 2(∂

s

η)

²

(1 + η)

³

0



 , (2.25)

∂

²

a

∂x

²₁

(X ) =









−∂

s

η

(1 + η)

²

0

y

2

∂

sss

η(1 + η)

²

− 6(1 + η)∂

s

η∂

ss

η + 6(∂

s

η)

³

(1 + η)

³

0







 , ∂

²

a

∂x

²₂

(X) = 0, (2.26)

∂

∂x

1

∇Y (X ) =





0 0

y

2

−∂

ss

η(1 + η) + 2(∂

s

η)

²

(1 + η)

²

−∂

s

η (1 + η)

²



 , ∂

∂x

2

∇Y (X) =





0 0

− ∂

s

η (1 + η)

²

0



 , (2.27)

∂

t

a(X ) =







−∂

t

η

(1 + η)

²

0

y

2

∂

ts

η(1 + η) − 2∂

s

η∂

t

η

(1 + η)

²

0





 , ∂

t

Y (X ) =



 0

−y

2

∂

t

η 1 + η



 . (2.28)

3 The linear system

In this section, we consider a linear system associated with (2.20). This linear system is similar to the one introduced in [1] in the case of a damped beam equation and we use several results of this previous work.

We recall that F, F

#

, Γ, Γ

#

, Γ

b

, Γ

b,#

are defined by (2.1), (2.2) and (2.3), that L

^α

and H

^k

stand for Lebesgue spaces and Sobolev spaces respectively, and that we use the bold notation for the spaces of vector fields: L

^α

= (L

^α

)

²

, H

^k

= (H

^k

)

²

etc. We also recall that L

^p_#

(0, L), L

^p_#,0

(0, L) and H

#,0^α

(0, L) are defined by (1.12)–(1.14).

We consider the following functional spaces:

L

^p_#

(Γ

b

)

^def

= {f ∈ L

^p_loc

(Γ

b,#

) ; f(· + Le

1

) = f} (p ∈ [1, ∞]), (3.1) L

^p_#,0

(Γ

b

)

^def

=

f ∈ L

^p_#

(Γ

b

) ; Z

Γ_b

f · n dγ = 0

(p ∈ [1, ∞]), (3.2)

H

#^α

(Γ

b

)

^def

= H

loc^α

(Γ

b,#

) ∩ L

²#

(Γ

b

) (α > 0), (3.3) V

^α#

(Γ

b

)

^def

= H

^α#

(Γ

b

) ∩ L

²#,0

(Γ

b

) (α > 0), (3.4) L

^p_#

(F )

^def

= {f ∈ L

^p_loc

(F

#

) ; f(· + Le

1

) = f} (p ∈ [1, ∞]), (3.5) H

#^α

(F)

^def

= H

loc^α

(F

#

) ∩ L

²#

(F) (α > 0), (3.6)

V

^α#

(F)

^def

=

f ∈ H

^α#

(F) ; div f = 0 , (3.7)

V

_#,n^α

(F)

^def

=

f ∈ H

^α_#

(F) ; div f = 0, f · n = 0 on Γ

b

(α ∈ [0, 1/2)), (3.8) V

#,n^α

(F)

^def

=

f ∈ H

^α#

(F) ; div f = 0, f = 0 on Γ

b

(α ∈ (1/2, 1]). (3.9)

We introduce the operator Λ : L

²_#

(0, L) → L

²_#

(Γ

b

) defined by (Λη)(y) = (M η(y

1

)) e

2

if y ∈ Γ,

(Λη)(y) = 0 if y ∈ Γ

fix

. (3.10)

The adjoint Λ

^∗

: L

²_#

(Γ

b

) → L

²_#

(0, L) of Λ is given by

(Λ

^∗

v)(s) = M (v(s, 1) · e

2

) (s ∈ (0, L)). (3.11) Note that for any α ∈ [0, 4],

Λ ∈ L(H

#^α

(0, L), V

^α#

(Γ

b

)) (3.12) and

Λ

^∗

∈ L(H

^α#

(Γ

b

)), D(A

^α/4₁

)). (3.13) We recall that A

1

is defined in (1.15), (1.16) and satisfies (1.17). Moreover,

kΛηk

H^α_#(Γ_b)

> c(α)kA

^α/4₁

ηk

H_S

(η ∈ D(A

^α/4₁

)). (3.14) Note that H

0

(see (2.17)) can be written in a simpler way as

H

0

(w, q) = Λ

^∗

n

T (w, q)n

|Γ_b,#

o

. (3.15)

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 Γ

b,#

, div w = 0 in F o

, (3.16) V

^def

= n

[w, η

1

, η

2

] ∈ H

¹#

(F) × D(A

^3/4₁

) × D(A

^1/4₁

) ; w = Λη

2

on Γ

b,#

, 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

:

Lemma 3.1. For any s ∈ [0, 1],

P

0

∈ L(H

^s#

(F ) × D(A

^1/2+s/4₁

) × D(A

^s/4₁

)), (3.17) and

P

0

∈ L(L

²#

(F) × D(A

^3/8₁

) × D(A

^1/8₁

)

⁰

). (3.18) Proof. We have proven in [1, Proposition 3.1 and Proposition 3.2] relation (3.17) and that for any [w, η

1

, η

2

] ∈ L

²

(F) × D(A

^1/2₁

) × H

S

,

P

0



 w η

1

η

2



 =





w − ∇p η

1

η

2

+ Λ

^∗

(pn)





where the pressure function p ∈ H

¹

(F) obeys Z

F

p dy = 0 and is solution to the Neumann problem:



 

 

∀q ∈ H

¹

(F ) such that Z

F

q dy = 0, Z

F

∇p · ∇q dy +

Λ

^∗

(pn), Λ

^∗

(qn)

H_S

= Z

F

w · ∇q dy −

η

2

, Λ

^∗

(qn)

H_S

.

(3.19)

From (3.19) and by using the trace inequality kpk

_H1/2(Γ)

6 Ck∇pk

_L2(F)

and (3.13) for α = 1/2, we deduce that

k∇pk

²_L2(F)

+ kΛ

^∗

(pn)k

²H_S

6 C(kwk

L²(F)

k∇pk

L²(F)

+ kA

^−1/8₁

η

2

k

H_S

kA

^1/8₁

Λ

^∗

(pn)k

H_S

),

6 C(kwk

L²(F)

+ kA

^−1/8₁

η

2

k

H_S

)k∇pk

L²(F)

,

from which, we obtain (3.18).

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

, (3.20)

and for

w, η

1

, η

2

∈ D(A

0

), we set

A e

0



 w η

1

η

2





def

=











ν∆w η

2

−A

1

η

1

− Λ

^∗

(2νD(w)n)











(3.21)

and

A

0

def

= P

0

A e

0

. (3.22)

Remark 3.2. As already pointed out in Remark 2.2, by using the particular geometry of F , we can simplify the above expression since Λ

^∗

(2νD(w)n) = 0. This simplification is not used in this paper and several results of the next sections remain true for a general geometry. The important consequence of our geometry in our work corresponds to the commutativity of some operators (see Proposition 4.6).

By using the above operators, we can rewrite the following linear system



 

 



 

 



∂

t

w − div T (w, q) = F in (0, ∞) × F

#

, div w = 0 in (0, ∞) × F

#

,

w(t, s, 1) = (∂

t

η)(t, s)e

2

(t, s) ∈ (0, ∞) × R , w(t, ·) Le

1

− periodic t ∈ (0, ∞),

∂

tt

η + A

1

η = − H

0

(w, q) + G in (0, ∞),

(3.23)

with the initial conditions

w(0, ·) = w

⁰

, η(0, ·) = η

⁰₁

, ∂

t

η(0, ·) = η

⁰₂

, (3.24) 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



 . (3.25)

We have the following result (see [1, Proposition 3.4, Proposition 3.5 and Remark 3.6]).

Proposition 3.3. The operator A

0

defined by (3.20)–(3.22) 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 [1, Proposition 3.8]).

Proposition 3.4. 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) , (3.26)

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 Γ

b,#

o

if θ ∈ (1/4, 1]. (3.27)

4 Gevrey type resolvent estimates for A ₀

We introduce the notation

C

^+def

= {λ ∈ C ; Re(λ) > 0} , (4.1)

and

C

⁺α def

=

λ ∈ C

⁺

; |λ| > α . (4.2)

The goal of this section it to prove the following result on the operator A

0

defined by (3.20)–(3.22).

Theorem 4.1. There exists C > 0 such that for all λ ∈ C

⁺

|λ|

^1/2

(λ − A

0

)

⁻¹

_L(H)

6 C. (4.3)

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

⁺

and for all f, g, h

∈ H ∩

L

²#

(F) × D(A

^5/8₁

) × D(A

^1/8₁

) , the following estimates hold

(λ − A

0

)

⁻¹



 f g h





H²_#(F)×D(A^7/8₁ )×D(A^3/8₁ )

6 C



 f g h





L²_#(F)×D(A^5/8₁ )×D(A^1/8₁ )

(4.4)

and

|λ|

(λ − A

0

)

⁻¹



 f g h





L²_#(F)×D(A^3/8₁ )×D(A^1/8₁ )⁰

6 C



 f g h





L²_#(F)×D(A^5/8₁ )×D(A^1/8₁ )

. (4.5)

Remark 4.2. A consequence of (4.3) is the following resolvent estimate, sup

τ∈R

|τ|

^1/2

k(ıτ − A

0

)

⁻¹

k

L(H)

< +∞,

which implies in particular that (e

^tA0

)

_t>0

is of Gevrey class δ for all δ > 2 (see [19]), namely, for all compact K ⊂ (0, +∞) and θ > 0 there exists C = C(θ, K) such that for all t ∈ K and all n ∈ N ,

d

ⁿ

e

^tA0

dt

ⁿ

L(H)

6 Cθ

ⁿ

(n!)

^δ

.

In order to prove Theorem 4.1, we rewrite the resolvent equation in a more convenient way. Assume λ ∈ C

⁺

and [f, g, h] ∈ H. We set [v, η

1

, η

2

] := (λ − A

0

)

⁻¹

[f, g, h] so that



 

 

 

 

 

 

 

 

λv − div T (v, p) = f in F

#

, div v = 0 in F

#

, v = Λη

2

on Γ

b,#

,

v Le

1

− periodic, λη

1

− η

2

= g, λη

2

+ A

1

η

1

= −Λ

^∗

n

T (v, p)n

|Γ_b,#

o + h.

(4.6)

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 Γ

b,#

, w

η

Le

1

− periodic.

(4.7)

We also define the operator

D

0

(λ)η

^def

= w

η

. (4.8)

We denote by A the Stokes operator:

D( A )

^def

= V

¹_#,n

(F) ∩ H

²_#

(F), A

^def

= ν P ∆ : D( A ) → V

_#,n⁰

(F), (4.9) where P : L

²#

(F) → V

⁰#,n

(F ) is the Leray projection operator.

Using the above notation, we can decompose the solution of (4.6) as

v = D

0

(λ)η

2

+ (λ − A )

⁻¹

P f,

that is (v, p) = (w

η₂

, q

η₂

) + ( b v, p) where (w b

η₂

, q

η₂

) satisfies (4.7) with η = η

2

and where ( v, b p) satisfies b



 



 



λ b v − div T ( v, b p) = b f in F

#

, div b v = 0 in F

#

,

b v = 0 on Γ

b,#

, b v Le

1

− periodic.

(4.10)

In that case, system (4.6) becomes

( λη

1

− η

2

= g λη

2

+ A

1

η

1

= −Λ

^∗

n

T ( b v, p)n b

|Γ_b,#

o − Λ

^∗

n

T (w

η2

, q

η2

)n

|Γ_b,#

o

+ h. (4.11)

This leads us to define the operators T (λ) ∈ L(L

²#

(F), D(A

^1/8₁

)) and L(λ) ∈ L(D(A

^3/8₁

), D(A

^1/8₁

)) (see Proposition 4.3 and (4.31) below) by

T (λ)f

^def

= −Λ

^∗

n

T ( b v, p)n b

|Γ_b,#

o

, (4.12)

L(λ)η

^def

= Λ

^∗

n

T (w

η

, q

η

)n

|Γ_b,#

o

, (4.13)

so that system (4.11) can be written

λη

1

− η

2

= g

λη

2

+ L(λ)η

2

+ A

1

η

1

= T (λ)f + h. (4.14) We thus introduce the operator

A(λ)

^def

=

0 −I A

1

L(λ)

(4.15) and study the equation

(λ + A(λ)) η

1

η

2

= g

b h

, (4.16)

where

b h

^def

= T (λ)f + h.

In what follows, we need the following operator:

V (λ) = λ

²

I + λL(λ) + A

1

. (4.17)

We will prove that V (λ) is invertible (see Proposition 4.8 below), so we can compute the inverse of λ + A(λ) (λ + A(λ))

⁻¹

=





I − V

⁻¹

(λ)A

1

λ V

⁻¹

(λ)

−V

⁻¹

(λ)A

1

λV

⁻¹

(λ)



 , (4.18)

and the inverse of λ − A

0

:

(λ − A

0

)

⁻¹

=











(λ − A )

⁻¹

P + λD

0

(λ)V

⁻¹

(λ)T (λ) −D

0

(λ)V

⁻¹

(λ)A

1

λD

0

(λ)V

⁻¹

(λ) V

⁻¹

(λ)T (λ) I − V

⁻¹

(λ)A

1

λ V

⁻¹

(λ)

λV

⁻¹

(λ)T (λ) −V

⁻¹

(λ)A

1

λV

⁻¹

(λ)











. (4.19)

4.1 Preliminaries

We first recall a standard result on the operator A (see (4.9)) and on the operator T (see (4.12)).

Proposition 4.3. Let θ ∈ [0, 1] and f ∈ L

²_#

(F ). There exists C > 0 such that for any solution ( b v, p) b of (4.10) and for any λ ∈ C

⁺

,

k b vk

_H2θ

#(F)

6 C|λ|

^θ−1

kfk

_L2

#(F)

. (4.20)

In particular, there exists C > 0 such that for any λ ∈ C

⁺

, k(λ − A )

⁻¹

P k

_L(L2

#(F),H²_#(F))

+ |λ|k(λ − A )

⁻¹

P k

_L(L2

#(F))

6 C, (4.21)

and

kT (λ)k

_L(L2

#(F),D(A^1/8₁ ))

6 C. (4.22)

Proof. Using that the Stokes operator A (defined by (4.9)) is the infinitesimal generator of an analytic semigroup and that C

⁺

⊂ ρ( A ), we have the following properties

k(− A )

^θ

(λ − A )

⁻¹

gk

_L2

#(F)

6 C|λ|

^θ−1

kgk

_L2

#(F)

(g ∈ V

#,n⁰

(F), λ ∈ C

⁺

, θ ∈ [0, 1]).

This can be proven by interpolation by showing the case θ = 0 and θ = 1. We deduce (4.20) (and then (4.21)) from the above estimate and from the embedding D((− A )

^θ

) , → H

^2θ#

(F).

We deduce in particular k∇ pk b

_L2

#(F)

6 Ck vk b

_H2

#(F)

+ |λ|k b vk

_L2

#(F)

+ kfk

_L2

#(F)

6 Ckfk

_L2

#(F)

. Using (3.13) and (4.12), we obtain

kT (λ)fk

_D(A1/8

1 )

6 Ck T ( b v, b p)nk

H^1/2_# (Γ_b,#)

6 Ckfk

_L2

#(F)

.

Next we show the following result on the operator D

0

(λ) defined by (4.8).

Proposition 4.4. For any λ ∈ C

⁺

, the operator D

0

(λ) satisfies

D

0

(λ) ∈ L(D(A

^3/8₁

), H

²_#

(F )) ∩ L(D(A

^1/8₁

)

⁰

, L

²_#

(F)).

More precisely, for θ ∈ [0, 1], there exists a constant C > 0 such that the operator D

0

(λ) defined by (4.8) satisfies

kD

0

(λ)ηk

_H2θ

#(F)

6 C

kA

^θ/2−1/8₁

ηk

H_S

+ |λ|

^θ

kA

^−1/8₁

ηk

H_S

(η ∈ D(A

^θ/2−1/8₁

), λ ∈ C

⁺

). (4.23) Proof. Using (3.12) and standard Stokes regularity results, we obtain the first part of the proof.

In order to prove (4.23), we first decompose the solution (w

η

, q

η

) of (4.7) as (w

η

, q

η

) = (v

η

, p

η

) + (z

η

, ζ

η

)

with 

 



 



− div T (v

η

, p

η

) = 0 in F

#

, div v

η

= 0 in F

#

, v

η

= Λη on Γ

b,#

, v

η

Le

1

− periodic,

and



 



 



λz

η

− div T (z

η

, ζ

η

) = −λv

η

in F

#

, div z

η

= 0 in F

#

,

z

η

= 0 on Γ

b,#

, z

η

Le

1

− periodic.

(4.24)

Using again standard Stokes regularity results, we deduce that for any θ ∈ [0, 1], there exists a positive constant C independent of λ such that

kv

η

k

_H2θ

#(F)

6 CkA

^θ/2−1/8₁

ηk

H_S

. (4.25)

On the other hand, from (4.20) in Proposition 4.3 we deduce kz

η

k

_H2θ

#(F)

6 C|λ|

^θ

kv

η

k

_L2

#(F)

. Combining the above estimates with (4.25), we obtain

kz

η

k

_H2θ

#(F)

6 C|λ|

^θ

kA

^−1/8₁

ηk

H_S

. Then (4.23) follows by combining the above inequalities with (4.25).

Using Proposition 4.4, we can define for λ ∈ C

⁺

,

K(λ) ∈ L(D(A

^1/8₁

)

⁰

, D(A

^1/8₁

)), G(λ) ∈ L(D(A

^1/8₁

), D(A

^1/8₁

)

⁰

) by

hK(λ)η, ζi

_D(A1/8

1 ),D(A^1/8₁ )⁰ def

=

Z

F

w

η

· w

ζ

dy (4.26)

and

hG(λ)η, ζi

D(A^1/8₁ )⁰,D(A^1/8₁ ) def

= 2

Z

F

Dw

η

: Dw

ζ

dy. (4.27)

Note that we have

K(λ)η = −Λ

^∗

{ T (ϕ

η

, π

η

)n|

Γ_b,#

} (4.28) where



 



 



λϕ

η

− div T (ϕ

η

, π

η

) = w

η

in F

#

, div ϕ

η

= 0 in F

#

,

ϕ

η

= 0 on Γ

b,#

, ϕ

η

Le

1

− periodic,

(4.29)

and where w

η

is the solution of (4.7).

If η ∈ D(A

^3/8₁

), then we can write 2

Z

F

Dw

η

: Dw

ζ

dy = 2 Z

L

0

Λ

^∗

((Dw

η

)n) ζ ds − Z

F

∆w

η

· w

ζ

dy, and, with Proposition 4.4, we deduce that

G(λ) ∈ L(D(A

^3/8₁

), D(A

^1/8₁

)). (4.30) The operators K(λ) and G(λ) are related to the operator L(λ) defined by (4.13): multiplying (4.7) by w

ζ

and integrating by part, we deduce that

L(λ) = λK(λ) + G(λ). (4.31)

Proposition 4.5. The operators K(λ) ∈ L(D(A

^1/8₁

)

⁰

, D(A

^1/8₁

)) and G(λ) ∈ L(D(A

^1/8₁

), D(A

^1/8₁

)

⁰

) defined above are non negative and self-adjoint. There exist 0 < ρ

1

< ρ

2

such that for any λ ∈ C

⁺

, we have

ρ

1

kA

^1/8₁

ηk

²H_S

6 hG(λ)η, ηi

D(A^1/8₁ )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

)), (4.32) 0 6 hK(λ)η, ηi

_D(A1/8

1 ),D(A^1/8₁ )⁰

6 ρ

2

kA

^−1/8₁

ηk

²H_S

(η ∈ D(A

^1/8₁

)

⁰

). (4.33) Proof. First, by definition, K(λ), G(λ) are symmetric and



 



 



hG(λ)η, ηi

D(A^1/8₁ )⁰,D(A^1/8₁ )

= 2 Z

F

|Dw

η

|

²

dy (η ∈ D(A

^1/8₁

)), hK(λ)η, ηi

D(A^1/8₁ ),D(A^1/8₁ )⁰

= Z

F

|w

η

|

²

dy (η ∈ D(A

^1/8₁

)).

(4.34)

In particular, they are non negative and since K(λ) is bounded, it is self-adjoint.

From (4.34), (3.14) and from the trace theorem, there exists a constant C > 0 such that hG(λ)η, ηi

H_S

> Ckw

η

k

²

H^1/2_# (Γ)

= CkΛηk

²

H^1/2_# (Γ_b)

> CkA

^1/8₁

ηk

²H_S

. This yields the left inequality in (4.32).

The other estimates are a consequence of (4.34) and (4.23).

Finally, from the Lax-Milgram lemma we deduce that G(λ) : D(A

^1/8₁

) → D(A

^1/8₁

)

⁰

is onto. Since it is also symmetric we conclude that G(λ) is self-adjoint (see e.g. [21, Proposition 3.2.4]).

The next result is crucial in our analysis and is due to the particular shape of the domain F . Proposition 4.6. Assume λ ∈ C

⁺

. Then for α ∈ R

L(λ)A

^α1

= A

^α1

L(λ), K(λ)A

^α1

= A

^α1

K(λ), G(λ)A

^α1

= A

^α1

G(λ). (4.35)

Proof. Using the definition (3.10) of the operator Λ and (3.11), we first notice that

∂

y₁

Λη = Λ(∂

s

η) (η ∈ H

_#¹

(0, L)), ∂

s

Λ

^∗

v = Λ

^∗

(∂

y₁

v) (v ∈ H

¹_#

(Γ

b

)).

Thus, we can differentiate system (4.7) with respect to y

1

and we obtain that

∂

y₁

w

η

= w

∂_sη

.

We thus deduce L(λ)A

1

= A

1

L(λ). Using (4.28), we show similarly that K(λ)A

1

= A

1

K(λ) and from (4.31), we finally deduce G(λ)A

1

= A

1

G(λ).

Corollary 4.7. Let ρ

2

the constant introduced in Proposition 4.5. For any λ ∈ C

⁺

, we have kG(λ)ηk

²H_S

6 2ρ

²2

kA

^1/4₁

ηk

²H_S

+ |λ|

²

kA

^−1/4₁

ηk

²H_S

(η ∈ D(A

^1/4₁

)). (4.36) Proof. First, since G(λ) : D(A

^1/8₁

) → D(A

^1/8₁

)

⁰

is positive and self-adjoint, we can define its square root G(λ)

^1/2

. Using that [D(A

^1/8₁

), D(A

^1/8₁

)

⁰

]

1/2

= H

S

, we have G(λ)

^1/2

∈ L(H

S

, D(A

^1/8₁

)

⁰

) and G(λ)

^1/2

∈ L(D(A

^1/8₁

), H

S

).

The right inequality in (4.32) yields kG(λ)

^1/2

ηk

²H_S

6 ρ

2

kA

^1/8₁

ηk

²H_S

+ |λ|kA

^−1/8₁

ηk

²H_S

(η ∈ D(A

^1/8₁

)), and using the identity G(λ)

^1/2

G(λ)

^1/2

= G(λ) and the fact that G(λ) and A

1

commute, we find

kG(λ)ηk

²H_S

6 ρ

²2

kA

^1/4₁

ηk

²H_S

+ 2|λ|kηk

²H_S

+ |λ|

²

kA

^−1/4₁

ηk

²H_S

(η ∈ D(A

^1/4₁

)).

The conclusion follows from 2kηk

²H_S

= 2(A

^1/4₁

η, A

^−1/4₁

η)

H_S

6 (|λ|

⁻¹

kA

^1/4₁

ηk

H_S

+ |λ|kA

^−1/4₁

ηk

H_S

).

According to (4.31), for λ ∈ C

⁺

the operator V (λ) that is (formally) introduced in (4.17) can be defined as the following unbounded operator on H

S

:

D(V (λ)) = D(A

1

) and V (λ)η = λ

²

(η + K(λ)η) + λG(λ)ζ + A

1

ζ. (4.37) Proposition 4.8. For all λ ∈ C

⁺

the operator V (λ) is an isomorphism from D(A

1

) onto H

S

.

Proof. The case λ = 0 is straightforward since V (0) = A

1

. In what follows we suppose λ ∈ C

⁺

and λ 6= 0.

We can write

V (λ) =

λ

²

(A

⁻¹₁

+ K(λ)A

⁻¹₁

) + λG(λ)A

⁻¹₁

+ I A

1

, with K(λ)A

⁻¹₁

, G(λ)A

⁻¹₁

∈ L(H

S

, D(A

^1/8₁

)) (see (4.30)).

It is sufficient to show that V (λ)A

⁻¹₁

∈ L(H

S

) is invertible. Since λ

²

(A

⁻¹₁

+ K(λ)A

⁻¹₁

) + λG(λ)A

⁻¹₁

is a compact operator, we can use the Fredholm alternative: assume ξ ∈ ker(V (λ)A

⁻¹₁

) and let us write η

^def

= A

⁻¹₁

ξ. Then

0 = Re V (λ)A

⁻¹₁

ξ, λA

⁻¹₁

ξ

H_S

= Re(V (λ)η, λη)

H_S

= Re λ|λ|

²

(η + K(λ)η, η)

H_S

+ |λ|(G(λ)η, η)

H_S

+ Re λkA

^1/2₁

ηk

²H_S

> ρ

1

|λ|kA

^1/8₁

ηk

²H_S

, (4.38)

and thus ξ = 0. We thus deduce that V (λ) is an isomorphism from D(A

1

) onto H

S

.

4.2 Estimation of V ⁻¹ (λ)

The following section is devoted to the estimation (in terms of λ) of the inverse of the operator V (λ) for λ ∈ C

⁺1

. We recall that the notation C

⁺1

is introduced in (4.2).

We also recall that K and G are defined by (4.26) and (4.27) and from Proposition 4.5, we have sup

λ∈C⁺

kKk

L(H_S)

+ kA

^1/8₁

K(λ)A

^1/8₁

k

L(H_S)

< +∞. (4.39) To obtain estimates for V (λ) we first consider the following “approximation”

V

K

(λ)

^def

= λ

²

(I + K(λ)) + 2ρλA

^1/4₁

+ A

1

, (4.40) where ρ > 0 is a given parameter. The estimates on V (λ) will then be deduced by a perturbation argument.

Theorem 4.9. For all λ ∈ C

⁺1

the operator V

K

(λ) : D(A

1

) → H

S

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

sup

λ∈C⁺₁

|λ|

^3/2−2θ

kA

^θ1

V

_K⁻¹

(λ)k

L(H_S)

< +∞. (4.41)

Proof. The proof of the invertibility of V

K

(λ) can be done in the same way as for V (λ) (see Proposition 4.8).

We only prove (4.41) for θ = 0 and θ = 1, the other cases are obtained by interpolation. Let us consider λ ∈ C

⁺1

and η ∈ D(A

1

). We first develop the expression of V

K

(λ) in (4.40):

V

K

(λ)η λ

²

2

H_S

= kη + K(λ)ηk

²_H

S

+ 4ρ

²

A

^1/4₁

η λ

2

H_S

+

A

1

η λ

²

2

H_S

+ 2 Re

η + K(λ)η, A

1

η λ

²

H_S

+ 4ρ Re K(λ)η, A

^1/4₁

η λ

!

H_S

+ 4ρ Re η, A

^1/4₁

η λ

!

H_S

+ 4ρ Re A

^1/4₁

η λ , A

1

η

λ

²

!

H_S

. (4.42) Since Re λ > 0 we have Re(1/λ) > 0 and we deduce,

Re A

^1/4₁

η λ , A

1

η

λ

²

!

H_S

+ Re η, A

^1/4₁

η λ

!

H_S

= Re 1

λ

A

^5/8₁

η λ

2

H_S

+ Re 1

λ

A

^1/8₁

η λ

2

H_S

> 0. (4.43)

Using the fact that K(λ) and A

^1/8₁

commute (see Proposition 4.6), Re(1/λ) > 0 and (4.33), we deduce Re K(λ)η, A

^1/4₁

η

λ

!

H_S

= Re 1

λ

K(λ)A

^1/8₁

η, A

^1/8₁

η

H_S

> 0. (4.44)

Using again that K(λ) and A

^1/8₁

commute, Proposition 4.5 and (4.39), we deduce

(η + K(λ)η, A

1

η)

_H

S

=

A

^1/4

η + A

^1/8₁

K(λ)A

^1/8₁

η, A

^3/4₁

η

H_S

6 C

1

kA

^1/4₁

ηk

H_S

kA

^3/4₁

ηk

H_S

. (4.45) Combining (4.42), (4.43), (4.44), (4.45) yields

V

K

(λ)η λ

²

2

H_S

> kη + K(λ)ηk

²_HS

+

A

1

η λ

²

2

H_S

− 2

η + K(λ)η, A

1

η λ

²

H_S

1 − 2ρ

²

C

1

kA

^1/4₁

ηk

H_S

kA

^3/4₁

ηk

H_S

!

. (4.46)

Note that if kA

^3/4₁

ηk

H_S

6 2ρ

²

C

1

kA

^1/4₁

ηk

H_S

, then we deduce from (4.46), (4.33) and |λ| > 1 that kV

K

(λ)ηk

²_HS

> |λ|

³

kηk

²H_S

+ |λ|

⁻¹

kA

1

ηk

²_HS

(λ ∈ C

⁺1

),

which yields (4.41) for θ = 0 and θ = 1.