Gevrey regularity for a system coupling the Navier-Stokes system with a beam: the non-flat case
aa r X i v : . [ m a t h . A P ] O c t Gevrey regularity for a system coupling the Navier-Stokes systemwith a beam: the non-flat case
Mehdi Badra and Tak´eo Takahashi Institut de Math´ematiques de Toulouse ; UMR5219; Universit´e de Toulouse ; CNRS ; UPSIMT, F-31062 Toulouse Cedex 9, France, [email protected] Universit´e de Lorraine, CNRS, Inria, IECL, F-54000 Nancy, France, [email protected]
October 17, 2019
Abstract
We consider a bi-dimensional viscous incompressible fluid in interaction with a beam located at itsboundary. We show the existence of strong solutions for this fluid-structure interaction system, extending aprevious result [3] where we supposed that the initial deformation of the beam was small. The main point ofthe proof consists in the study of the linearized system and in particular in proving that the correspondingsemigroup is of Gevrey class.
Keywords: fluid-structure, Navier-Stokes system, Gevrey class semigroups
Contents e V − λ
166 Proof of Theorem 2.4 20 V − λ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.2 Proof of Theorem 2.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 η Γ − −− −− Γ η L Figure 1: Our geometry
This work is devoted to the mathematical analysis of a fluid-structure interaction system where the fluid ismodeled 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 whereas thebeam domain is an interval. Another important assumption for our analysis is to assume periodic boundaryconditions in the direction orthogonal to the beam deformation. To be more precise, let L > I def = R /L Z . (1.1)For any deformation η : I → ( − , ∞ ), we also consider the corresponding fluid domain F η def = { ( x , x ) ∈ I × R ; x ∈ (0 , η ( x )) } . (1.2)The boundary of F η can be splitted into a “deformable” partΓ η def = { ( s, η ( s )) , s ∈ I} , and a “fixed” part Γ − def = I × { } . 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 theinteraction between the viscous incompressible fluid and the beam is ∂ t v + ( v · ∇ ) v − div T ( v, p ) = 0 , t > , x ∈ F η ( t ) , div v = 0 , t > , x ∈ F η ( t ) ,v ( t, s, η ( t, s )) = ( ∂ t η )( t, s ) e , t > , s ∈ I ,v = 0 t > , x ∈ Γ − ,∂ tt η + α ∂ ssss η − α ∂ ss η = − e H η ( v, p ) , t > , s ∈ I , (1.3)with the initial conditions η (0 , · ) = η , ∂ t η (0 , · ) = η and v (0 , · ) = v in F η . (1.4)The two first equations correspond to the Navier-Stokes system, whereas the last equation is the beamequation. We have considered the no-slip boundary conditions (third and forth equations). The canonicalbasis of R is denoted by ( e , e ) and we have also used the following notations: T ( v, p ) def = 2 νD ( v ) − pI , D ( v ) = 12 ( ∇ v + ( ∇ v ) ∗ ) , (1.5) e H η ( v, p ) def = n (1 + | ∂ s η | ) / [ T ( v, p ) n ] ( t, s, η ( t, s )) · e o . (1.6)We assume that the constants satisfy ν > , α > , α > . inally, the vector fields n is the unit exterior normal to F η ( t ) : n = − e on Γ − and on Γ η ( t ) , n ( t, x , x ) = 1 p | ∂ s η ( t, x ) | (cid:20) − ∂ s η ( t, x )1 (cid:21) . (1.7)An important remark is that a solution to (1.3) satisfies ddt Z L η ( t, s ) ds = 0 . By assuming that the mean value of η is zero, this leads to Z L η ( t, s ) ds = 0 ( t > . (1.8)We denote by M the orthogonal projection from L ( I ) onto L ( I ) where L ( I ) def = (cid:26) f ∈ L ( I ) ; Z L f ( s ) ds = 0 (cid:27) . (1.9)Taking the projection of the last equation of (1.3) on L ( I ) gives ∂ tt η + A η = − H η ( v, p ) , t > , s ∈ I , (1.10)where H η ( v, p ) def = M e H η ( v, p ) , (1.11)and where A is the operator for the structure defined by H S def = L ( I ) , D ( A ) def = H ( I ) ∩ L ( I ) , (1.12) A : D ( A ) → H S , η α ∂ ssss η − α ∂ ss η. (1.13)One can check that for any θ > D ( A θ ) = H θ ( I ) ∩ L ( I ) . (1.14)The projection of the last equation of (1.3) on L ( 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 thepressure 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 for thespace of continuous maps and C b for the space of continuous and bounded maps. We use the bold notationfor the spaces of vector fields: L α = ( L α ) , H k = ( H k ) etc. Since the fluid domain is moving, we introducespaces of the form H (0 , T ; L q ( F η )), L (0 , T ; H k ( F η )), etc. with T ∞ . If η ( t, · ) > − t ∈ (0 , T )), then v ∈ H (0 , T ; L q ( F η )) if y v ( t, y , y (1 + η ( t, y )) ∈ H (0 , T ; L q ( F ))and similarly, for the other spaces. We also write H α ( I ) def = H α ( I ) ∩ L ( I ) ( α > . 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 > η ∈ W , ∞ ( I ) ∩ L ( I ) , η ∈ H ε ( I ) η > − I , (1.15) v ∈ H ( F η ) , (1.16)with div v = 0 in F η , v ( s, η ( s )) = η ( s ) e s ∈ I , v = 0 on Γ − . (1.17)Our main result on (1.3) is the existence and uniqueness of strong solutions for small times: heorem 1.1. For any [ v , η , η ] satisfying (1.15) – (1.17) , there exist T > and a strong solution ( η, v, p ) of (1.3) with η ( t, · ) > − t ∈ [0 , T ] , (1.18) v ∈ L (0 , T ; H ( F η ) ∩ C ([0 , T ]; H ( F η )) ∩ H (0 , T ; L ( F η )) , p ∈ L (0 , T ; H ( F η )) , (1.19) η ∈ L (0 , T ; H / ( I )) ∩ C ([0 , T ]; H / ( I )) ∩ H (0 , T ; H / ( I )) ,∂ t η ∈ L (0 , T ; H / ( I )) ∩ C ([0 , T ]; H / ( I )) ∩ H (0 , T ; ( H / ( I )) ′ ) , (1.20) the first four equations of (1.3) are satisfied almost everywhere or in the trace sense and (1.10) holds in L (0 , T ; H / ( I ) ′ ) .This solution is unique locally: if ( η ( ∗ ) , v ( ∗ ) , p ( ∗ ) ) is another solution with the same regularity, there exists T ∗ > 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 inthe fixed spatial domain F def = F η , (1.21)by using a change of variables. Then, one of the main ingredients to obtain Theorem 1.1 is a result on alinear system associated with (1.3): ∂ t w − div T ( w, q ) = F, t > , y ∈ F , div w = 0 t > , y ∈ F ,w ( t, s, η ( t, s )) = ( ∂ t η )( t, s ) e t > , s ∈ I ,∂ tt η + A η = − H η ( w, q ) + G, t > , s ∈ I , (1.22)with the initial conditions w (0 , · ) = w , η (0 , · ) = ζ , ∂ t η (0 , · ) = ζ . (1.23)For this system, we have the following result Theorem 1.2.
Assume η ∈ W , ∞ ( I ) , η > − in I . Suppose F ∈ L (0 , ∞ ; L ( F )) and G ∈ L (0 , ∞ ; D ( A / )) , ε > , ζ ∈ H ε ( I ) , ζ ∈ H ε ( I ) , w ∈ H ( F ) , (1.24)div w = 0 in F , w ( s, ζ ( s )) = ζ ( s ) e s ∈ I , w = 0 on Γ − . (1.25) Then (1.22) - (1.23) admits a unique solution w ∈ L (0 , ∞ ; H ( F )) ∩ C b ([0 , ∞ ); H ( F )) ∩ H (0 , ∞ ; L ( F )) , q ∈ L (0 , ∞ ; H ( F ) / R ) , (1.26) η ∈ L (0 , ∞ ; D ( A / )) ∩ C b ([0 , ∞ ); D ( A / )) ∩ H (0 , ∞ ; D ( A / )) , (1.27) and ∂ t η ∈ L (0 , ∞ ; D ( A / )) ∩ C b ([0 , ∞ ); D ( A / )) ∩ H (0 , ∞ ; D ( A / ) ′ ) . (1.28) Moreover, there exists C > such that k w k L (0 , ∞ ; H ( F )) ∩ C b ([0 , ∞ ); H ( F )) ∩ H (0 , ∞ ; L ( F )) + k q k L (0 , ∞ ; H ( F ) / R ) + k η k L (0 , ∞ ; D ( A / )) ∩ C b ([0 , ∞ ); D ( A / )) ∩ H (0 , ∞ ; D ( A / )) + k ∂ t η k L (0 , ∞ ; D ( A / )) ∩ C b ([0 , ∞ ); D ( A / )) ∩ H (0 , ∞ ; D ( A / ) ′ ) C (cid:16) k w k H ( F ) + k ζ k D ( A / ε ) + k ζ k D ( A / ε ) + k F k L (0 , ∞ ; L ( F )) + k G k L (0 , ∞ ; D ( A / )) (cid:17) . (1.29)In [3], we obtained Theorem 1.2 only in the case η = 0 so that the result on (1.3) was reduced to thecase of small initial deformations. Here we are no longer restricted to this hypothesis. As in [3], the proofof Theorem 1.2 relies on resolvent estimates and results on semigroup of Gevrey class. More precisely, it isa consequence of Theorem 2.4. emark 1.3. As explained above, the main novelty here is to remove the restriction of smallness of η thatwas 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 , ∞ instead of H ε ). Even without this condition, we have as in our previous result a loss of regularityfor ( η, ∂ t η ) : the continuity of ( η, ∂ t η ) lies in H / (0 , L ) × H / (0 , L ) but we need to impose that at initialtime, it belongs to W , ∞ (0 , L ) × H ε (0 , L ) for some ε > . This is due to this model that couples twodynamical systems of different nature and in particular the linear system (1.22) couples the Stokes systemand the beam equation and the corresponding semigroup is not analytic but only of Gevrey class as stated inTheorem 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 replacedby ∂ tt η + α ∂ ssss η − α ∂ ss η − δ∂ tss η = − e H η ( v, p ) , (1.30) with δ > .Several works analyze such a model: [6] (existence of weak solutions), [4], [10] and [8] (existence of strongsolutions), [12] (stabilization of strong solutions), [2] (stabilization of weak solutions around a stationarystate). In all these works, the damping term − δ∂ tss η is crucial. Few works have tackled the case withoutdamping: the existence of weak solutions is proved in [7]. In [9], the existence of local strong solutions isobtained for a structure described by either a wave equation ( α = δ = 0 and α > in (1.30) ) or a beamequation with inertia of rotation ( α > , α = 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 ofan 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 towrite system (1.3) in a cylindrical domain and then linearize it. Section 3 is devoted to the introduction ofseveral useful operators together with their properties. In order to prove Theorem 1.2 we need to estimatecommutators appearing due the fact that our initial domain F is not flat. Such estimates allows us todeduce resolvent estimates in Section 6 by estimating the inverse of the operator V λ (see (6.6)). At first, wefirst estimate an approximation of V − λ in Section 5. Finally, in Section 7 we recall the idea of the proof ofTheorem 1.1 based on Theorem 1.2, by using a fixed point argument. In this section, we defined and use a standard change of variables to rewrite system (1.3) in a cylindricaldomain. We set X η ,η : F η → F η , ( y , y ) (cid:18) y , y η ( y )1 + η ( y ) (cid:19) , (2.1)whose inverse is X η ,η . In our case, we consider X ( t, · ) def = X η ,η ( t ) : ( y , y ) (cid:18) y , y η ( t, y )1 + η ( y ) (cid:19) , (2.2) Y ( t, · ) def = X ( t, · ) − = X η ( t ) ,η : ( x , x ) (cid:18) x , x η ( x )1 + η ( t, x ) (cid:19) , (2.3)so that X ( t, · ) transforms F = F η 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) fter some calculation (see for instance [3]), system (1.3), (1.4) rewrites, ∂ t w − div T ( w, q ) = b F ( ξ, w, q ) in (0 , ∞ ) × F , div w = 0 in (0 , ∞ ) × F ,w ( t, s,
1) = ( ∂ t η )( t, s ) e t > , s ∈ I ,w = 0 t > , y ∈ Γ − ,∂ tt η + A η = − H η ( w, q ) + b G η ( ξ, w ) , t > , (2.7)with the initial conditions η (0 , · ) = η , ∂ t η (0 , · ) = η and w (0 , y ) = w ( y ) def = b (0 , y ) v ( X (0 , y )) ( y ∈ F ) , (2.8)where we have the following definitions: b F α ( η, w, q ) def = ν X i,j,k b αi ∂ a ik ∂x j ( X ) w k + 2 ν X i,j,k,ℓ b αi ∂a ik ∂x j ( X ) ∂w k ∂y ℓ ∂Y ℓ ∂x j ( X )+ ν X j,ℓ,m ∂ w α ∂y ℓ ∂y m (cid:18) ∂Y ℓ ∂x j ( X ) ∂Y m ∂x j ( X ) − δ ℓ,j δ m,j (cid:19) + ν X j,ℓ ∂w α ∂y ℓ ∂ Y ℓ ∂x j ( X ) − X k,i ∂q∂y k (cid:18) det( ∇ X ) ∂Y α ∂x i ( X ) ∂Y k ∂x i ( X ) − δ α,i δ k,i (cid:19) − X i,j,k,m b αi ∂a ik ∂x j ( X ) a jm ( X ) w k w m − ∇ X ) [( w · ∇ ) w ] α − [ b ( ∂ t a )( X ) w ] α − [( ∇ w )( ∂ t Y )( X )] α , (2.9) b G η ( η, w )( t, s ) = νM ( X k,ℓ (cid:20) δ ,k δ ,ℓ − a k ( X ) ∂Y ℓ ∂x ( X ) (cid:21) ∂w k ∂y ℓ + ∂ s ( η − η ) (cid:18) ∂w ∂y + ∂w ∂y (cid:19) + ∂ s η X k,ℓ (cid:18) a k ( X ) ∂Y ℓ ∂x ( X ) − δ ,k δ ,ℓ (cid:19) ∂w k ∂y ℓ + X k,ℓ (cid:18) a k ( X ) ∂Y ℓ ∂x ( X ) − δ ,k δ ,ℓ (cid:19) ∂w k ∂y ℓ + X k (cid:18) ∂ s η (cid:20) ∂a k ∂x ( X ) + ∂a k ∂x ( X ) (cid:21) − ∂a k ∂x ( X ) (cid:19) w k ) ( t, s, η ( s )) . (2.10)Moreover, we recall that H η ( w, q )( t, s ) = M n (1 + | ∂ s η | ) / [ T ( v, p ) n ] ( t, s, η ( s )) · e o . (2.11) 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 ) > } . (2.12) C + α def = (cid:8) λ ∈ C + ; | λ | > α (cid:9) . (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 , / , (2.15) V θn ( F ) def = n f ∈ H θ ( F ) ; div f = 0 , f = 0 on ∂ F o ( θ ∈ (1 / , , (2.16) θ ( ∂ F ) def = (cid:26) f ∈ H θ ( ∂ F ) ; Z ∂ F f · n dγ = 0 (cid:27) ( θ > . (2.17)We introduce the operator Λ : L ( I ) → L ( ∂ F ) defined by(Λ η )( y ) = ( Mη ( s )) e if y = ( s, η ( s )) ∈ Γ η , (Λ η )( y ) = 0 if y ∈ Γ − . (2.18)The adjoint Λ ∗ : L ( ∂ F ) → L ( I ) of Λ is given by(Λ ∗ v )( s ) = M (cid:16) (1 + | ∂ s η ( s ) | ) / v ( s, η ( s )) · e (cid:17) . (2.19)Since η ∈ W , ∞ ( I ), then for any θ ∈ [0 , H θ ( I )) ⊂ V θ ( ∂ F ) (2.20)and Λ ∗ ( H θ ( ∂ F )) ⊂ D ( A θ/ ) . (2.21)In particular k Λ η k H θ ( ∂ F ) > c ( θ ) k A θ/ η k H S ( η ∈ D ( A θ/ )) . (2.22)We can also define the Stokes operator D ( A ) def = V n ( F ) ∩ H ( F ) , A def = P ∆ : D ( A ) → V n ( F ) , (2.23)where P : L ( F ) → V n ( F ) is the Leray projection operator.We consider the space L ( F ) × D ( A / ) × H S equipped with the scalar product: Dh w (1) , η (1)1 , η (1)2 i , h w (2) , η (2)1 , η (2)2 iE = Z F w (1) · w (2) dy + (cid:16) A / η (1)1 , A / η (2)1 (cid:17) H S + (cid:16) η (1)2 , η (2)2 (cid:17) H S , and we introduce the following spaces: H def = n [ w, η , η ] ∈ L ( F ) × D ( A / ) × H S ; w · n = (Λ η ) · n on ∂ F , div w = 0 in F o , (2.24) V def = n [ w, η , η ] ∈ H ( F ) × D ( A / ) × D ( A / ) ; w = Λ η on ∂ F , div w = 0 in F o . We denote by P the orthogonal projection from L ( F ) × D ( A / ) × H S onto H . We have the followingregularity result on P (see [3]): Lemma 2.1.
For any θ ∈ [0 , , P ∈ L ( H θ ( F ) × D ( A / θ/ ) × D ( A θ/ )) , (2.25) and P ∈ L ( L ( F ) × D ( A / ) × D ( A / ) ′ ) . (2.26)We now define the linear operator A : D ( A ) ⊂ H → H : D ( A ) def = V ∩ h H ( F ) × D ( A ) × D ( A / ) i , (2.27)and for (cid:2) w, η , η (cid:3) ∈ D ( A ), we set e A wη η def = ∆ wη − A η − Λ ∗ (2 D ( w ) n ) (2.28)and A def = P e A . (2.29) y using the above operators, we can rewrite the linear system (1.22), as follows ddt wη∂ t η = A wη∂ t η + P F G , wη∂ t η (0) = w η η . (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 defined by (2.27) – (2.29) has compact resolvents, it is the infinitesimalgenerator 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 , , the following equalities hold D (( − A ) θ ) = h H θ ( F ) × D ( A / θ/ ) × D ( A θ/ ) i ∩ H if θ ∈ (0 , / , (2.31) D (( − A ) θ ) = n [ w, η , η ] ∈ h H θ ( F ) × D ( A / θ/ ) × D ( A θ/ ) i ∩ H ; w = Λ η on ∂ F o if θ ∈ (1 / , . (2.32)One of the main goals of this article is to show the following result: Theorem 2.4.
There exists
C > such that for all λ ∈ C + | λ | / (cid:13)(cid:13) ( λI − A ) − (cid:13)(cid:13) L ( H ) C. (2.33) Moreover, there exists a constant
C > such that for all λ ∈ C + (cid:13)(cid:13) ( λI − A ) − z (cid:13)(cid:13) H ( F ) ×D ( A / ) ×D ( A / ) + | λ | (cid:13)(cid:13) ( λI − A ) − z (cid:13)(cid:13) L ( F ) ×D ( A / ) ×D ( A / ) ′ C k z k L ( F ) ×D ( A / ) ×D ( A / ) (cid:16) z ∈ H ∩ (cid:16) L ( F ) × D ( A / ) × D ( A / ) (cid:17)(cid:17) . (2.34)Using the above theorem and Theorem 5.1 in [3], we deduce Theorem 1.2. This section is devoted to the introduction of several operators that are used to prove the resolvent estimatesin Theorem 2.4. In this section we assume η ∈ W , ∞ ( I ). It implies in particular that the domain F is ofclass C , .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 = Λ η 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 ifwe define the operators W λ η def = w η , Q λ η def = q η , (3.2)since F is of class C , , we have W λ ∈ L ( D ( A / ) , H ( F )) ∩ L ( D ( A / ) , H ( F )) ∩ L ( D ( A / ) ′ , L ( F )) (3.3)and Q λ ∈ L ( D ( A / ) , H ( F ) / R ) . (3.4)We also define the operator L λ ∈ L ( D ( A / ) , D ( A / )) y L λ η def = Λ ∗ (cid:8) T ( w η , q η ) n | ∂ F (cid:9) . (3.5)We decompose L λ with the operators K λ ∈ L ( D ( A / ) ′ , D ( A / )) , G λ ∈ L ( D ( A / ) , D ( A / ) ′ ) ∩ L ( D ( A / ) , D ( A / ))defined by h K λ η, ζ i D ( A / ) , D ( A / ) ′ def = Z F w η · w ζ dy (3.6)and h G λ η, ζ i D ( A / ) ′ , D ( A / ) def = 2 ν Z F Dw η : Dw ζ dy = 2 ν Z L Λ ∗ (( Dw η ) n ) ζ ds − ν Z F ∆ w η · w ζ dy. (The second relation holds if η ∈ D ( A / )).The operators K λ and G λ are related to the operator L λ defined by (3.5): multiplying (3.1) by w ζ andintegrating 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 thereexist < ρ < ρ such that for any λ such that Re λ > , we have ρ k A / η k H S h G λ η, η i D ( A / ) ′ , D ( A / ) ρ (cid:16) k A / η k H S + | λ |k A − / η k H S (cid:17) ( η ∈ D ( A / )) , (3.8)0 h K λ η, η i D ( A / ) , D ( A / ) ′ ρ k A − / η k H S ( η ∈ D ( A / ) ′ ) . (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 I + λL λ + A = λ ( I + K λ ) + λG λ + A , (3.12)and an “approximation”: e V λ def = λ ( I + K λ ) + 2 ρλA / + A , (3.13)where ρ > λ b v − div T ( b v, b p ) = f in F , div b v = 0 in F , b v = 0 on ∂ F . (3.14) Proposition 3.2.
Let γ ∈ [0 , / , θ ∈ [ γ, .1. There exists C > such that for any f ∈ H γ ( F ) and for any λ ∈ C +0 , the solution ( b v, b p ) of (3.14) satisfies k b v k H θ ( F ) C | λ | θ − γ − k f k H γ ( F ) . (3.15)
2. There exists
C > such that for any f ∈ H θ ( F ) and for any λ ∈ C +0 , the solution ( b v, b p ) of (3.14) satisfies k b v k H θ ( F ) + k b p k H θ ( F ) C (cid:16) | λ | θ − γ k f k H γ ( F ) + k f k H θ ( F ) (cid:17) . (3.16) roof. Using that the Stokes operator A (defined by (2.23)) is the infinitesimal generator of an analyticsemigroup and that C +0 ⊂ ρ ( A ), we have the existence of a constant C such that k ( − A ) α ( λI − A ) − g k L ( F ) C | λ | α − k g k L ( F ) ( g ∈ V n ( F ) , λ ∈ C +0 , α ∈ [0 , . Using that for γ ∈ [0 , / P ∈ L ( H γ ( F ) , D (( − A ) γ )) (see [1, Section 2.1]), we have k ( − A ) γ P f k L ( F ) C k f k H γ ( F ) ( γ ∈ [0 , / , f ∈ H γ ( F )) . Gathering the two above estimates with the fact that D (( − A ) θ ) ⊂ H θ ( F ), we can deduce k b v k H θ ( F ) C (cid:13)(cid:13)(cid:13) ( − A ) θ ( λI − A ) − P f (cid:13)(cid:13)(cid:13) L ( F ) C | λ | θ − γ − k f k H γ ( F ) . For the second estimate, we use the following classical estimate for Stokes system: k b v k H θ ( F ) + k b p k H θ ( F ) C (cid:0) | λ |k b v k H θ ( F ) + k f k H θ ( F ) (cid:1) , and we combine it with (3.15).Using the above proposition, we can define the following operator T λ ∈ L ( L ( F ) , D ( A / )) , T λ f def = − Λ ∗ (cid:8) T ( b v, b p ) n | ∂ F (cid:9) . (3.17)We have in particular that the norm of T λ in L ( L ( F ) , D ( A / )) is independent of λ . Proposition 3.3.
For θ ∈ [0 , , ε ∈ (0 , / and λ ∈ C +0 , the operators W λ and Q λ defined by (3.2) satisfy k W λ η k H θ ( F ) C k A θ/ − / η k H S ( θ < / , (3.18) k W λ η k H θ ( F ) C (cid:16) k A θ/ − / η k H S + | λ | θ k A − / η k H S (cid:17) ( θ > / , (3.19) k W λ η k H θ ( F ) C (cid:16) k A θ/ − / η k H S + | λ | θ − / ε k η k H S (cid:17) , θ > / − ε, (3.20) k W λ η k H θ ( F ) + k Q λ η k H θ ( F ) C (cid:16) k A θ/ / η k H S + | λ | θ k A − / η k H S (cid:17) , (3.21) k W λ η k H θ ( F ) + k Q λ η k H θ ( F ) C (cid:16) k A θ/ / η k H S + | λ | / ε + θ k η k H S (cid:17) . (3.22) Proof.
We write W λ η = W η + z η , Q λ η = Q η + ζ η , with λz η − div T ( z η , ζ η ) = − λW 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 k W η k H θ ( F ) C k A θ/ − / η k H S ( θ ∈ [0 , , η ∈ D ( A θ/ − / )) . (3.24)Combining the above relation with (3.15) we deduce the following relations: k z η k H θ ( F ) C | λ | θ k A − / η k H S , k z η k H θ ( F ) C k A θ/ − / η k H S if θ ∈ [0 , / , k z η k H θ ( F ) C | λ | θ − / ε k η k H S ( θ > / − ε ) . Then (3.18), (3.19) and (3.20) follow by combining the above inequalities with (3.24).From (3.16) we deduce k z η k H θ ( F ) + k ζ η k H θ ( F ) C (cid:16) | λ | / ε + θ k W η k H / − ε ( F ) + | λ |k W η k H θ ( F ) (cid:17) , (3.25) nd k z η k H θ ( F ) + k ζ η k H θ ( F ) C (cid:16) | λ | θ k W η k L ( F ) + | λ |k W η k H θ ( F ) (cid:17) . (3.26)Moreover, from (3.24), we have | λ |k W η k H θ ( F ) C (cid:16) | λ | θ k W η k L ( F ) (cid:17) θ (cid:0) k W η k H θ ( F ) (cid:1) θ θ C (cid:16) k W η k H θ ( F ) + | λ | θ k W η k L ( F ) (cid:17) C (cid:16) k A θ/ / η k H S + | λ | θ k A − / η k H S (cid:17) , and | λ |k W η k H θ ( F ) C (cid:16) | λ | θ − / ε k W η k H / − ε ( F ) (cid:17) θ − / ε (cid:0) k W η k H θ ( F ) (cid:1) θ − / ε θ − / ε C (cid:16) k W η k H θ ( F ) + | λ | / ε + θ k W η k H / − ε ( F ) (cid:17) C (cid:16) k A θ/ / η k H S + | λ | / ε + θ k η k H S (cid:17) . Combining the above estimates with (3.25) and (3.26), we deduce (3.21) and (3.22).
Proposition 3.4.
Let θ ∈ [1 / , / and λ ∈ C +0 , then k A θ/ K λ η k H S C k A θ/ − / η k H S . (3.27) Let ε ∈ (0 , / , λ ∈ C +0 and θ ∈ [1 / , , then k A θ/ K λ η k H S C (cid:16) k A θ/ − / η k H S + | λ | θ − / ε k η k H S (cid:17) . (3.28) Proof.
From (3.10) and properties on the trace operator and on Λ ∗ , k A θ/ K λ η k H S C (cid:16) k ϕ η k H / θ ( F ) + k π η k H / θ ( F ) (cid:17) . (3.29)Using (3.11), (3.16) and (3.18), we deduce from the above estimate that k A θ/ K λ η k H S C k W λ η k H θ − / ( F ) C k A θ/ − / η k H S if θ ∈ [1 / , / θ ∈ [1 / , k A θ/ K λ η k H S C (cid:16) | λ | θ − / ε k W λ η k H / − ε ( F ) + k W λ η k H θ − / ( F ) (cid:17) C (cid:16) | λ | θ − / ε k η k H S + k A θ/ − / η k H S (cid:17) . Proposition 3.5.
Assume α > . Let θ ∈ ( − / , / , then k A θ/ ( I + K λ ) − η k H S C k A θ/ η k H S ( λ ∈ C + ) . (3.30) Let ε > and θ ∈ [1 / , , then k A θ/ ( I + K λ ) − η k H S C (cid:16) k A θ/ η k H S + | λ | θ − / ε k η k H S (cid:17) ( λ ∈ C + α ) . (3.31) Proof.
We write ζ = ( I + K λ ) − η, ζ = η − K λ ζ. First, using the positivity of K λ stated in (3.9), we find k ζ k H S k η k H S . (3.32)Moreover, we have the relation k A θ/ ζ k H S k A θ/ η k H S + k A θ/ K λ ζ k H S . (3.33) ssume first θ ∈ [1 / , / θ ∈ [0 , / θ ∈ ( − / ,
0) we usea duality argument. First, for θ ∈ (0 , /
2) (3.30) can be rewritten as k A θ/ ( I + K λ ) − A − θ/ k L ( H S ) < + ∞ . Thus, noticing that ( A θ/ ( I + K λ ) − A − θ/ ) ∗ = A − θ/ ( I + K λ ) − A θ/ we also have k A − θ/ ( I + K λ ) − A θ/ k L ( H S ) < + ∞ , which is equivalent to (3.30) for θ ∈ ( − / , θ ∈ [1 / ,
2] and ε >
0. We deduce from (3.33) and (3.28) that k A θ/ ζ k H S k A θ/ η k H S + C (cid:16) k A θ/ − / ζ k H S + | λ | θ − / ε k ζ k H S (cid:17) . (3.34)If θ ∈ [1 / , θ − / ∈ [0 , /
2) and we can use (3.30) and (3.32) in the above relation to deduce (3.31).If θ ∈ [1 , / θ − / ∈ [1 / ,
1) and we can use (3.31) and (3.32) in (3.34) to deduce k A θ/ ζ k H S k A θ/ η k H S + C (cid:16) k A θ/ − / η k H S + | λ | θ − ε k ζ k H S + | λ | θ − / ε k ζ k H S (cid:17) . Since | λ | > α we have | λ | θ − ε α − / | λ | θ − / ε and it yields (3.31). We can then repeat the sameargument for θ ∈ [3 / ,
2) and θ = 2. The aim of this section is to show the following result:
Lemma 4.1.
Assume η ∈ W , ∞ ( I ) . For ε ∈ (0 , / , there exists a constant C > such that for any λ ∈ C +0 , k [ A / , K λ ] η k H S C ( | λ | − k A / ε η k H S + | λ | ε k η k H S ) . Here we have denoted by [
A, B ] the commutator of A and B : [ A, B ] = AB − BA . We transform the systems (3.1) and (3.11) written in F = F η into systems written in the domain F = I × (0 , . We use the change of variables e X : F → F , ( y , y ) (cid:0) y , y (1 + η ( y )) (cid:1) , e Y : F → F , ( x , x ) (cid:18) x , x η ( x ) (cid:19) . We write e a def = Cof( ∇ e Y ) ∗ , e b def = Cof( ∇ e X ) ∗ . We set e w ( y ) def = e b ( y ) w ( e X ( y )) and e q ( y ) def = q ( e X ( y )) , (4.1)so that w ( x ) = e a ( x ) e w ( e Y ( x )) and q ( x ) = e q ( e Y ( x )) . (4.2)We set[ L e w ] α def = X i,j,k e b αi ∂ e a ik ∂x j ( e X ) e w k + 2 X i,j,k,ℓ e b αi ∂ e a ik ∂x j ( e X ) ∂ e w k ∂y ℓ ∂ e Y ℓ ∂x j ( e X )+ X j,ℓ,m ∂ e w α ∂y ℓ ∂y m ∂ e Y ℓ ∂x j ( e X ) ∂ e Y m ∂x j ( e X ) + X j,ℓ ∂ e w α ∂y ℓ ∂ e Y ℓ ∂x j ( e X ) , (4.3) G e q ] α def = det( ∇ e X ) X k,i ∂ e q∂y k ∂ e Y α ∂x i ( e X ) ∂ e Y k ∂x i ( e X ) . (4.4)Then some calculation yields he b ∆ w ( e X ) i α = [ L e w ] α , he b ∇ q ( e X ) i α = [ G e 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 newstates e w η def = e b ( w η ◦ e X ) , e q η def = q η ◦ e X, e ϕ η def = e b ( ϕ η ◦ e X ) , e π η def = π η ◦ e X. From the above relations, systems (3.1) and (3.11) are transformed into the following systems λ e w η − ν L e w η + G e q η = 0 in F , div e w η = 0 in F , e w η = Λ η on ∂ F (4.6)and λ e ϕ η − ν L e ϕ η + G e π η = e w η in F , div e ϕ η = 0 in F , e ϕ η = 0 on ∂ F . (4.7)Here, we have also transformed the operator Λ defined by (2.18) into the operator Λ : L ( I ) → L ( ∂ F )defined by ( (Λ η )( s,
1) = ( Mη ( s )) e (Λ η )( s,
0) = 0 , s ∈ I . (4.8)From (3.10) and (2.19), we have the following formula K λ η = − M (cid:18) νD ( ϕ η )( s, η ( s )) (cid:20) − ∂ s η ( s )1 (cid:21) · e − π η ( s, η ( s )) (cid:19) . (4.9)Thus K λ η = − M ( ν D f ϕ η ( s, − e π η ( s, , (4.10)with D f ϕ η = 12 ( − ∂ s η ) X k ∂ e a k ∂x ( e X )( f ϕ η ) k + X k,ℓ e a k ( e X ) ∂ ( f ϕ η ) k ∂y ℓ ∂ e Y ℓ ∂x ( e X ) + X k,ℓ e a k ( e X ) ∂ ( f ϕ η ) k ∂y ℓ ∂ e Y ℓ ∂x ( e X ) + X k ∂ e a k ∂x ( e X )( f ϕ η ) k + X k,ℓ e a k ( e X ) ∂ ( f ϕ η ) k ∂y ℓ ∂ e Y ℓ ∂x ( e X ) . (4.11)In what follows, we write the above operators by splitting the derivatives with respect to y and y . Moreprecisely, we introduce the set O βα ( β > α ) of operators of the form f X i α c ( β − i ) i ∂ i f, (4.12)where c ( k ) i are functions of the form c ( k ) i = b c i ( η ( y ) , . . . , ∂ ks η ( y ) , y ) , (4.13)with b c i a smooth function and k ∈ N . These operators are thus depending on y but it can be seen as aparameter.For instance, using (A.1)–(A.8), we deduce D f = D (1) f + D (0) ∂ f, L f = L (2) f + L (1) ∂ f + L (0) ∂ f, G f = G (1) f + G (0) ∂ f, (4.14)where L (2) ∈ O , D (1) , L (1) , G (1) ∈ O , D (0) , L (0) , G (0) ∈ O . .2 Commutator estimate First we show the following result:
Proposition 4.2.
Assume B ∈ O βα and η ∈ W β, ∞ ( I ) . For any θ ∈ (0 , and for any s > , if s > θ + α − , then there exists C > such that (cid:13)(cid:13)(cid:13) [ A θ M, B ] f (cid:13)(cid:13)(cid:13) L ( I ) C k f k H s ( I ) ( f ∈ H s ( I )) . (4.15) Proof.
We write B f = X i α c ( β − i ) i ∂ i f. Then we recall the following formula (see for instance [13, p. 98]): A θ = 1Γ( θ )Γ(1 − θ ) Z ∞ t θ − A ( tI + A ) − dt. In particular,[ A θ M, B ] = A θ M B − B A θ M = A θ M B M − M B A θ M − ( I − M ) B A θ M + A θ M B ( I − M ) . (4.16)By using the identity A ( tI + A ) − = I − t ( tI + A ) − we deduce that A ( tI + A ) − M B M − M B A ( tI + A ) − M = − t ( tI + A ) − M B M + tM B ( tI + A ) − M = t ( tI + A ) − ( − M B ( tI + A )+( tI + A ) M B )( tI + A ) − M = t ( tI + A ) − ( − M B A + A M B )( tI + A ) − M and the first two terms in (4.16) give A θ M B M − M B A θ M = 1Γ( θ )Γ(1 − θ ) Z ∞ t θ ( tI + A ) − M [ A M, B ]( tI + A ) − M dt.
Moreover,[ A M, B ] f = (cid:0) α ∂ − α ∂ (cid:1) M X i α c ( β − i ) i ∂ i f − X i α c ( β − i ) i ∂ i (cid:0) α ∂ − α ∂ (cid:1) Mf = (cid:0) α ∂ − α ∂ (cid:1) X i α c ( β − i ) i ∂ i f − X i α c ( β − i ) i ∂ i (cid:0) α ∂ − α ∂ (cid:1) f and thus [ A M, B ] ∈ O β +4 α +3 . (4.17)We deduce that for f ∈ H s ( I ), k ( A θ M B M − M B A θ M ) f k L ( I ) θ )Γ(1 − θ ) Z ∞ t θ k ( tI + A ) − M [ A M, B ]( tI + A ) − Mf k L ( I ) dt C Z ∞ t θ − k [ A M, B ]( tI + A ) − Mf k L ( I ) dt C Z ∞ t θ − k ( tI + A ) − Mf k H α +3 ( I ) dt C (cid:18)Z t θ − k A ( tI + A ) − A α − Mf k L ( I ) dt + Z ∞ t θ − k A α +3 − s ( tI + A ) − A s Mf k L ( I ) dt (cid:19) C (cid:18)Z t θ − k Mf k H α − ( I ) dt + Z ∞ t θ − t − ( α +3 − s ) / k Mf k H s ( I ) dt (cid:19) C k Mf k H s ( I ) . (4.18)For the third term in (4.16), we write( I − M ) B A θ Mf = 1 L Z L X i α c ( β − i ) i ∂ i A θ Mf ! ds = 1 L Z L X i α ( − i A θ M∂ i c ( β − i ) i ! f ds L Z L A θ − X i α ( − i M ( α ∂ i +41 − α ∂ i +21 ) c ( β − i ) i ! f ds (4.19) hat yields k ( I − M ) B A θ Mf k L ( I ) C X i α k c ( β − i ) i k H i +4 ( I ) ! k f k L ( I ) . (4.20)For the last term in (4.16), we simply write k ( A θ M B )( I − M ) f k L ( I ) = (cid:13)(cid:13)(cid:13)(cid:13) ( A θ Mc ( β )0 ) 1 L Z L f ds (cid:13)(cid:13)(cid:13)(cid:13) L ( I ) C k c ( β )0 k H θ ( I ) k f k L ( 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 ∈ W , ∞ ( I ) then for all ε > C = C ( ε ) > (cid:13)(cid:13)(cid:13) [ A / M, D ] f (cid:13)(cid:13)(cid:13) L ( I ) C k f k H / ε ( I ) (4.21)and (cid:13)(cid:13)(cid:13) [ A / M, L ] f (cid:13)(cid:13)(cid:13) L ( F ) C k f k H / ε ( F ) , (cid:13)(cid:13)(cid:13) [ A / M, G ] f (cid:13)(cid:13)(cid:13) L ( F ) C k f k H / ε ( F ) . (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 ofvariables, we have formulas (4.10), (4.11) and (4.14) and thus[ A / , K λ ] η = M h ν D , A / M i e ϕ η ( s, − Mν D e ϕ ( s,
1) + M e π ( s,
1) (4.23)where e ϕ = A / M e ϕ η − e ϕ A / η and e π = A / M e π η − e π A / η . (4.24)From (4.6) and (4.7), we have λ e ϕ − ν L e ϕ + G e π = e w + [ A / M, ν L ] e ϕ η − [ A / M, G ] e π η in F , div e ϕ = 0 in F , e ϕ = 0 on ∂ F (4.25)where e w = A / M e w η − e w A / η and e q = A / M e q η − e q A / η satisfy λ e w − ν L e w + G e q = [ A / M, ν L ] e w η − [ A / M, G ] e q η in F , div e w = 0 in F , e w = 0 on ∂ F . (4.26)From (3.15), (3.16) and using the change of variables in Section 4.1, we deduce k e w k L ( F ) C | λ | − (cid:13)(cid:13)(cid:13) [ A / M, ν L ] e w η − [ A / M, G ] e q η (cid:13)(cid:13)(cid:13) L ( F ) (4.27)and k e ϕ k H ( F ) + k e π k H ( F ) C (cid:13)(cid:13)(cid:13) e w + [ A / M, ν L ] e ϕ η − [ A / M, G ] e π η (cid:13)(cid:13)(cid:13) L ( F ) . (4.28)From (4.27), (4.22), (3.22) and using the change of variables in Section 4.1, we deduce k e w k L ( F ) C | λ | − (cid:16) k e w η k H / ε ( F ) + k e q η k H / ε ( F ) (cid:17) C (cid:16) | λ | − k A / ε/ η k H S + | λ | ε k η k H S (cid:17) . (4.29)Using (4.28) and (4.22), we find k e ϕ k H ( F ) + k e π k H ( F ) C (cid:16) k e w k L ( F ) + k e ϕ η k H / ε ( F ) + k e π η k H / ε ( F ) (cid:17) . (4.30)From (3.16), (3.18), (3.20) and using the change of variables in Section 4.1, we deduce k e ϕ η k H / ε ( F ) + k e π η k H / ε ( F ) C (cid:16) | λ | ε k e w η k H / − ε ( F ) + k e w η k H / ε ( F ) (cid:17) C (cid:16) | λ | ε k η k H S + k A ε/ η k H S (cid:17) . (4.31) ombining the above equation with (4.29) and (4.30), we deduce k e ϕ k H ( F ) + k e π k H ( F ) C (cid:16) | λ | − k A / ε/ η k H S + | λ | ε k η k H S + k A ε/ η k H S (cid:17) . (4.32)By using the above estimate, (4.23), (4.31), (4.21), and trace estimates, (cid:13)(cid:13)(cid:13) [ A / , K λ ] η (cid:13)(cid:13)(cid:13) H S (cid:13)(cid:13)(cid:13) M h D , A / M i e ϕ η ( · , (cid:13)(cid:13)(cid:13) H S + k M D e ϕ ( · , k H S + k M e π ( · , k H S C (cid:0) k e ϕ η k H ε ( F ) + k e ϕ k H ( F ) + k e π k H ( F ) (cid:1) C (cid:16) | λ | − k A / ε/ η k H S + | λ | ε k η k H S + k A ε/ η k H S (cid:17) . The conclusion follows from k A ε/ η k H S ( | λ | ε k η k H S ) ε (cid:16) | λ | − k A / ε/ η k H S (cid:17) ε ε C (cid:16) | λ | ε k η k H S + | λ | − k A / ε/ η k H S (cid:17) . e V − λ The aim of this section is to estimate e V − λ where e V λ is defined by (3.13). We recall that the notation C + α isintroduced in (2.13). Theorem 5.1.
There exists α > such that for all λ ∈ C + α the operator e V λ : D ( A ) → H S is an isomorphismand for θ ∈ [0 , the following estimates hold sup λ ∈ C + α | λ | / − θ k A θ e V − λ k L ( H S ) < + ∞ . (5.1) Moreover, sup λ ∈ C + α | λ | / − θ k A θ e V ∗− λ 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 byinterpolation.Let us consider λ ∈ C + α and η ∈ D ( A ). Then from (3.13) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) e V λ ηλ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H S = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( I + K λ ) η + 2 ρA / ηλ + A ηλ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H S . (5.3)Using (3.9) and Proposition 3.4, we deduce that k η k H S k ( I + K λ ) / η k H S C k η k H S . (5.4)Thus, combining (5.3) and (5.4), we deduce (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) e V λ ηλ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H S > (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( I + K λ ) / η + 2 ρ ( I + K λ ) − / A / ηλ + ( I + K λ ) − / A ηλ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H S = (cid:13)(cid:13)(cid:13)(cid:13) ( I + K λ ) − / (cid:18) η + K λ η + A ηλ (cid:19)(cid:13)(cid:13)(cid:13)(cid:13) H S + 4 ρ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( I + K λ ) − / A / ηλ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H S + 4 ρ Re (cid:18) λ (cid:19) (cid:13)(cid:13)(cid:13) A / η (cid:13)(cid:13)(cid:13) H S + 4 ρ Re (cid:18) | λ | λ D ( I + K λ ) − A η, A / η E H S (cid:19) . (5.5)We can write D ( I + K λ ) − A η, A / η E H S = k ( I + K λ ) − / A / η k H S + Dh ( I + K λ ) − , A / i A / η, A / η E H S = k ( I + K λ ) − / A / η k H S + D ( I + K λ ) − A / η, h A / , K λ i ( I + K λ ) − A / η E H S . (5.6) et use introduce the following notation, C λ def = 4 ρ Re (cid:18) | λ | λ D ( I + K λ ) − A / η, h A / , K λ i ( I + K λ ) − A / η E H S (cid:19) . (5.7)Using (5.4) and (5.6) in (5.5) we deduce, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) e V λ ηλ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H S > C (cid:13)(cid:13)(cid:13)(cid:13) η + K λ η + A ηλ (cid:13)(cid:13)(cid:13)(cid:13) H S + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) A / ηλ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H S + C λ (5.8)and thus (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) e V λ ηλ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H S > C k ( I + K λ ) η k H S + (cid:13)(cid:13)(cid:13)(cid:13) A ηλ (cid:13)(cid:13)(cid:13)(cid:13) H S − (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:28) A ηλ , η + K λ η (cid:29) H S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) A / ηλ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H S + C λ . (5.9)Let us estimate (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:28) A ηλ , ( I + K λ ) η (cid:29) H S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . To do this, we start by noticing that (cid:12)(cid:12)(cid:12) h A η, η i H S (cid:12)(cid:12)(cid:12) C k A / η k H S k A / η k H S . On the other hand, using (3.28), (cid:12)(cid:12)(cid:12) h A η, K λ η i H S (cid:12)(cid:12)(cid:12) | λ | ε k η k H S k A / η k H S , so that (cid:12)(cid:12)(cid:12) h A η, ( I + K λ ) η i H S (cid:12)(cid:12)(cid:12) C k A / η k H S k A / η k H S + | λ | ε k η k H S ! . (5.10)From interpolation inequality and Young inequality we deduce (cid:13)(cid:13)(cid:13) A / η (cid:13)(cid:13)(cid:13) H S | λ | / C (cid:18) k A η k H S | λ | (cid:19) / k η k / H S C (cid:13)(cid:13)(cid:13)(cid:13) A ηλ (cid:13)(cid:13)(cid:13)(cid:13) H S + k η k H S ! . (5.11)Moreover, from the interpolation inequality k A / η k H S C k A / η k / H S k A η k / H S , we also deduce | λ | ε k A / η k H S k η k H S C | λ | ε k A / η k / H S k A η k / H S k η k H S C | λ | / ε k A / η k H S (cid:13)(cid:13)(cid:13)(cid:13) A ηλ (cid:13)(cid:13)(cid:13)(cid:13) / H S k η k / H S C | λ | / ε k A / η k H S (cid:13)(cid:13)(cid:13)(cid:13) A ηλ (cid:13)(cid:13)(cid:13)(cid:13) H S + k η k H S ! . (5.12)Hence, combining (5.10), (5.11) and (5.12) with ε < / (cid:12)(cid:12)(cid:12) h A η, ( I + K λ ) η i H S (cid:12)(cid:12)(cid:12) C | λ | / k A / η k H S (cid:13)(cid:13)(cid:13)(cid:13) A ηλ (cid:13)(cid:13)(cid:13)(cid:13) H S + k η k H S ! , (5.13)and we thus obtain (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:28) A ηλ , ( I + K λ ) η (cid:29) H S (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) C | λ | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) A / ηλ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H S + 14 (cid:13)(cid:13)(cid:13)(cid:13) A ηλ (cid:13)(cid:13)(cid:13)(cid:13) H S + k ( I + K λ ) η k H S ! . Then with (5.9) it yields |C λ | + | λ | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) A / ηλ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H S + (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) e V λ ηλ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H S > C k ( I + K λ ) η k H S + (cid:13)(cid:13)(cid:13)(cid:13) A ηλ (cid:13)(cid:13)(cid:13)(cid:13) H S ! , nd with (5.8), since | λ | > α with α >
0, we deduce | λ ||C λ | + | λ | (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) e V λ ηλ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H S > C k ( I + K λ ) η k H S + (cid:13)(cid:13)(cid:13)(cid:13) A ηλ (cid:13)(cid:13)(cid:13)(cid:13) H S ! , (5.14)where C λ is the product defined in (5.7).Next, let us now estimate | λ ||C λ | . Using Lemma 4.1 and Proposition 3.5, we first deduce, (cid:13)(cid:13)(cid:13)h K λ , A / i ( I + K λ ) − A / η (cid:13)(cid:13)(cid:13) H S C ( | λ | − k A / ε ( I + K λ ) − A / η k H S + | λ | ε k ( I + K λ ) − A / η k H S ) C ( | λ | − k A / ε η k H S + | λ | − / ε k A / η k H S + | λ | ε k A / η k H S ) . (5.15)Thus, with (5.7) we obtain, | λ ||C λ | C | λ | − k A / η k H S (cid:16) | λ | − k A / ε η k H S + | λ | ε k A / η k H S (cid:17) . (5.16)From interpolation inequality and Young inequality we deduce (cid:13)(cid:13)(cid:13) A / η (cid:13)(cid:13)(cid:13) H S | λ | / C (cid:18) k A η k H S | λ | (cid:19) / k η k / H S C (cid:13)(cid:13)(cid:13)(cid:13) A ηλ (cid:13)(cid:13)(cid:13)(cid:13) H S + k η k H S ! , (cid:13)(cid:13)(cid:13) A / ε η (cid:13)(cid:13)(cid:13) H S | λ | / ε C (cid:18) k A η k H S | λ | (cid:19) / ε k η k / − ε H S C (cid:13)(cid:13)(cid:13)(cid:13) A ηλ (cid:13)(cid:13)(cid:13)(cid:13) H S + k η k H S ! , and (cid:13)(cid:13)(cid:13) A / η (cid:13)(cid:13)(cid:13) H S | λ | / C (cid:18) k A η k H S | λ | (cid:19) / k η k / H S C (cid:13)(cid:13)(cid:13)(cid:13) A ηλ (cid:13)(cid:13)(cid:13)(cid:13) H S + k η k H S ! . The above estimates with (5.16) yield, | λ ||C λ | C | λ | − / ε (cid:13)(cid:13)(cid:13)(cid:13) A ηλ (cid:13)(cid:13)(cid:13)(cid:13) H S + k η k H S ! , (5.17)and by combining (5.14) and (5.17), for | λ | > α and α > (cid:13)(cid:13)(cid:13) e V λ η (cid:13)(cid:13)(cid:13) H S > C (cid:16) | λ | k η k H S + | λ | − k A η k H S (cid:17) . (5.18)Then (5.1) is proved for θ = 0 and θ = 1 if we show that e V λ is invertible.For that, we first deduce from Proposition 3.1 that e V ∗ λ = λ ( I + K λ ) + 2 ρλA / + A . (5.19)Since λ ∈ C + α if λ ∈ C + α , we perform the same calculations as above that led to (5.18) to find (cid:13)(cid:13)(cid:13) e V ∗ λ η (cid:13)(cid:13)(cid:13) H S > C (cid:16) | λ | k η k H S + | λ | − k A η k H S (cid:17) . (5.20)Relation (5.20) yields that the image of e V λ is dense in H S and relation (5.18) implies that the image of e V λ is closed in H S and that e V λ is injective. We then deduce that e V λ is invertible.Finally, (5.20) gives (5.2). Corollary 5.2.
Let α > be given in Theorem 5.1. For θ ∈ [0 , and β ∈ [0 , such that θ + β thefollowing estimate holds sup λ ∈ C + α | λ | / − θ − β k A θ e V − λ A β k L ( H S ) < + ∞ . (5.21) roof. From (5.2) we deduce sup λ ∈ C + α | λ | − / k ( e V − λ ) ∗ k L ( H S , D ( A )) < + ∞ , from which we obtain by duality, sup λ ∈ C + α | λ | − / k e V − λ k L ( D ( A ) ′ , H S ) < + ∞ . (5.22)By combining (5.22) with (5.1) for θ = 1 with an interpolation argument yields,sup λ ∈ C + α | λ | − / k e V − λ k L ( D ( A β ) ′ , D ( A − β )) < + ∞ ( β ∈ [0 , . (5.23)By combining (5.22) with (5.1) for θ = 0 with an interpolation argument yields for β ∈ [0 , λ ∈ C + α | λ | / − β k e V − λ k L ( D ( A β ) ′ , H S ) < + ∞ ( β ∈ [0 , . (5.24)Then (5.21) follows by interpolating (5.23) and (5.24). Proposition 5.3.
Let α > be given in Theorem 5.1. For θ ∈ [ − / , the following estimates hold sup λ ∈ C + α | λ | / − θ k A − / e V − λ A θ k L ( H S ) < + ∞ , (5.25)sup λ ∈ C + α | λ | / − θ k A θ e V − λ A − / k L ( H S ) < + ∞ . (5.26) Proof.
In a first step, we prove the case θ = 0. For that we first observe that (3.30) with θ = − / k A − / e V − λ k L ( H S ) C k A − / ( I + K λ ) e V − λ k L ( H S ) . (5.27)Next, we make the following calculations: λ A − / ( I + K λ ) e V − λ = A − / − (2 ρλA / + A / ) e V − λ (5.28)which yields with (5.1) and (5.27), | λ | k A − / e V − λ k L ( H S ) | λ | k A − / ( I + K λ ) e V − λ k L ( H S ) C (cid:16) k A − / k L ( H S ) + | λ |k A / e V − λ k L ( H S ) + k A / e V − λ k L ( H S ) (cid:17) C (1 + | λ | − / + | λ | / ) . This leads to (5.25) for θ = 0. Moreover, extimate (5.25) for θ = 0 but for e V ∗ λ instead of e V λ followsanalogously from (5.2), and then (5.26) for θ = 0 by a duality argument.In a second step, let us prove the case θ = 1. For that we first observe, A e V − λ A − / = A − / − λ ( I + K λ ) e V − λ A − / − ρλA / e V − λ A − / . Using (5.26) for θ = 0 and (5.1) for θ = 1 / k A e V − λ A − / k L ( H S ) k A − / k L ( H S ) + C (cid:16) | λ | k e V − λ A − / k L ( H S ) + | λ |k A / e V − λ A − / k L ( H S ) (cid:17) C (1 + | λ | / ) C | λ | / , where in the last inequality we have used the fact that λ ∈ C + α with α >
0. Then (5.26) for θ = 1 is proved.Finally, (5.25) for θ = 1 follows by duality, and (5.25), (5.26) for θ ∈ (0 ,
1) follow by interpolation.It remains to prove the case θ = − /
8. The case θ ∈ ( − / ,
0) will then follow by interpolation. Wecome back to (5.28) from which we deduce λ A − / ( I + K λ ) e V − λ A − / = A − / − (2 ρλA / + A / ) e V − λ A − / , (5.29) nd with (5.27), (5.25) with θ = 1 / θ = 7 /
8, we get | λ | k A − / e V − λ A − / k L ( H S ) | λ | k A − / ( I + K λ ) e V − λ A − / k L ( H S ) C (cid:16) k A − / k L ( H S ) + | λ |k A / e V − λ A − / k L ( H S ) + k A / e V − λ A − / k L ( H S ) (cid:17) C (1 + | λ | − / ) C. Then the case θ = − / Corollary 5.4.
Let α > be given in Theorem 5.1. For θ ∈ [ − / , and β ∈ [ − / , such that θ + β the following estimate holds sup λ ∈ C + α | λ | / − θ − β k A θ e V − λ A β k L ( H S ) < + ∞ . (5.30) Proof.
First, assume θ ∈ [0 , β = 0 and (5.26) with an interpolation argument we deduce(5.30) for θ ∈ [0 ,
1] and β ∈ [ − / , θ ∈ [0 ,
1] and β ∈ [ − / , θ = 0 and β ∈ [ − / ,
1] and (5.25) with θ = β allows us to obtain(5.30) for θ ∈ [ − / ,
0] and β ∈ [ − / , The goal of this section it to prove the Gevrey type resolvent estimates for the operator A defined by(2.28)–(2.29). In order to prove Theorem 2.4, we rewrite the resolvent equation in a more convenient way.Assume λ ∈ C + α for α > f, g, h ] ∈ H . We set [ v, η , η ] def = ( λ − A ) − [ f, g, h ] sothat λv − div T ( v, p ) = f in F , div v = 0 in F ,v = Λ η on ∂ F ,λη − η = gλη + A η = − Λ ∗ (cid:8) T ( v, p ) n | ∂ F (cid:9) + h. (6.1)Using W λ and A introduced in (3.2) and (2.23), we can decompose the fluid velocity of (6.1) as v = W λ η + ( λI − A ) − P f, and using L λ ∈ L ( D ( A / ) , D ( A / )) and T λ ∈ L ( L ( F ) , D ( A / )) defined by (3.5) and (3.17) we can rewritesystem (6.1) as (cid:26) λη − η = gλη + A η + L λ η = T λ f + h. (6.2)This writes ( λI + A λ ) (cid:20) η η (cid:21) = (cid:20) g T λ f + h (cid:21) (6.3)with A λ def = (cid:20) − IA L λ (cid:21) . (6.4)We recall that V λ defined by (3.12) is invertible. It is a consequence of the following result proved inProposition 4.8 of [3]. Proposition 6.1.
For all λ ∈ C + the operator V λ is an isomorphism from D ( A ) onto H S . Hence, direct calculations lead to the following formulas for the inverse of λI + A λ and of λI − A :( λI + A λ ) − = I − V − λ A λ V − λ − V − λ A λV − λ , (6.5)and ( λI − A ) − = ( λI − A ) − P + λW λ V − λ T λ − W λ V − λ A λW λ V − λ V − λ T λ I − V − λ A λ V − λ λV − λ T λ − V − λ A λV − λ . (6.6) .1 Estimation of V − λ In this section, we estimate the inverse of the operator V λ defined in (3.12) for λ ∈ C + α and α > ρ < ρ / ρ is defined in Proposition 3.1. The main result of thissection is the following: Theorem 6.2.
Let α > be given in Theorem 5.1. For θ ∈ [ − / , / and β ∈ [ − / , / such that θ + β the following estimate holds sup λ ∈ C + α | λ | / − θ − β k A θ V − λ A β k L ( H S ) < + ∞ . (6.7) Proof.
Comparing (3.12) and (3.13), we see that V λ − e V λ = λS λ , S λ def = G λ − ρA / and thus [ I + λ e V − λ S λ ] V − λ = e V − λ . (6.8)We thus need to estimate the inverse of [ I + λ e V − λ S λ ].From Proposition 3.1 and in particular (3.8), we have S λ is a positive self-adjoint operator satisfying k S / λ η k H S C (cid:16) k A / η k H S + | λ | / k A − / η k H S (cid:17) ( η ∈ D ( A / )) . Combining the above inequality with (5.21) and (5.25) we obtain for β ∈ [ − / , / k S / λ e V − λ A β η k H S C | λ | − / β k η k H S . (6.9)Analogously we can prove (6.9) but for e V ∗ λ instead of e V λ . Then a duality argument yield for θ ∈ [ − / , / k A θ e V − λ S / λ η k H S C | λ | − / θ k η k H S . (6.10)From (3.13), we obtain that for any λ ∈ C + and for any ζ ∈ D ( A ),Re h e V λ ζ, λζ i H S = Re λ k λ ( I + K λ ) / ζ k H S + 2 ρ k λA / ζ k H S + Re λ k A / ζ k H S > . (6.11)In particular, for any λ ∈ C + and for any ζ ∈ H S ,Re h λ e V − λ ζ, ζ i H S > . (6.12)Let us now consider the equation η + λ e V − λ S λ η = f. (6.13)If we multiply (6.13) by S λ η , take the real part and use (6.12), we obtain k S / λ η k H S k S / λ f k H S and applying such a result to equality (6.8) yields ∀ η ∈ H S , k S / λ V − λ η k H S k S / λ e V − λ η k H S . Thus, coming back to equality (6.8) we deduce that for η ∈ H S , θ ∈ [ − / , / β ∈ [ − / , / θ + β k A θ V − λ A β η k H S k A θ e V − λ A β η k H S + | λ |k A θ e V − λ S λ V − λ A β η k H S k A θ e V − λ A β η k H S + | λ |k A θ e V − λ S / λ k L ( H S ) k S / λ V − λ A β η k H S k A θ e V − λ A β η k H S + | λ |k A θ e V − λ S / λ k L ( H S ) k S / λ e V − λ A β η k H S Then using estimates (5.30), (6.9) and (6.10) yields (6.7). .2 Proof of Theorem 2.4 Proof.
First, the exponential stability of ( e A t ) t > (see Proposition 2.2) and standard results (see [5, p.101,Theorem 2.5]) yield that k ( λ − A ) − k L ( H ) is uniformly bounded for λ ∈ C + . This implies thatsup λ ∈ C + , | λ | α (cid:8) k A ( λ − A ) − k L ( H ) + | λ |k ( λ − A ) − k L ( H ) (cid:9) < ∞ . Using (2.24) and (2.27), we deduce (2.33) and (2.34) for λ ∈ C + with | λ | α . In the remaining part of theproof, we can thus assume λ ∈ C + α (see (2.13)) for α > | λ |k ( λ − A ) − P f k L ( F ) C k f k L ( F ) . (6.14)From (3.18), (6.7) with ( θ, β ) = (0 ,
0) and (3.17), | λ | / k λW λ V − λ T λ f k L ( F ) C | λ | / k V − λ T λ f k H S C kT λ f k H S C k f k L ( F ) . (6.15)From (6.7) with ( θ, β ) = (1 / ,
0) and (3.17), | λ | / k A / V − λ T λ f k H S C k f k L ( F ) . (6.16)From (6.7) with ( θ, β ) = (0 ,
0) and (3.17), | λ | / k λV − λ T λ f k H S C k f k L ( F ) . (6.17)From (3.18) and (6.7) with ( θ, β ) = (0 , / | λ | / k W λ V − λ A g k L ( F ) C | λ | / k V − λ A g k H S C k A / g k H S . (6.18)From (6.7) with ( θ, β ) = (1 / , / | λ | / (cid:13)(cid:13)(cid:13)(cid:13) A / I − V − λ A λ g (cid:13)(cid:13)(cid:13)(cid:13) H S C k A / g k H S . (6.19)From (6.7) with ( θ, β ) = (0 , / | λ | / (cid:13)(cid:13) V − λ A g (cid:13)(cid:13) H S C k A / g k H S . (6.20)From (3.18) and (6.7) with ( θ, β ) = (0 , | λ | / k λW λ V − λ h k L ( F ) C | λ | / k V − λ h k H S C k h k H S . (6.21)From (6.7) with ( θ, β ) = (1 / , | λ | / (cid:13)(cid:13)(cid:13) A / V − λ h (cid:13)(cid:13)(cid:13) H S C k h k H S . (6.22)From (6.7) with ( θ, β ) = (0 , | λ | / (cid:13)(cid:13) λV − λ h (cid:13)(cid:13) H S C k h k H S . (6.23)Using (6.6), we deduce (2.33).Next, let us prove (2.34). From Proposition 3.2, (3.17), (3.19) with θ = 1, (6.7) with ( θ, β ) = (3 / , − / θ, β ) = ( − / , − / θ, β ) = (7 / , − / θ, β ) = (3 / , − / k ( λ − A ) − P f k H ( F ) + k λW λ V − λ T λ f k H ( F ) + k V − λ T λ f k D ( A / ) + k λV − λ T λ f k D ( A / ) C k f k L ( F ) . From (3.19) with θ = 1, (6.7) with ( θ, β ) = (3 / , /
8) and ( θ, β ) = ( − / , / θ, β ) = (3 / , / k W λ V − λ A g k H ( F ) + k V − λ A g k D ( A / ) C k g k D ( A / ) . (6.24) rom the relation A ( λ − A ) − = − I + λ ( λ − A ) − with (2.33) we first first obtainsup λ ∈ C + α | λ | − / k A ( λ − A ) − k L ( H ) < + ∞ , and by interpolation with (2.33) we getsup λ ∈ C + α k ( − A ) / ( λ − A ) − k L ( H ) < + ∞ . (6.25)Thus using (6.25) we obtain (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( − A ) / ( λ − A ) − g (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( − A ) / ( λ − A ) − ( − A ) / g (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( − A ) / g (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H and thus from Proposition 2.3 and (6.6) k W λ V − λ A g k H / ( F ) + (cid:13)(cid:13)(cid:13)(cid:13) I − V − λ A λ g (cid:13)(cid:13)(cid:13)(cid:13) D ( A / ) + k V − λ A g k D ( A / ) C k g k D ( A / ) which gives (cid:13)(cid:13)(cid:13)(cid:13) I − V − λ A λ g (cid:13)(cid:13)(cid:13)(cid:13) D ( A / ) C k g k D ( A / ) . Then, since we also have (6.24), we have proved k W λ V − λ A g k H ( F ) + (cid:13)(cid:13)(cid:13)(cid:13) I − V − λ A λ g (cid:13)(cid:13)(cid:13)(cid:13) D ( A / ) + k V − λ A g k D ( A / ) C k g k D ( A / ) . From (3.19) with θ = 1, (6.7) with ( θ, β ) = (3 / , − /
8) and ( θ, β ) = ( − / , − / θ, β ) =(7 / , − / θ, β ) = (3 / , − / k λW λ V − λ h k H ( F ) + (cid:13)(cid:13) V − λ h (cid:13)(cid:13) D ( A / ) + k λV − λ h k D ( A / ) C k h k D ( A / ) . Then combining the above estimates we have proved (cid:13)(cid:13) ( λI − A ) − z (cid:13)(cid:13) H ( F ) ×D ( A / ) ×D ( A / ) C k z k L ( F ) ×D ( A / ) ×D ( A / ) (cid:16) z ∈ H ∩ (cid:16) L ( F ) × D ( A / ) × D ( A / ) (cid:17)(cid:17) . (6.26)It remains to estimate in (2.34) the term | λ | (cid:13)(cid:13) ( λ − A ) − z (cid:13)(cid:13) L ( F ) ×D ( A / ) ×D ( A / ) ′ . Assume[ w, ξ , ξ ] ∈ H ∩ (cid:16) H ( F ) × D ( A / ) × D ( A / ) (cid:17) . First, from the continuity of Λ ∗ : H / ( ∂ F ) → D ( A / ) and a trace inequality we have, k Λ ∗ (2 D ( w ) n ) k D ( A / ) ′ C k Λ ∗ (2 D ( w ) n ) k D ( A / ) C k D ( w ) n k H / ( ∂ F ) C k w k H ( F ) . From (2.26) and the above estimate we deduce, (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) A wξ ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( F ) ×D ( A / ) ×D ( A / ) ′ = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) P ∆ wξ − A ξ − Λ ∗ (2 D ( w ) n ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( F ) ×D ( A / ) ×D ( A / ) ′ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∆ wξ − A ξ − Λ ∗ (2 D ( w ) n ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) L ( F ) ×D ( A / ) ×D ( A / ) ′ C (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) wξ ξ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H∩ (cid:16) H ( F ) ×D ( A / ) ×D ( A / ) (cid:17) . hen with formula λ ( λ − A ) − = A ( λ − A ) − + I and (6.26) we deduce | λ | (cid:13)(cid:13) ( λ − A ) − z (cid:13)(cid:13) L ( F ) ×D ( A / ) ×D ( A / ) ′ C k z k L ( F ) ×D ( A / ) ×D ( A / ) , which gives the result. We prove here Theorem 1.1 by a fixed point argument. The proof is quite similar to the same proof for the“flat” case considered in [3]. For sake of completeness, we give here the main ideas of the proof.For
R > , T > B R,T def = n ( F, G ) ∈ L (0 , T ; L ( F )) × L (0 , T ; H / ( I )) ; k ( F, G ) k L (0 ,T ; L ( F )) × L (0 ,T ; H / ( I ))) R o . For any (
F, G ) ∈ B R,T , we consider the solution ( w, η, q ) of system (2.30) given by Theorem 1.2. Inparticular w ∈ L (0 , T ; H ( F )) ∩ C ([0 , T ]; H ( F )) ∩ H (0 , T ; L ( F )) , q ∈ L (0 , T ; H ( F )) (7.1) η ∈ L (0 , T ; H / ( I )) ∩ C ([0 , T ]; H / ( I )) ∩ H (0 , T ; H / ( I )) , (7.2) ∂ t η ∈ L (0 , T ; H / ( I )) ∩ C ([0 , T ]; H / ( I )) ∩ H (0 , T ; H / ( I ) ′ ) , (7.3)with k w k L (0 ,T ; H ( F )) ∩ C ([0 ,T ]; H ( F )) ∩ H (0 ,T ; L ( F )) + k q k L (0 ,T ; H ( F )) , + k η k L (0 ,T ; H / ( I )) ∩ C ([0 ,T ]; H / ( I )) + k ∂ t η k L (0 ,T ; H / ( I )) ∩ C ([0 ,T ]; H / ( I )) C (cid:0) R + k [ w , η , η ] k H ( F ) × H ε ( I ) × H ε ( I ) (cid:1) . (7.4)Above and below, C denotes a positive constant that depends on k η k W , ∞ ( I ) only.In what follows, we take R (large enough) such that R > k [ w , η , η ] k H ( F ) × H ε ( I ) × H ε ( I ) . (7.5)First we notice that by interpolation, (7.4) yields k η k H / (0 ,T ; H ( I )) + k η k L (0 ,T ; H ( I )) + k ∂ t η k L (0 ,T ; H ( I )) + k w k L (0 ,T ; H / ( F )) C R. (7.6)The difference with respect to the proof in [3] is that here our formula for X and Y (see (2.2) and (2.3))involves η . Nevertheless, one can write X ( t, y , y ) = ( y , x (1 + ζ ( t, x ))) and Y ( t, x , x ) = (cid:18) x , y ζ ( t, y ) (cid:19) where ζ ( t, y ) def = η ( t, y ) − η ( y )1 + η ( y ) . Using that η ∈ H ε ( I ), η > − η ∈ H ε ( I ) . Combining this with Sobolev embeddings, we deduce that k ζ k C ([0 ,T ]; H / ( I )) + k ∂ t ζ k L (0 ,T ; H / ( I )) ∩ C ([0 ,T ]; H / ( I )) + k ζ k H / (0 ,T ; H ( I )) + k ζ k L (0 ,T ; H ( I )) + k ∂ t ζ k L (0 ,T ; H ( I )) C R. (7.7)In particular, we have the same estimates for ζ than for η except the estimate in L (0 , T ; H / ( I )). In theproof of [3], we only need the norms in (7.7) for the estimates associated with the change of variables (see A.1)–(A.8)) and this allows us to prove that for T small enough, we can construct the change of variablesdefined in Section 2.1 and consider the mapping Z : ( F, G ) ( b F ( η, w, q ) , b G η ( η, w )) (7.8)where the maps b F and b G η are defined by (2.9) and (2.10), and ( w, η, p ) is solution of system (1.22)-(1.23).We can also show that kZ ( F, G ) k L (0 ,T ; L ( F )) × L (0 ,T ; H / ( I )) C T / R N , (7.9)for some N >
2. More precisely, the main difference from [3] is the following: using Proposition A.1 in [3],(7.7) and ζ (0 , · ) = 0, we deduce that k ζ k L ∞ (0 ,T ; H ( I )) CT / k ζ k H / (0 ,T ; H ( I )) CT / R. This yields k∇ Y ( X ) − I k L ∞ (0 ,T ; L ∞ ( F ) ) + k det( ∇ X ) − k L ∞ (0 ,T ; L ∞ ( F )) CT / R instead of (6.16) in [3]. Then, following the computation in [3] we deduce (7.9).From (7.9), for all T C − R − N we have Z ( F, G ) ∈ B R,T . Similarly, taking T possibly smaller, we can also show that Z is a strict contraction on B R,T and usingthe Banach fixed point theorem, we deduce the existence and uniqueness of (
F, G ) ∈ B R,T such that Z (( F, G )) = (
F, G ) . The corresponding solution ( η, w, q ) of system (1.22)-(1.23) is a solution of (2.7)–(2.8).The proof of the uniqueness is similar to the proof of uniqueness given in [3].
A Formula for the change of variables
Let us give some formulas for the change of variables X ( t, y , y ) = ( y , x (1 + ζ ( t, x ))) and Y ( t, x , x ) = (cid:18) x , y ζ ( t, y ) (cid:19) that are used in Section 7 for the fixed point. We also need these formulas for the study of the linear system(see Section 4.1), and in that case, ζ = η , X = e X , a = ˜ a and b = ˜ b are independent of time. ∇ X ( t, y , y ) = (cid:20) y ∂ s ζ ζ (cid:21) , b ( t, y , y ) = (cid:20) ζ − y ∂ s ζ (cid:21) , (A.1) ∇ Y ( t, x , x ) = − x ∂ s ζ (1 + ζ )
11 + ζ , a ( t, x , x ) =
11 + ζ x ∂ s ζ (1 + ζ ) . (A.2) a ( X ) =
11 + ζ y ∂ s ζ ζ , ∇ Y ( X ) = − y ∂ s ζ ζ
11 + ζ , (A.3) ∇ Y ( X ) − I = − y ∂ s ζ ζ − ζ ζ , det( ∇ X ) = 1 + ζ, (A.4) ∂a∂x ( X ) = − ∂ s ζ (1 + ζ ) y ∂ ss ζ (1 + ζ ) − ∂ s ζ ) (1 + ζ ) , ∂a∂x ( X ) = ∂ s ζ (1 + ζ ) , (A.5) a∂x ∂x ( X ) = ∂ ss ζ (1 + ζ ) − ∂ s ζ ) (1 + ζ ) , (A.6) ∂ a∂x ( X ) = − ∂ s ζ (1 + ζ ) y ∂ sss ζ (1 + ζ ) − ζ ) ∂ s ζ∂ ss ζ + 6( ∂ s ζ ) (1 + ζ ) , ∂ a∂x ( X ) = 0 , (A.7) ∂∂x ∇ Y ( X ) = y − ∂ ss ζ (1 + ζ ) + 2( ∂ s ζ ) (1 + ζ ) − ∂ s ζ (1 + ζ ) , ∂∂x ∇ Y ( X ) = − ∂ s ζ (1 + ζ ) . (A.8) ∂ t a ( X ) = − ∂ t ζ (1 + ζ ) y ∂ ts ζ (1 + ζ ) − ∂ s ζ∂ t ζ (1 + ζ ) , ∂ t Y ( X ) = − y ∂ t ζ ζ . (A.9)We also recall here how to obtain formulas (4.5): differentiating (4.2), we deduce successively ∂w i ∂x j = X k ∂ e a ik ∂x j e w k ( e Y ) + X k,ℓ e a ik ∂ e w k ∂y ℓ ( e Y ) ∂ e Y ℓ ∂x j and ∂ w i ∂x j = X k ∂ e a ik ∂x j e w k ( e Y ) + 2 X k,ℓ ∂ e a ik ∂x j ∂ e w k ∂y ℓ ( e Y ) ∂ e Y ℓ ∂x j + X k,ℓ,m e a ik ∂ e w k ∂y ℓ ∂y m ( e Y ) ∂ e Y ℓ ∂x j ∂ e Y m ∂x j + X k,ℓ e a ik ∂ e w k ∂y ℓ ( e Y ) ∂ e Y ℓ ∂x j . Composing by e Y and multiplying by e b αi , we deduce the first formula of (4.5). The second one can be donein a similar way. References [1] M. Badra. Lyapunov function and local feedback boundary stabilization of the Navier-Stokes equations.
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