**Asymptotic Behavior of the Semigroup** **Associated with the Linearized Compressible**

**Navier-Stokes Equation in an Infinite Layer**

By

YoshiyukiKagei*∗*

**Abstract**

Asymptotic behavior of solutions to the linearized compressible Navier-Stokes
equation around a given constant state is considered in an infinite layer**R**^{n−1}*×*(0*, a*),
*n* *≥* 2, under the no slip boundary condition for the momentum. The *L** ^{p}* decay
estimates of the associated semigroup are established for all 1

*≤p≤ ∞*. It is also shown that the time-asymptotic leading part of the semigroup is given by an

*n−*1 dimensional heat semigroup.

**§****1.** **Introduction**

This paper is concerned with the large time behavior of solutions to the following system of equations:

(1.1) *∂*_{t}*u*+*Lu*= 0,

where *u*=

*φ*
*m*

with *φ*= *φ(x, t)* *∈***R** and *m* =* ^{T}*(m

^{1}(x, t), . . . , m

*(x, t))*

^{n}*∈*

**R**

*,*

^{n}*n≥*2, and

*L*is an operator deﬁned by

*L*=

0 *γdiv*
*γ∇ −ν*∆I_{n}*−ν∇*div

with positive constants*ν* and*γ* and a nonnegative constant*ν. Here* *t >*0 de-
notes the time variable and*x∈***R*** ^{n}*denotes the space variable; the superscript

^{T}*·*

Communicated by H. Okamoto. Received June 16, 2006. Revised November 9, 2006.

2000 Mathematics Subject Classiﬁcation(s): 35Q30, 76N15.

*∗*Faculty of Mathematics, Kyushu University, Fukuoka 812-8581, Japan.

stands for the transposition;*I** _{n}* is the

*n×n*identity matrix; and div,

*∇*and ∆ are the usual divergence, gradient and Laplacian with respect to

*x. We consider*(1.1) in an inﬁnite layer

*Ω*=**R**^{n−1}*×*(0, a) =

*x*=

*x*^{}*x*_{n}

;*x*^{}*∈***R**^{n−1}*,*0*< x*_{n}*< a*
under the boundary condition

(1.2) *m|** _{∂Ω}* = 0,

together with the initial condition

(1.3) *u|** _{t=0}*=

*u*

_{0}=

*φ*_{0}
*m*_{0}

*.*

Problem (1.1)–(1.3) is obtained by the linearization of the compressible Navier-
Stokes equation around a motionless state with a positive constant density,
where *φ*is the perturbation of the density and*m*is the momentum.

In [7] we showed that*−L*generates an analytic semigroup

*U*

^{(t) in}

^{W}^{1,p}

*×*

*L*

*for 1*

^{p}*< p <∞*. In this paper we establish an

*L*

*decay estimate of*

^{p}*U*

^{(t) for}

all 1*≤p≤ ∞*and give a more detailed description of the behavior of

*U*

^{(t) as}

*t→ ∞*.

One of the primary factors aﬀecting the large time behavior of solutions
to (1.1)–(1.3) is that (1.1) is a symmetric hyperbolic-parabolic system. Due to
this structure, solutions of (1.1) exhibit characters of solutions of both wave
and heat equations. In the case of the Cauchy problem on the whole space**R*** ^{n}*,
detailed descriptions of large time behavior of solutions have been obtained
([5, 6, 11, 13, 14]). Hoﬀ and Zumbrun [5, 6] showed that there appears some
interesting interaction of hyperbolic and parabolic aspects of (1.1) in the decay
properties of

*L*

*norms with 1*

^{p}*≤p≤ ∞*. It was shown in [5, 6] that the solution is asymptotically written in the sum of two terms, one is the solution of the heat equation and the other is given by the convolution of the heat kernel and the fundamental solution of the wave equation. The latter one is called the diﬀusion wave and it decays faster than the heat kernel in

*L*

*norm for*

^{p}*p >*2 while slower for

*p <*2. This decay property of the diﬀusion wave also appears in the exterior domain problem ([12]). In the case of the half space problem, it was shown in [8, 9] that not only the above mentioned behavior of the diﬀusion wave appears but also some diﬀerence to the Cauchy problem appears in the decay property of the spatial derivatives due to the presence of the unbounded boundary.

There is one more factor that aﬀects the large time behavior of solutions
to (1.1)–(1.3). In contrast to the domains mentioned above, the inﬁnite layer
*Ω* has a ﬁnite thickness in the *x** _{n}* direction. This implies that the Poincar´e
inequality holds. If one considers, for example, the incompressible Navier-
Stokes equation under the no slip boundary condition (1.2), then it is easy
to see that, by the Poincar´e inequality, the

*L*

^{2}norm of the solution tends to zero exponentially as

*t→ ∞*. In the case of problem (1.1)–(1.3), the Poincar´e inequality holds for

*m*but not for

*φ. This leads to that the spectrum reaches*the origin but it is like the one such as the

*n−*1 dimensional Laplace operator.

As a result, no hyperbolic feature appears in the leading part of the solution.

In fact, we will show that the solution *u*=

*U*

^{(t)u}0of (1.1)–(1.3) satisﬁes (1.4)

*u(t)*

*L*

*=*

^{p}*O(t*

^{−}

^{n−1}^{2}

^{(1−}

^{1}

^{p}^{)}),

*u(t)−u*

^{(0)}(t)

*L*

*=*

^{p}*O(t*

^{−}

^{n−1}^{2}

^{(1−}

^{1}

^{p}^{)−}

^{1}

^{2}) for any 1

*≤*

*p≤ ∞*as

*t*

*→ ∞*. Here

*u*

^{(0)}= (φ

^{(0)}(x

^{}*, t),*0) and

*φ*

^{(0)}(x

^{}*, t) is a*function satisfying

*∂*_{t}*φ*^{(0)}*−κ∆*^{}*φ*^{(0)}= 0, *φ*^{(0)}

*t=0*= 1
*a*

_{a}

0

*φ*_{0}(x^{}*, x** _{n}*)

*dx*

_{n}*,*where

*κ*=

^{a}_{12ν}

^{2}

^{γ}^{2}and ∆

*=*

^{}*∂*

_{x}^{2}

1+*· · ·*+*∂*^{2}_{x}

*n−1*.

The proof of (1.4) is based on a detailed analysis of the resolvent (λ+
*L)** ^{−1}* associated with (1.1)–(1.3). We will consider the Fourier transform (λ+

*L*

*)*

_{ξ}*of the resolvent in*

^{−1}*x*

^{}*∈*

**R**

*, where*

^{n−1}*ξ*

^{}*∈*

**R**

*denotes the dual variable. The semigroup*

^{n−1}*U*

(t) generated by *−L*is then written as

*U*

^{(t) =}

*F*

^{−1}*ξ*

^{}1 2πi

*Γ**e** ^{λt}*(λ+

*L*

*)*

_{ξ}

^{−1}*dλ*

. Since (λ+*L** _{ξ}*)

*has diﬀerent characters be- tween the cases*

^{−1}*|ξ*

^{}*|>>*1 and

*|ξ*

^{}*|<<*1, we decompose the semigroup

*U*

^{(t)}

into the two parts according to the partition: *|ξ*^{}*| ≥r*_{0} and *|ξ*^{}*| ≤r*_{0} for some
*r*_{0}*>*0.

In [7] we established the estimates of (λ+*L** _{ξ}*)

*with*

^{−1}*|ξ*

^{}*| ≥r*

_{0}, which will lead to the exponential decay of the corresponding part of

*U*

(t). We derived an
integral representation for (λ+*L*

*)*

_{ξ}*and applied the Fourier multiplier the- orem as in [1, 2, 3], where*

^{−1}*L*

*estimates for the incompressible Stokes equation were established.*

^{p}In this paper we study (λ+*L** _{ξ}*)

*with*

^{−1}*|ξ*

^{}*|*

*<<*1. We regard

*L*

*as a perturbation from*

_{ξ}*L*

_{0}to investigate the spectrum of

*−L*near

*λ*= 0. We will ﬁnd that the spectrum near the origin is given by

*−κ|ξ*

^{}*|*

^{2}+

*O(|ξ*

^{}*|*

^{4}) with

*|ξ*^{}*|<<*1. It should be noted that the structure of the spectrum near the origin
is quite similar to that of the linearized operator appearing in the free surface
problem of viscous incompressible ﬂuid studied in [4]. As in [4] we will appeal

the analytic perturbation theory to compute the eigenvalue and the associated
eigenprojection of *λ*+*L** _{ξ}* for

*|ξ*

^{}*|*

*<<*1. We will then derive the estimates for the integral kernel of the eigenprojection which are used to obtain the

*L*

*estimates of the semigroup.*

^{p}This paper is organized as follows. In Section 2 we introduce some notation
and state the main result of this paper. In Section 3 we investigate (λ+*L** _{ξ}*)

*with*

^{−1}*|ξ*

^{}*|*

*<<*1. Section 4 is devoted to the proof of the main result. In the Appendix we will give the integral representation for (λ+

*L*

*)*

_{ξ}*obtained in [7] to estimate some part of the Dunford integral for the semigroup.*

^{−1}**§****2.** **Main Result**

We ﬁrst introduce some notation which will be used throughout the paper.

For a domain*D*and 1*≤p≤ ∞*we denote by*L** ^{p}*(D) the usual Lebesgue space
on

*D*and its norm is denoted by

*·*

*L*

*(D). Let be a nonnegative integer.*

^{p}The symbol *W** ^{ ,p}*(D) denotes the

*-th orderL*

*Sobolev space on*

^{p}*D*with norm

*·*_{W}^{,p}_{(D)}. When*p*= 2, the space*W** ^{ ,2}*(D) is denoted by

*H*(D) and its norm is denoted by

*·*

_{H}

^{}_{(D)}.

*C*

_{0}(D) stands for the set of all

*C*functions which have compact support in

*D. We denote byW*

_{0}

^{1,p}(D) the completion of

*C*

_{0}

^{1}(D) in

*W*

^{1,p}(D). In particular,

*W*

_{0}

^{1,2}(D) is denoted by

*H*

_{0}

^{1}(D).

We simply denote by *L** ^{p}*(D) (resp.,

*W*

*(D),*

^{ ,p}*H*(D)) the set of all vector ﬁelds

*m*=

*(m*

^{T}^{1}

*, . . . , m*

*) on*

^{n}*D*with

*m*

^{j}*∈L*

*(D) (resp.,*

^{p}*W*

*(D),*

^{ ,p}*H*(D)),

*j*= 1, . . . , n, and its norm is also denoted by

*·*

_{L}

^{p}_{(D)}(resp.,

*·*

_{W}

^{,p}_{(D)},

*·*

_{H}

^{}_{(D)}).

For *u* =

*φ*
*m*

with *φ* *∈* *W** ^{k,p}*(D) and

*m*=

*(m*

^{T}^{1}

*, . . . , m*

*)*

^{n}*∈*

*W*

*(D), we deﬁne*

^{ ,q}*u*

_{W}

^{k,p}_{(D)×W}

^{,q}_{(D)}by

*u*

_{W}

^{k,p}_{(D)×W}

^{,q}_{(D)}=

*φ*

_{W}

^{k,p}_{(D)}+

*m*

_{W}

^{,q}_{(D)}. When

*k*= and

*p*=

*q, we simply writeu*

_{W}

^{k,p}_{(D)×W}

^{k,p}_{(D)}=

*u*

_{W}

^{k,p}_{(D)}.

In case*D*=*Ω*we abbreviate*L** ^{p}*(Ω) (resp.,

*W*

*(Ω),*

^{ ,p}*H*(Ω)) as

*L*

*(resp.,*

^{p}*W*

*,*

^{ ,p}*H*). In particular, the norm

*·*

_{L}

^{p}_{(Ω)}=

*·*

*L*

*is denoted by*

^{p}*·*

*p*.

In case *D* = (0, a) we denote the norm of *L** ^{p}*(0, a) by

*| · |*

*. The inner product of*

_{p}*L*

^{2}(0, a) is denoted by

(f, g) =
_{a}

0

*f*(x* _{n}*)g(x

*)*

_{n}*dx*

_{n}*,*

*f, g∈L*

^{2}(0, a).

Here *g* denotes the complex conjugate of*g. Furthermore, we deﬁne* *·,· *and

*· *by

*f, g* = 1

*a*(f, g) and *f* =*f,*1 =1
*a*

_{a}

0

*f*(x* _{n}*)

*dx*

*for*

_{n}*f, g∈L*

^{2}(0, a), respectively.

The norms of *W** ^{ ,p}*(0, a) and

*H*(0, a) are denoted by

*| · |*

*W*

*and*

^{,p}*| · |*

*H*

*, respectively.*

^{}We often write*x∈Ω*as*x*=

*x*^{}*x*_{n}

,*x** ^{}* =

*(x*

^{T}_{1}

*, . . . , x*

*)*

_{n−1}*∈*

**R**

*. Partial derivatives of a function*

^{n−1}*u*in

*x,*

*x*

*,*

^{}*x*

*and*

_{n}*t*are denoted by

*∂*

_{x}*u,*

*∂*

_{x}*u,*

*∂*

_{x}

_{n}*u*and

*∂*

_{t}*u, respectively. We also write higher order partial derivatives ofu*in

*x*as

*∂*

^{k}

_{x}*u*= (∂

_{x}

^{α}*u;|α|*=

*k).*

We denote the*k×k*identity matrix by*I** _{k}*. In particular, when

*k*=

*n*+ 1, we simply write

*I*for

*I*

*. We also deﬁne (n+ 1)*

_{n+1}*×*(n+ 1) diagonal matrices

*Q*

_{0}and

*Q*by

*Q*_{0}= diag (1,0, . . . ,0), *Q*= diag (0,1, . . . ,1).

We then have, for *u*=

*φ*
*m*

*∈***R*** ^{n+1}*,

*Q*_{0}*u*=

*φ*
0

*,* *Qu* =

0
*m*

*.*

We next introduce some notation about integral operators. For a function
*f* =*f*(x* ^{}*) (x

^{}*∈*

**R**

*), we denote its Fourier transform by*

^{n−1}*f*or

*F*

^{f:}*f*(ξ* ^{}*) = (

*F*

^{f}^{)(ξ}

^{}^{) =}

^{}

**R**^{n−1}

*f*(x* ^{}*)e

^{−iξ}

^{}

^{·x}

^{}*dx*

^{}*.*

The inverse Fourier transform is denoted by

*F*

^{−1}^{:}

(

*F*

^{−1}*)(x) = (2π)*

^{f}

^{−(n−1)}**R**^{n−1}

*f*(ξ* ^{}*)e

^{iξ}

^{}

^{·x}

^{}*dξ*

^{}*.*

For a function *K(x*_{n}*, y** _{n}*) on (0, a)

*×*(0, a) we will denote by

*Kf*the integral operator

_{a}0 *K(x*_{n}*, y** _{n}*)f(y

*)*

_{n}*dy*

*.*

_{n}We denote the resolvent set of a closed operator *A*by*ρ(A) and the spec-*
trum of *A*by*σ(A). ForΛ∈***R**and*θ∈*(^{π}_{2}*, π) we will denote*

*Σ(Λ, θ) ={λ∈***C;***|*arg (λ*−Λ)| ≤θ}.*

We now state the main result of this paper. In [7] we showed that *−L*
generates an analytic semigroup

*U*

^{(t) on}

^{W}^{1,r}

^{(Ω)}

*×L*

*(Ω) (1*

^{r}*< r <∞*) and established the estimates of

*U*

^{(t) for 0}

*1. As for the large time behavior of*

^{< t}^{≤}*U*

(t), we have the following result.
**Theorem 2.1.** *Let*1*< r <∞and let*

*U*

^{(t)}

*be the semigroup generated*

*by*

*−L. Suppose that*

*u*

_{0}

*∈*

*L*

^{1}(Ω)

*∩*

*W*^{1,r}(Ω)*×L** ^{r}*(Ω)

*. Then the solution*
*u*=

*U*

^{(t)u}0

*of problem*(1.1)–(1.3)

*is decomposed as*

*U*

^{(t)u}0=

*U*

^{(0)}

^{(t)u}0+

*U*

^{(∞)}

^{(t)u}0

*,*

*where each term on the right-hand side has the following properties.*

(i)

*U*

^{(0)}

^{(t)u}0

*is written in the form*

*U*

^{(0)}

^{(t)u}0=

*W*

^{(0)}

^{(t)u}0+

*R*

^{(0)}

^{(t)u}0

*.*

*Here*

*W*

^{(0)}

^{(t)u}0=

*φ*^{(0)}(x^{}*, t)*
0

*andφ*^{(0)}(x^{}*, t)is a function independent ofx*_{n}*and satisfies the following heat equation on***R*** ^{n−1}*:

*∂*_{t}*φ*^{(0)}*−κ∆*^{}*φ*^{(0)}= 0, *φ*^{(0)}*|** _{t=0}*=

*φ*

_{0}(x

^{}*,·*)

*,*

*where*

*κ*=

^{a}_{12ν}

^{2}

^{γ}^{2}

*and*∆

*=*

^{}*∂*

_{x}^{2}

1+*· · ·*+*∂*_{x}^{2}

*n−1**. The function*

*R*

^{(0)}

^{(t)u}0

*satisfies*

*the following estimate. For any*1

*≤p≤ ∞and*= 0,1, there exists a positive

*constant*

*C*

*such that*

*∂*_{x}

*R*

^{(0)}

^{(t)u}0

_{p}*≤Ct*

^{−}

^{n−1}^{2}

^{(1−}

^{1}

^{p}^{)−}

^{1}

^{2}

*u*

_{0}

_{1}

*holds for*

*t≥*1. Furthermore, it holds that

*∂*_{x}

*R*

^{(0)}

^{(t)}

^{Qu}^{}0

*p*

*≤Ct*

^{−}

^{n−1}^{2}

^{(1−}

^{1}

^{p}^{)−1}

*Qu*

_{0}1

*and*

*R*

^{(0)}

^{(t)[∂}

*x*

*Qu*

_{0}]

*p*

*≤Ct*

^{−}

^{n−1}^{2}

^{(1−}

^{1}

^{p}^{)−}

^{1}

^{2}

*Qu*

_{0}1

*.*(ii)

*There exists a positive constantc*

*such that*

*U*

^{(∞)}

^{(t)u}0

*satisfies*

*∂*_{x}

*U*

^{(∞)}

^{(t)u}0

*r*

*≤Ce*

^{−ct}*u*

_{0}

_{W}

^{,r}

_{×L}

^{r}*,*= 0,1,

*for all*

*t≥*1. Furthermore, the following estimates

*∂*_{x}

*U*

^{(∞)}

^{(t)u}0

*∞*

*≤Ce*

^{−ct}*u*

_{0}

*[*

_{H}*n*

2 ]+1+*×H*^{[}^{n}^{2 ]+}^{}*,*
*∂*_{x}

*U*

^{(∞)}

^{(t)u}0

*p*

*≤Ce*

^{−ct}*u*

_{0}

_{W}

^{+1,p}

_{×W}

^{,p}*,*

*p*= 1,

*∞,*

*hold for all* *t≥*1, provided that*u*_{0}*belongs to the Sobolev spaces on the right of*
*the above inequalities. Here* [q] *denotes the greatest integer less than or equal*
*toq.*

*Remark* 2.1. Young’s inequality for convolution integral, together with a
direct computation of the*L** ^{p}*-norm of the heat kernel, shows that

*W*

^{(0)}

^{(t)u}0

*p*

decays exactly in the order *t*^{−}^{n−1}^{2} ^{(1−}^{1}^{p}^{)}. We thus have the optimal decay esti-
mate

*U*

^{(0)}

^{(t)u}0

*p*

*≤Ct*

^{−}

^{n−1}^{2}

^{(1−}

^{p}^{1}

^{)}

*u*

_{0}

_{1}

*.*Furthermore, noting that

*W*

^{(0)}

^{(t)}

^{Qu}^{}0= 0, we have the estimate

*∂*_{x}

*U*

^{(0)}

^{(t)}

^{Qu}^{}0

*p*

*≤Ct*

^{−}

^{n−1}^{2}

^{(1−}

^{1}

^{p}^{)−1}

*Qu*

_{0}

_{1}for

*t≥*1.

We will prove Theorem 2.1 in Section 4.

**§****3.** **Spectral Analysis for** *−L*

The proof of Theorem 2.1 is based on the analysis of the resolvent problem associated with (1.1)–(1.3), which takes the form

(3.1) *λu*+*Lu*=*f,*

where *L*is the operator on*H*^{1}*×L*^{2} deﬁned in (1.1) with domain of deﬁnition
*D(L) =H*^{1}*×*(H^{2}*∩H*_{0}^{1}). To investigate (3.1) we take the Fourier transform in
*x*^{}*∈***R*** ^{n−1}*. We then have the following boundary value problem for functions

*φ(x*

*) and*

_{n}*m(x*

*) on the interval (0, a):*

_{n}(3.2) *λu*+*L*_{ξ}*u*=*f,*

where *u*=

*φ(x** _{n}*)

*m*

*(x*

^{}*)*

_{n}*m*

*(x*

^{n}*)*

_{n}

,*f* =

*f*^{0}(x* _{n}*)

*f*

*(x*

^{}*)*

_{n}*f*

*(x*

^{n}*)*

_{n}

, and *L** _{ξ}* is the operator of the form

*L** _{ξ}* =

0 *iγ*^{T}*ξ*^{}*γ∂*_{x}_{n}

*iγξ*^{}*ν*(*|ξ*^{}*|*^{2}*−∂*_{x}^{2}

*n*)I* _{n−1}*+

*νξ*

^{}

^{T}*ξ*

^{}*−iνξ*

^{}*∂*

_{x}

_{n}*γ∂*

_{x}

_{n}*−iν*

^{T}*ξ*

^{}*∂*

_{x}

_{n}*ν*(

*|ξ*

^{}*|*

^{2}

*−∂*

_{x}^{2}

*n*)*−ν∂* _{x}^{2}

*n*

*,*

which is a closed operator on *H*^{1}(0, a)*×L*^{2}(0, a) with domain of deﬁnition
*D(L** _{ξ}*) =

*H*

^{1}(0, a)

*×*(H

^{2}(0, a)

*∩H*

_{0}

^{1}(0, a)).

In [7] we studied (λ+*L** _{ξ}*)

*with*

^{−1}*|ξ*

^{}*| ≥r*for any

*r >*0. In this section we investigate the spectrum of

*−L*

*for*

_{ξ}*|ξ*

^{}*|<<*1. We analyze it regarding the problem as a perturbation from the one with

*ξ*

*= 0.*

^{}We write*L** _{ξ}* in the following form:

*L** _{ξ}* =

*L*

_{0}+

*n−1*

*j=1*

*ξ*_{j}*L*^{(1)}* _{j}* +

*n−1*

*j,k=1*

*ξ*_{j}*ξ*_{k}*L*^{(2)}_{jk}*,*

where *ξ** ^{}*=

*(ξ*

^{T}_{1}

*, . . . , ξ*

*),*

_{n−1}*L*_{0}=

0 0 *γ∂*_{x}_{n}

0 *−ν∂*_{x}^{2}

*n**I** _{n−1}* 0

*γ∂*

_{x}*0*

_{n}*−ν*

_{1}

*∂*

^{2}

_{x}*n*

*, ν*_{1}=*ν*+*ν,*

*L*^{(1)}* _{j}* =

0 *iγ*^{T}**e**^{}* _{j}* 0

*iγ*

**e**

^{}*0*

_{j}*−iν*

**e**

^{}

_{j}*∂*

_{x}

_{n}0 *−iν*^{T}**e**^{}_{j}*∂*_{x}* _{n}* 0

*,*

*L*^{(2)}* _{jk}* =

0 0 0

0 *νδ*_{jk}*I** _{n−1}*+

*ν*

**e**

^{}

_{j}

^{T}

**e**

^{}*0*

_{k}0 0 *νδ*_{jk}

*.*

We will treat *L** _{ξ}* as a perturbation from

*L*

_{0}. We begin with the analysis of (3.2) with

*ξ*

*= 0:*

^{}(λ+*L*_{0})u=*f.*

We introduce some quantities. For *k* = 1,2, . . ., we set *a** _{k}* =

*kπ/a. We*deﬁne

*λ*

_{1,k}and

*λ*

*by*

_{±,k}*λ*_{1,k}=*−νa*^{2}* _{k}*
and

*λ** _{±,k}*=

*−ν*

_{1}2

*a*

^{2}

_{k}*±*1

2

*ν*_{1}^{2}*a*^{4}_{k}*−*4γ^{2}*a*^{2}_{k}

for*k*= 1,2, . . .. An elementary observation shows that*λ** _{±,k}*are the two roots
of

*λ*

^{2}+

*ν*

_{1}

*a*

^{2}

_{k}*λ*+

*γ*

^{2}

*a*

^{2}

*= 0;*

_{k}*λ*

*=*

_{−,k}*λ*

_{+,k}with Im

*λ*

_{+,k}=

*γa*

_{k}

1*−*_{4γ}^{ν}^{1}^{2}2*a*^{2}* _{k}* when

*a*

_{k}*<*2γ/ν

_{1}and

*λ*

_{±,k}*∈*

**R**when

*a*

_{k}*>*2γ/ν

_{1}; and it holds that

(3.3) *λ*_{+,k}=*−γ*^{2}

*ν*_{1} +*O(k** ^{−2}*), λ

*=*

_{−,k}*−ν*

_{1}

*a*

^{2}

*+*

_{k}*O(1)*as

*k→ ∞*. (See [7, Remarks 3.2 and 3.5].)

**Lemma 3.1.** (i)*The spectrumσ(−L*_{0})*is given by*

*σ(−L*_{0}) =*{*0*} ∪ {λ*_{1,k}*}*^{∞}_{k=1}*∪ {λ*_{+,k}*, λ*_{−,k}*}*^{∞}_{k=1}*∪ {−*^{γ}_{ν}^{2}_{1}*}.*
*Here* 0 *is an eigenvalue with eigenspace spanned by* * ^{T}*(1,0, . . . ,0).

(ii) *There exist positive numbersη*_{0} *andθ*_{0} *with* *θ*_{0}*∈*(^{π}_{2}*, π)* *such that the*
*following estimates hold uniformly for* *λ∈ρ(−L*_{0})*∩Σ(−η*_{0}*, θ*_{0}):

(λ+*L*_{0})^{−1}*f*

*H*^{}*×L*^{2} *≤* *C*

*|λ||f|*_{H}^{}_{×L}^{2}*, *= 0,1,
*∂*_{x}

*n**Q(λ* +*L*_{0})^{−1}*f*

2*≤* *C*

(*|λ|*+ 1)^{1−}* ^{}*2

*|f|*

_{H}*−1*

*×L*

^{2}

*,*= 1,2,

*∂*

_{x}^{2}

_{n}*Q*

_{0}(λ+

*L*

_{0})

^{−1}*f*

2*≤* *C*

(*|λ|*+ 1)^{1}2*|f|*_{H}^{2}_{×H}^{1}*.*
*Proof.* We write (3.2) with*ξ** ^{}*= 0 as

(3.4) *λm*^{}*−ν∂*^{2}_{x}_{n}*m** ^{}* =

*f*

^{}*,*

*m*

^{}*|*

_{x}

_{n}_{=0,a}= 0, and

(3.5)

*λφ*+*γ∂*_{x}_{n}*m** ^{n}* =

*f*

^{0}

*,*

*λm*^{n}*−ν*_{1}*∂*_{x}^{2}_{n}*m** ^{n}*+

*γ∂*

_{x}

_{n}*φ*=

*f*

^{n}*,*

*m*

^{n}*|*

_{x}

_{n}_{=0,a}= 0.

By using the Fourier series expansion, it is easy to see that (3.4) has a
unique solution *m*^{}*∈* *H*^{2}(0, a)*∩H*_{0}^{1}(0, a) for any *f*^{}*∈* *L*^{2}(0, a) if and only if
*λ* =*λ*_{1,k} for any*k* = 1,2, . . .. Furthermore, it is also possible to deduce the
estimates

*∂*_{x}_{n}*m*^{}

2*≤* *C*

(*|λ|*+ 1)^{1−}^{2}^{}*|f*^{}*|*_{2}*,* = 0,1,2,

uniformly in *λ*=*−*^{νπ}_{2a}2^{2}+*ηe** ^{±iθ}* with

*η≥*0 and

*θ∈*[0, θ

_{0}). Here

*θ*

_{0}is any ﬁxed constant in (

^{π}_{2}

*, π) andC*is a positive constant depending only on

*θ*

_{0}.

We next consider (3.5). Let *λ* = 0 and *f*^{0} = *f** ^{n}* = 0 in (3.5). We see
from the ﬁrst equation of (3.5) that

*∂*

_{x}

_{n}*m*

*= 0. Then the boundary condition*

^{n}*m*

^{n}*|*

_{x}

_{n}_{=0,a}= 0 implies that

*m*

*= 0. It follows from the second equation of (3.5) that*

^{n}*φ*is a constant. Therefore, 0 is an eigenvalue and the geometric eigenspace is spanned by

*ψ*

^{(0)}=

*(1,0, . . . ,0).*

^{T}Let*λ*= 0 in (3.5). We then see that problem (3.5) is equivalent to

(3.6) *φ*= 1

*λ*

*f*^{0}*−γ∂*_{x}_{n}*m*^{n}*,*

(3.7) *λ*^{2}*m*^{n}*−*(ν_{1}*λ*+*γ*^{2})∂_{x}^{2}

*n**m** ^{n}*=

*λf*

^{n}*−γ∂*

_{x}

_{n}*f*

^{0}

*,*

*m*

^{n}*|*

_{x}

_{n}_{=0,a}= 0.

In case *ν*_{1}*λ*+*γ*^{2} = 0, it is easy to see that problem (3.6)–(3.7) has only the
trivial solution *φ* = *m** ^{n}* = 0 for

*f*

^{0}=

*f*

*= 0. For general*

^{n}*f*

^{0}

*∈*

*H*

^{1}(0, a) and

*f*

^{n}*∈L*

^{2}(0, a), (3.7) implies that

*m*

*=*

^{n}*λ*

^{−2}*λf*^{n}*−γ∂*_{x}_{n}*f*^{0}

which is not
necessarily in*H*^{1}(0, a). This implies that*−*^{γ}_{ν}_{1}^{2} *∈σ(−L*_{0}).

Let us consider the case *λ*= 0 and *ν*_{1}*λ*+*γ*^{2} = 0. In this case, (3.7) is
equivalent to

(3.8) *σm*^{n}*−∂*_{x}^{2}

*n**m** ^{n}*= 1

*ν*

_{1}

*λ*+

*γ*

^{2}

*λf*^{n}*−γ∂*_{x}_{n}*f*^{0}

*,* *m*^{n}*|*_{x}_{n}_{=0,a}= 0,
where *σ*=_{ν}^{λ}^{2}

1*λ+γ*^{2}. Since*λf*^{n}*−γ∂*_{x}_{n}*f*^{0}*∈L*^{2}(0, a), problem (3.8) has a unique
solution*m*^{n}*∈H*^{2}(0, a)*∩H*_{0}^{1}(0, a) if and only if*σ*=*−a*^{2}* _{k}* for any

*k*= 1,2, . . ., namely, (λ

*−λ*

_{+,k})(λ

*−λ*

*) = 0 for any*

_{−,k}*k*= 1,2, . . . . If (3.8) has a solu- tion

*m*

^{n}*∈*

*H*

^{2}(0, a)

*∩H*

_{0}

^{1}(0, a), then (3.6) determines

*φ*which is in

*H*

^{1}(0, a).

Consequently we see that*σ(−L*_{0}) =*{*0*}∪{λ*_{1,k}*}*^{∞}_{k=1}*∪{λ*_{+,k}*, λ*_{−,k}*}*^{∞}_{k=1}*∪{−*^{γ}_{ν}^{2}_{1}*}*.
We next derive estimates for*φ*and*m** ^{n}* uniformly in

*λ∈ρ(−L*

_{0})

*∩Σ(−η*

_{0}

*,*

*θ*

_{0}) with suitable

*η*

_{0}and

*θ*

_{0}. To do so, we expand the solution

*m*

*of (3.8) into the Fourier sine series*

^{n}*m*

*=*

^{n}

_{∞}*k=1**m*^{n}* _{k}*sin

*a*

_{k}*x*

*. It is easy to see that the Fourier coeﬃcients*

_{n}*m*

^{n}*are given by*

_{k}*m*^{n}* _{k}* = 1

*σ*+

*a*

^{2}

_{k}1
*ν*_{1}*λ*+*γ*^{2}

*λf*_{k}* ^{n}*+

*γa*

_{k}*f*

_{k}^{0}

for*k*= 1,2, . . ., where*f*_{k}^{0}and*f*_{k}* ^{n}* are the coeﬃcients of the Fourier cosine and
sine series expansion of

*f*

^{0}and

*f*

*, respectively.*

^{n}Since (σ+*a*^{2}* _{k}*)(ν

_{1}

*λ*+

*γ*

^{2}) = (λ

*−λ*

_{+,k})(λ

*−λ*

*), we have*

_{−,k}*|m*^{n}*|*^{2}_{2}*≤C*
*∞*
*k=1*

1

*|*(λ*−λ*_{+,k})(λ*−λ** _{−,k}*)

*|*

^{2}

*|λ|*^{2}*|f*_{k}^{n}*|*^{2}+*a*^{2}_{k}*f*_{k}^{0}^{2}
*.*

It then follows from (3.3) that there are positive numbers *η*_{0} and *θ*_{0} *∈*(^{π}_{2}*, π)*
such that, for *λ*with*|*arg (λ+*η*_{0})*| ≤θ*_{0},

*|m*^{n}*|*^{2}_{2}*≤C*
*∞*
*k=1*

1

(*|λ|*+ 1)^{2}(*|λ|*+*k*^{2})^{2}

*|λ|*^{2}*|f*_{k}^{n}*|*^{2}+*a*^{2}* _{k}*f

_{k}^{0}

^{2}

*≤* *C|f|*^{2}_{2}
(*|λ|*+ 1)^{2}*.*

This, together with (3.8), then implies that

∂_{x}^{2}

*n**m*^{n}

2*≤ |σ| |m*^{n}*|*_{2}+ *|λ|*

*|ν*_{1}*λ*+*γ*^{2}*||f*^{n}*|*_{2}+ *γ*

*|ν*_{1}*λ*+*γ*^{2}*|*∂_{x}_{n}*f*^{0}

2

*≤C|f|**H*^{1}*×L*^{2}

uniformly in *λ*with *|*arg (λ+*η*_{0})*| ≤θ*_{0}. Taking the*L*^{2} inner product of (3.8)
with *m** ^{n}* and integrating by parts, we have

*|∂*_{x}_{n}*m*^{n}*|*^{2}_{2}*≤C*

*|σ| |m*^{n}*|*^{2}_{2}+*|f*^{n}*|*_{2}*|m*^{n}*|*_{2}+ 1

*|λ|*+ 1f^{0}

2*|∂*_{x}_{n}*m*^{n}*|*_{2}

*≤* *C|f|*^{2}_{2}

*|λ|*+ 1 +1

2*|∂*_{x}_{n}*m*^{n}*|*^{2}_{2}

uniformly in*λ*with*|*arg (λ+*η*_{0})*| ≤θ*_{0}, and hence,*|∂*_{x}_{n}*m*^{n}*|*_{2}*≤* ^{C|f|}^{2}

(|λ|+1)^{1}^{2}. Con-
sequently, we have

(3.9) *∂*_{x}_{n}*m*^{n}

2*≤* *C|f|*_{H}^{(−}1)+*×L*^{2}

(*|λ|*+ 1)^{1−}^{2}^{}

for = 0,1,2 uniformly in*λ*with*|*arg (λ+*η*_{0})*| ≤θ*_{0}. It then follows from (3.6)
and (3.9) that

*|φ|*_{2}*≤* 1

*|λ|*f^{0}

2+*γ|∂*_{x}_{n}*m*^{n}*|*_{2}

*≤* *C*

*|λ||f|*2*.*

We next estimate the derivatives of*φ. Diﬀerentiating the ﬁrst equation of*
(3.5) we have

(3.10) *λ∂*_{x}_{n}*φ*+*γ∂*_{x}^{2}_{n}*m** ^{n}* =

*∂*

_{x}

_{n}*f*

^{0}

*.*We see from the second equation of (3.5) that (3.11)

*−ν*

_{1}

*∂*

_{x}^{2}

*n**m** ^{n}*+

*γ∂*

_{x}

_{n}*φ*=

*f*

^{n}*−λm*

^{n}*.*By adding (3.11)

*×*

_{ν}

^{γ}_{1}to (3.10) we obtain

*λ*+*γ*^{2}
*ν*_{1}

!

*∂*_{x}^{ +1}_{n}*φ*=*∂*_{x}^{ +1}_{n}*f*^{0}+ *γ*
*ν*_{1}

*∂*_{x}_{n}*f*^{n}*−λ∂*_{x}_{n}*m*^{n}

*, *= 0,1.

This, together with (3.9), implies that
*∂*_{x}^{ +1}_{n}*φ*

2*≤* *C*

*|λ|*+ 1∂_{x}^{ +1}

*n* *f*^{0}

2+*∂*_{x}_{n}*f*^{n}

2+*|λ|∂*_{x}_{n}*m*^{n}

2

*≤* *C*

(*|λ|*+ 1)^{1−}^{}^{2}*|f|*_{H}^{+1}_{×H}^{}*, *= 0,1,

for *λ*with *|*arg (λ+*η*_{0})*| ≤θ*_{0}, by changing *η*_{0} *>*0 and *θ*_{0} *∈*(^{π}_{2}*, π) suitably if*
necessary. This completes the proof.

We next investigate the eigenvalue 0 of *−L*_{0}.

**Lemma 3.2.** *The eigenvalue* 0 *of* *−L*_{0} *is simple and the associated*
*eigenprojection is given by*

*Π*^{(0)}*u*=
*φ*

0

*for* *u*=
*φ*

*m*

*.*

*Proof.* To show the simplicity of the eigenvalue 0, let us ﬁrst consider the
problem *L*_{0}*u* = *ψ*^{(0)}, where *ψ*^{(0)} = * ^{T}*(1,0, . . . ,0) is an eigenfunction for the
eigenvalue 0. This problem is equivalent to (3.4)–(3.5) with

*λ*= 0,

*f*

*= 0,*

^{}*f*

^{0}= 1,

*f*

*= 0. By (3.4), we have*

^{n}*m*

*= 0, and by the ﬁrst equation of (3.5), we have*

^{}*m*

*=*

^{n}^{1}

*γ**x** _{n}*+

*c*for some constant

*c. There is no suchm*

*satisfying the boundary condition*

^{n}*m*

^{n}*|*

_{x}

_{n}_{=0,a}= 0. Therefore, 0 is a simple eigenvalue.

Let us prove that the eigenprojection *Π*^{(0)} has the desired form. Since
dim Range*Π*^{(0)}= 1, we have *Π*^{(0)}*u*=*c*_{u}*ψ*^{(0)} for some *c*_{u}*∈***C. It then follows**
that

(3.12) *Π*^{(0)}*u, ψ*^{(0)} =*c*_{u}*.*
Consider now the formal adjoint problem

*λu*+*L*^{∗}_{0}*u*= 0,
where

*L*^{∗}_{0}=

0 0 *−γ∂*_{x}_{n}

0 *−ν∂*_{x}^{2}

*n**I** _{n−1}* 0

*−γ∂*_{x}* _{n}* 0

*−ν*

_{1}

*∂*

^{2}

_{x}*n*

with domain of deﬁnition*D(L*^{∗}_{0}) =*D(L*_{0}). Similarly to above, we can see that
*σ(−L*^{∗}_{0}) =*σ(−L*_{0}), and, in particular, 0 is a simple eigenvalue and*L*^{∗}_{0}*ψ*^{(0)}= 0.

Furthermore, let*Π*^{(0)∗}be the eigenprojection for the eigenvalue 0 of*−L*^{∗}_{0}. Then
we have

*Π*^{(0)}*u*= 1
2πi

*Γ*

(λ+*L*_{0})^{−1}*u dλ,* *Π*^{(0)∗}*u*= 1
2πi

*Γ*

(λ+*L*^{∗}_{0})^{−1}*u dλ,*
where *Γ* is a circle with center 0 and suﬃciently small radius. By integration
by parts, we have

(λ+*L*_{0})*u,* *v* =*u,*(λ+*L*^{∗}_{0})*v*

for*u,* *v∈D(L*_{0}). Taking*u*= (λ+*L*_{0})^{−1}*u*and*v*= (λ+*L*^{∗}_{0})^{−1}*v, we have*
(λ+*L*_{0})^{−1}*u, v* =*u,*(λ+*L*^{∗}_{0})^{−1}*v*

for*u, v∈H*^{1}(0, a)*×L*^{2}(0, a). We then obtain

"

*Π*^{(0)}*u, ψ*^{(0)}

#

=

$ 1 2πi

*Γ*

(λ+*L*_{0})^{−1}*u dλ, v*

%

=

$
*u,* 1

2πi

*Γ*

(λ+*L*^{∗}_{0})^{−1}*v dλ*

%

=

"

*u,Π*^{(0)∗}*ψ*^{(0)}

#

=&

*u, ψ*^{(0)}'

=*φ*

for*u*=

*φ*
*m*

. This, together with (3.12), gives the desired expression of*Π*^{(0)}.
This completes the proof.

We next estimate (λ+*L** _{ξ}*)

*for small*

^{−1}*ξ*

*. Based on Lemma 3.1 we obtain the following estimates.*

^{}**Theorem 3.1.** *Letη*_{0}*andθ*_{0}*be the numbers given in Lemma*3.1. Then
*there exists a positive number* *r*_{0} = *r*_{0}(η_{0}*,θ*_{0}) *such that the set* *Σ(−η*_{0}*, θ*_{0})*∩*
*λ;|λ| ≥* ^{η}_{2}^{0}

*is in* *ρ(−L** _{ξ}*)

*for*

*|ξ*

^{}*| ≤*

*r*

_{0}

*. Furthermore, the following esti-*

*mates hold for any multi-index*

*α*

^{}*with|α*

^{}*| ≤n*

*uniformly inλ∈Σ(−η*

_{0}

*, θ*

_{0})

*∩*

*λ;|λ| ≥*

^{η}_{2}

^{0}

*andξ*^{}*with* *|ξ*^{}*| ≤r*_{0}:
*∂*_{ξ}^{α}* ^{}*(λ+

*L*

*)*

_{ξ}

^{−1}*f*

*H*^{}*×L*^{2} *≤* *C*

*|λ||f|*_{H}^{}_{×L}^{2}*,* = 0,1,
*∂*^{α}_{ξ}^{}*∂*_{x}

*n**Q(λ* +*L** _{ξ}*)

^{−1}*f*

2*≤* *C*

(*|λ|*+ 1)^{1−}* ^{}*2

*|f|*

_{H}

^{−1}

_{×L}^{2}

*,*= 1,2,

*∂*

_{ξ}

^{α}

^{}*∂*

^{2}

_{x}

_{n}*Q*

_{0}(λ+

*L*

*)*

_{ξ}

^{−1}*f*

2*≤* *C*

(*|λ|*+ 1)^{1}2*|f|**H*^{2}*×H*^{1}*.*
*Proof.* In the following we will write

*L*^{(1)}(ξ* ^{}*) =

*n−1*

*j=1*

*ξ*_{j}*L*^{(1)}* _{j}* and

*L*

^{(2)}(ξ

*) =*

^{}*n−1*

*j,k=1*

*ξ*_{j}*ξ*_{k}*L*^{(2)}_{jk}*.*

We ﬁrst observe that
(3.13) *L*^{(1)}_{j}*u*

*H*^{}*×H*^{(−}^{1)+} *≤C*

*|Q*_{0}*u|*_{H}^{(−}1)++*|Qu* *|*_{H}^{(−}1)++1

and

(3.14) *L*^{(2)}_{jk}*u*

*H*^{}*×H*^{(−}^{1)+} *≤C|Qu* *|*_{H}^{(−}1)+*.*