MI Preprint Series
Mathematics for IndustryKyushu University
Asymptotic behavior of solutions of
the compressible Navier-Stokes
equations in a cylinder under the
slip boundary condition
Abulizi Aihaiti
& Yoshiyuki Kagei
MI 2018-4
( Received April 27, 2018 )
Institute of Mathematics for Industry
Graduate School of Mathematics
Asymptotic behavior of solutions of the compressible
Navier-Stokes equations in a cylinder under
the slip boundary condition
Abulizi Aihaiti
1and Yoshiyuki Kagei
21 Graduate School of Mathematics,
Kyushu University, Fukuoka 819-0395, Japan
2 Faculty of Mathematics,
Kyushu University, Fukuoka 819-0395, Japan
Abstract
The large time behavior of solutions to the compressible Navier-Stokes equations around the motionless state is considered in a cylinder under the slip boundary condition. It is shown that if the initial data is sufficiently small, the global solution uniquely exists and the large time behavior of the solution is described by a superposition of one-dimensional nonlinear diffusion waves and a diffusive rigid rotation.
Keywords: Compressible Navier-Stokes equations, cylinder, slip bound-ary condition, asymptotic behavior, nonlinear diffusion waves, diffu-sive rigid rotation.
1
Introduction
This paper studies the large time behavior of solutions of the compressible Navier-Stokes equations
∂tρ+ div(ρv) = 0, (1.1)
ρ(∂tv+v· ∇v)−µdivD(v)−µ′∇divv+∇p(ρ) = 0 (1.2)
in an infinite cylinder Ωℓ =R×Dℓ :
Dℓ={x′ = (x2, x3); x22+x23 ≤ℓ2},
under the slip boundary condition
v·n|∂Ωℓ = 0, D(v)·n−
(
D(v)n·n)n
∂Ωℓ =0. (1.3)
Here ρ = ρ(x, t) and v = ⊤(v1(x, t), v2(x, t), v3(x, t)) denote the unknown
density and the unknown velocity field, respectively, at time t≥0 and posi-tionx∈Ωℓ; p=p(ρ) is the pressure that is assumed to be a smooth function
of ρ and satisfies
p′(ρ∗)>0
for a given constant ρ∗ > 0; µ and µ′ are viscosity coefficients that are
assumed to be constants and satisfy
µ >0, 2
3µ+µ
′ >0;
div and ∇ denote the usual divergence and gradient with respect to x; D(·) denotes the deformation tensor whose (j, k)-components (j, k = 1,2,3) are given by
D(v)jk =∂xjv
k+∂ xkv
j;
n is the unit outward normal vector to ∂Ωℓ, which is given by n=⊤(0,n′)
with n′ = 1 ℓx
′ = 1 ℓ
⊤(x
2, x3) being the unit outward normal vector to ∂Dℓ.
Here and in what follows ⊤·stands for the transposition.
We impose the initial condition
ρ|t=0 =ρ0, v|t=0 =v0. (1.4)
Here ρ0 = ρ0(x) and v0 = v0(x) satisfy ρ0(x) → ρ∗ and v0(x) → 0 as
|x| → ∞.
In this paper we will consider the stability of the motionless state us =
⊤(ρ
∗,0) and will investigate the large time behavior of solutions around us.
We thus rewrite (1.1)-(1.2) into the following equations for the perturbation
∂tϕ+γdivw=f0(ϕ,w), (1.5)
∂tw−νdivD(w)−ν′∇divw+γ∇ϕ =f(ϕ,w). (1.6)
Hereu=⊤(ϕ,w) with ϕ= 1
ρ∗(ρ−ρ∗) andw= 1
γv denotes the perturbation
of us =⊤(ρ∗,0); ν, ν′, γ are parameters given by
ν = µ
ρ∗
, ν′ = µ
′
ρ∗
and f(ϕ,w) =⊤(f0(ϕ,w),f(ϕ,w)) denotes the nonlinear terms:
f0(ϕ,w) = −γdiv(ϕw),
f(ϕ,w) = −γw· ∇w− ϕ
1 +ϕ{νdivD(w) +ν
′
∇divw}+ γϕ 1 +ϕ∇ϕ
− ρ∗p
′′(ρ ∗)
2γ(1 +ϕ)∇(ϕ
2)
− ρ
2
∗
2γ(1 +ϕ)∇
(
p(3)(ϕ)ϕ3),
where
p(3)(ϕ) =
∫ 1
0
(1−θ)2p′′′(
ρ∗(1 +θϕ)
)
dθ.
The boundary condition (1.3) and initial condition (1.4) are transformed into
w·n|∂Ωℓ = 0, D(w)·n−(D(w)n·n)n
∂Ωℓ =0, (1.7)
and
u|t=0 =u0 =⊤(ϕ0,w0). (1.8)
Here u0 satisfiesu0(x)→0 as |x| → ∞.
Large time behavior of solutions of the compressible Navier-Stokes equa-tions in unbounded domains have been studied in detail in various contexts; see, e.g., [5, 6, 7, 10, 13, 15, 18, 20, 21, 25, 26, 27, 30, 33] for the cases of the multi-dimensional whole space, half space and exterior domains. In ad-dition to these domains, problems in infinite layers and cylindrical domains have been also studied, e.g., in [2, 3, 4, 8, 9, 11, 12, 16] under the non-slip boundary condition v|x2=0,1 = 0. It was shown in [11, 16] that the large
time behavior of perturbations of the motionless state is described by a one-dimensional linear heat equation. This kind of purely diffusive behaviors has been also observed when background flows such as stationary/time-periodic parallel flows and spatially periodic patterns appear, although in these cases the mass of perturbations not only decays diffusively but also is transported by the background flows; see [2, 3, 4, 8, 9, 12]. We also mention the work [23] by H.-L. Li and X. Zhang, where the problem under the Navier-slip boundary condition was considered and an interesting observation on the effect of the slip at the boundary was also made.
In the case of the problem under the slip boundary condition (1.7), the large time behavior of solutions to the compressible Navier-Stokes equations was considered in an infinite layer of R2 in [1]. It was shown in [1] that if
In contrast to the cases of the non-slip or the Navier-slip boundary conditions, a hyperbolic aspect of the equations (1.5)-(1.6), i.e., a wave propagation phenomenon, appears in the asymptotic leading part of the solution.
In this paper we show that the solution u(t) of (1.5)-(1.6) under the slip boundary condition (1.7) with (1.8) behaves like a superposition of one-dimensional nonlinear diffusion waves and a diffusive rigid rotation as t →
∞. More precisely, we prove that, under appropriate conditions for u0, the
solution u(t) satisfies
∥∂xk(u−κ+a+−κ−a−−κrigarig)(t)∥L2(Ω
ℓ)≤C(1 +t) −1
2−
k
2, k = 0,1, (1.9)
where a± = 12⊤(1,±1,0,0) and κ± = κ±(x1, t) are the nonlinear diffusion
waves given by
κ±(x1, t) =Z±(x1 ±γt, t). (1.10)
Here Z±=Z±(x1, t) are the self-similar solutions of the Burgers equations
∂tZ±−
2ν+ν′
2 ∂
2
x1Z±±c∂x1(Z±2) = 0 (1.11)
satisfying
∫
R
Z±(x1, t)dx1 =
1 2
∫
Ωℓ
(
ϕ0(x)±(1 +ϕ0(x))w01(x)
)
dx (1.12)
for some constant c∈R; and
arig=⊤(0,arig), arig =
1
ℓ2
√
2
π
⊤(0,−x
3, x2), (1.13)
κrig(x1, t) = w0,rig(4πt)− 1 2e−
x21
4t (1.14)
with w0,rig=
∫
Ωℓw0·arigdx.We note that, in addition to the wave
propaga-tion part κ+a++κ−a−, the diffusive rigid motion part κrigarig also appears
in the asymptotic leading part of the solution in the case of the cylinder. We also note that the diffusive rigid motion partκrigarig gives the incompressible
part of the asymptotic leading part of u since div(κrigarig) = 0. It should
Zajaczkowski, where the global existence on a bounded domain was proved based on the energy method.
We briefly explain a sketch of the proof of the main results. To prove (1.9), we first establish the decay estimates for the solution u(t) of (1.5)-(1.8). We decompose u(t) into its low and high frequency parts. As for the low frequency part, we investigate the spectrum of the low-frequency part of the linearized semigroup and show that the leading part is decomposed into the linear diffusion waves part and the diffusive rigid motion part. As a result, the low frequency part decays like a one-dimensional heat kernel, namely, k-th order derivative decays in the order O(t−1
4−
k
2) in the L2 norm.
To establish suitable decay estimates for the nonlinear problem, we introduce the momentum formulation for the low frequency part, which makes the equations a conservation form. This enables us to deal with a slowly decaying part caused by the interaction between the diffusion waves and the diffusive rigid motion. For the high frequency part, we apply the Matsumura-Nishida energy method ([27]) and show that the high frequency part decays in the order O(t−34) in the H2 norm. To this end, a Korn type inequality plays
an important role. Combining the estimates for the low and high frequency parts, we establish the decay estimate of u(t) in H2 norm. Based on the spectral properties of the low frequency part of the linearized semigroup and the decay estimate for u(t), we deduce the asymptotic behavior (1.9) by applying the argument of Kawashima [19].
This paper is organized as follows. In Section 2 we introduce notations and rewrite the problem in a non-dimensional form. In Section 3 we state the main results of this paper. In Section 4 we study the spectral properties of the linearized operator and derive a Korn type inequality. In Section 5 we rewrite the problem (1.5)-(1.8) into a problem for a system of equations for the low and high frequency parts and introduce the momentum formulation for the low frequency part. Section 6 is devoted to estimating the low frequency part, while the high frequency part is estimated in Section 7. In Section 8, we give necessary estimates for the nonlinearities. In Section 9 we study the asymptotic behavior of the solution of (1.5)-(1.8).
2
Preliminaries
2.1
Notation
For 1 ≤p≤ ∞we denote byLp(E) the usual Lebesgue space on a domainE
of Rn and its norm is denoted by∥ · ∥
Lp(E).Let m be a nonnegative integer.
The symbol Hm(E) denotes the m-th order L2-Sobolev space on E with norm ∥ · ∥Hm(E).
We simply denote by Lp(E) (resp., Hm(E)) the set of all vector fields w =⊤(w1,· · · , wn) onE withwj ∈Lp(E) (resp,Hm(E)), j = 1,· · · , n, and
its norm is also denoted by ∥ · ∥Lp(E) (resp.,∥ · ∥Hm(E)). Foru=⊤(ϕ,w) with
ϕ ∈ Hk(E) and w = ⊤(w1,· · · , wn) ∈ Hm(E), we define ∥u∥
Hk(E)×Hm(E)
by ∥u∥Hk(E)×Hm(E) = ∥ϕ∥Hk(E) +∥w∥Hk(E). When k = m, we simply write
∥u∥Hk(E)×Hk(E) =∥u∥Hk(E).
Partial derivatives of a functionvinx, x1, x′andtare denoted by∂xv, ∂x1v, ∂x′v
and ∂tv. We also write the higher order partial derivatives of v in x as
∂xlv = (∂xαv;|α|=l).
In Section 4 we will consider the complex-valued functions. We set Ω := Ω1. In the case where E = Ω we abbreviate Lp(Ω) (resp., Hm(Ω)) as Lp
(resp., Hm). In particular, the norm ∥ · ∥Lp(Ω) =∥ · ∥Lp is denoted by ∥ · ∥p.
We denote the inner product of L2(Ω) by
(f, g) =
∫
Ω
f(x)g(x)dx, f, g∈L2(Ω).
Here ¯g denotes the complex conjugate of g.
We setD:=D1.The inner product of L2(D) is denoted by
(f, g)L2(D) =
∫
D
f(x′)g(x′)dx′, f, g
∈L2(D).
We also define a weighted inner product ⟨f, g⟩ by
⟨f, g⟩= 1
|D|(f, g)L2(D)=
1
|D| ∫
D
f(x′)g(x′)dx′,
where |D| = ∫
Ddx
′. For f ∈ L1(D) we denote the mean value of f over D
by ⟨f⟩:
⟨f⟩=⟨f,1⟩= 1
|D| ∫
D
fdx′.
Forα∈R, we denote byL1α =L1α(Ω) the weightedL1 space with weight (1 +|x1|)α, and its norm is denoted by
∥f∥L1
α =
∫
Ω
We denote the Fourier transform off =f(x1) (x1 ∈R) by ˆf orF[f] :
ˆ
f(ξ) =F[f](ξ) =
∫
R
f(x1)e−iξx1dx1, ξ ∈R.
The inverse Fourier transform is denoted by F−1 :
F−1[f](x1) = (2π)−1
∫
R
f(ξ)eiξx1dξ, x1 ∈R.
We denote the resolvent set of a closed operator A by ρ(A) and the spectrum of A byσ(A).
For operators A, B, we denote the commutator ofA and B by [A, B] :
[A, B]f =A(Bf)−B(Af).
2.2
Non-dimensionalization
In this subsection we rewrite the problem into the one in a non-dimensional form. We introduce the following non-dimensional variables:
˜
x= 1
ℓx, ˜t= γ
ℓt, ρ˜= ρ ρ∗
, v˜= 1
γv, p˜=
1
ρ∗γ2
p,
where
γ =√p′(ρ ∗).
The problem (1.1)-(1.2) is then transformed into the following non-dimensional problem on Ω =R×D:
∂˜tρ˜+ divx˜(˜ρv˜) = 0, (2.1)
˜
ρ(∂˜tv˜+ ˜v· ∇˜xv˜)−νdivx˜
(
D˜x(˜v)
)
−ν′∇x˜divx˜v˜+∇x˜p˜(˜ρ) =0. (2.2)
Here ν and ν′ are non-dimensional parameters given by
ν = µ
ρ∗γℓ
, ν′ = µ
′
ρ∗γℓ
.
The boundary condition (1.3) and initial condition (1.4) are transformed into
˜
v·n˜|∂Ω= 0, D(˜v)·n˜ −(D(˜v) ˜n·n˜) ˜n|∂Ω =0, (2.3)
and
In what follows, for simplicity, we omit the tildes of ˜x,˜t,ρ,˜ v˜ and ˜p and write them as x, t, ρ,v and p, respectively. Observe that, due to the non-dimensionalization, we have
D={x′ = (x2, x3);x22+x23 ≤1}, |D|=
∫
D
dx′ =π,
and thus
⟨f⟩= 1
π ∫
D
f(x′)dx′.
The perturbation equations for the non-dimensionalized problem (2.1)-(2.2) is given by
∂tϕ+ divw=f0(ϕ,w), (2.5)
∂tw−νdivD(w)−ν′∇divw+∇ϕ =f(ϕ,w). (2.6)
Hereu=⊤(ϕ,w) withϕ =ρ−1 andw=vdenotes the non-dimensional
per-turbation, and f(ϕ,w) = ⊤(f0(ϕ,w),f(ϕ,w)) denotes the non-dimensional
nonlinear terms:
f0(ϕ,w) = −div(ϕw),
f(ϕ,w) = −w· ∇w− ϕ
1 +ϕ{νdivD(w) +ν
′
∇divw}+ ϕ 1 +ϕ∇ϕ
− p
′′(1)
2(1 +ϕ)∇(ϕ
2)
− 2(1 +1 ϕ)∇(
p(3)(ϕ)ϕ3) ,
where
p(3)(ϕ) =
∫ 1
0
(1−θ)2p′′′(1 +θϕ)dθ.
The non-dimensional form of the boundary and initial conditions are
w·n|∂Ω = 0, D(w)·n−(D(w)n·n)n
∂Ω =0, (2.7)
with n=⊤(0,n′), n′ =⊤(x
2, x3), and
u|t=0 =u0 =⊤(ϕ0,w0). (2.8)
3
Main results
In this section we state the main results of this paper. We begin with the global existence result. We set
H∗2 =H∗2(Ω) ={w∈H2(Ω)3;w·n|∂Ω = 0,D(w)·n−(D(w)n·n)n
∂Ω =0}. Theorem 3.1 There exists a positive constantε0 such that ifu0 =⊤(ϕ0,w0)∈
(H2 ×H2
∗)∩(L1 ×L1) and ∥u0∥H2∩L1 ≤ε0, then problem (1.5)-(1.8) has a unique global solution
u(t) = ⊤(ϕ(t),w(t))∈C([0,∞);H2×H∗2)
and u(t) satisfies
∥∂x1k u(t)∥2 ≤C(1 +t)− 1 4−
k
2∥u
0∥H2∩L1 (3.1)
for t ≥0, k = 0,1.
We next consider the asymptotic behavior of solutions.
Theorem 3.2 In addition to the assumption of Theorem 3.1, assume that
u0 ∈L11/2×L11/2. Then
∥∂x1k (u−χ+b+−χ−b−−χrigbrig)(t)∥2 ≤C(1 +t)− 1 2−
k
2, k= 0,1. (3.2)
Here b± = 12⊤(1,±1,0,0); χ± = χ±(x1, t) are the diffusion waves given by
χ±(x1, t) = z±(x1±t, t) with z±(x1, t) = Z(ℓx1, ℓt) and χrig is the diffusive rigid rotation given by χrig(x1, t) =κrig(ℓx1, ℓt),where Z±andκrig are defined in (1.10)-(1.12) and (1.13)-(1.14), respectively.
The proof of Theorem 3.1 will be given in Sections 4-8, and Theorem 3.2 will be proved in Section 9.
4
Spectral properties of the linearized
oper-ator
In this section we investigate the spectral properties of the linearized opera-tors and give a Korn type inequality, which will be used in the proof Theorem 3.1.
We consider the linearized equation
∂tu+Lu= 0, u
t=0 =u0 =
⊤(ϕ
where u=⊤(ϕ,w) and Lis an operator on L2×L2 of the form
L=
(
0 div
∇ −νdivD(·)−ν′∇div
)
with domain D(L) = H1 ×H2
∗. It was shown in [28] that −L generates an
analytic semigroup e−tL onL2×L2.
To investigate the spectrum of L, we consider the Fourier transform of (4.1) in x1 variable, which take the form
∂tϕˆ+iξwˆ1+∇′·wˆ′ = 0, (4.2)
∂twˆ1+ν(ξ2−∆′) ˆw1−i(ν+ν′)ξ(iξwˆ1+∇′·wˆ′) +iξϕˆ= 0, (4.3)
∂twˆ′+ν
(
ξ2wˆ′− ∇′·D′( ˆw′))
−ν′∇′∇′·wˆ′−i(ν+ν′)ξ∇′wˆ1+∇′ϕˆ=0′.
(4.4)
Here ∇′ =⊤(∂
x2, ∂x3), ∆′ =∂x22 +∂x32 ; and D′(w′) is the 2×2 matrix with
(j, k)-components D′(w′)
j,k =∂xjw
k+∂ xkw
j, j, k = 2,3, and 0′ =⊤(0,0).
We thus arrive at the following problem
∂tuˆ+ ˆLξuˆ= 0, uˆ|t=0 = ˆu0 (4.5)
with a parameter ξ∈R.Here ˆLξ is an operator onX :=H1(D)×L2(D)3 of
the form
ˆ
Lξ =
0 iξ ⊤∇′
iξ (2ν+ν′)ξ2 −ν∆′ −i(ν+ν′)ξ⊤∇′
∇′ −iξ(ν+ν′)∇′ ν(
ξ2− ∇′·D′(·))
−ν∇′⊤∇′
with domain D( ˆLξ) = H1(D)×H∗2(D), where
H∗2(D) = {w=⊤(w1,w′)∈H2(D)3; ∂w
1
∂n′
∂D = 0,
w′·n′|∂D = 0,
D′(w′)·n′−(D′(w′)n′·n′)n′
∂D =0
′}.
It was shown in [28] that−Lˆ0 =−Lˆξ|ξ=0 generates an analytic semigroup.
Since
∥( ˆLξ−Lˆ0)u∥H1(D)×L2(D) ≤C|ξ|
(
1 +|ξ|)∥u∥L2(D)×H1(D),
one can see from a standard perturbation argument that −Lˆξ generates an
To investigate the spectral properties of −Lˆξ,we introduce the following
notations. For u = ⊤(ϕ,w), w =⊤(w1,w′), we define Π
0u, Π1u and Πrigu
by
Π0u=
(
⟨ϕ⟩
0
) ,
Π1u=
(
0
Π1w
)
, Π1w =
(
⟨w1⟩
0′
)
,
Πrigu=
(
0
Πrigw
)
, Πrigw =⟨w,brig⟩brig,
where
brig =
(
0
b′rig
)
, b′rig =√2
(
−x3
x2
) .
Note that ⟨brig,brig⟩ = ⟨b′rig,b′rig⟩ = 1 and that Πj (j = 0,1,rig) are
pro-jections satisfying ΠjΠk = 0 (j ̸= k). We also define the projection Π′rig
by
Π′rigw′ =⟨w′,b′rig⟩b′rig.
By a direct computation, one can verify the following lemma.
Lemma 4.1 (i) For any ξ∈R, −Lˆξ has the following eigenvalues
λ±(ξ) = −
2ν+ν′
2 ξ
2
± 12√(2ν+ν′)2ξ4−4ξ2,
λrig(ξ) = −νξ2.
(ii) Let Πˆ±(ξ) be the projection operator given by
ˆ
Π±(ξ) =
1
λ+(ξ)−λ−(ξ)
∓λ∓(ξ) ±iξ ⊤0′
±iξ ±λ±(ξ) ⊤0′
0′ 0′ O′
(Π0+ Π1)
for ξ ̸=± 2
2ν+ν′, where
O′ =
(
0 0 0 0
) ,
and let Πˆrig(ξ) be the projection operator given by
ˆ
Πrig(ξ) = Πrig for ξ ∈R. Then for j, k =±,rig,
ˆ
and
−Πˆj(ξ) ˆLξ ⊂ −LˆξΠˆj(ξ) =λj(ξ) ˆΠj(ξ).
(iii) If |ξ|< 2ν+ν2 ′, then
λ+(ξ) =λ−(ξ), Re λ±(ξ) = −
2ν+ν′
2 ξ
2.
Furthermore there exists a positive constant R0 such that
Im λ±(ξ) =±i(ξ+ ˜λ(ξ)),
ˆ
Π±(ξ) = Π±,0+ ˜Π±(ξ)
for |ξ| ≤R0, where
Π±,0 =
1 2
1 ±1 ⊤0′
±1 1 ⊤0′
0′ 0′ O′
(Π0 + Π1);
˜
λ(ξ) and Π˜±(ξ) satisfy
|λ˜(ξ)| ≤C|ξ|3, ∥Π˜±(ξ)∥ ≤C|ξ|
uniformly for |ξ| ≤R0. Here ∥Π˜±(ξ)∥ denotes the matrix norm of Π˜±(ξ).
Let F be a smooth function on [0, T] with values in X, and let u0 be in
X. Then u(t) defined by
u(t) =e−tLˆξu
0+
∫ t
0
e−(t−s) ˆLξF(s)ds
is a unique solution of
∂tu+ ˆLξu=F, u|t=0 =u0. (4.6)
We decompose (4.6) into Πj(ξ) (j =±,rig) andI−(Π++ Π−+ Πrig)(ξ)
parts; applying these projections to (4.6), we have
∂tuj −λj(ξ)uj =Fj, uj|t=0 =u0,j, j =±,rig, (4.7)
∂tu˜+ ˆLξu˜= ˜F , u˜|t=0 = ˜u0. (4.8)
Here
and
˜
u(t) = (
I−Πˆ+(ξ)−Πˆ−(ξ)−Πˆrig(ξ)
) u(t),
and so on. It then follows from (4.7) that
uj(t) = eλj(ξ)tu0,j+
∫ t
0
eλj(ξ)(t−s)F
j(s)ds,
ˆ
Πj(ξ)e−t ˆ Lξu
0 =eλj(ξ)tΠˆj(ξ)u0.
By Lemma 4.1, we deduce that for any nonnegative integers k and l,
∥ξkΠˆ±(ξ)e−t
ˆ Lξu
0∥L2(D) ≤Ck(1 +t)−
k
2e− 2ν+ν′
k ξ2t∥u0∥
L1(D) (4.9)
uniformly in |ξ| ≤R0 and t≥0,and
∥ξk∂xl′Πrige−t ˆ Lξu
0∥L2(D) ≤Ck,l(1 +t)−
k
2e−
ν
kξ2t∥u0∥
L1(D)
uniformly in |ξ| ≤ R1 and t ≥ 0, where R1 is an arbitrarily fixed positive
constant.
We next introduce a projection ona low frequency (slowly decaying) part. For a given positive number R, let 1R be a function on R given by
1R(ξ) =
{
1 (|ξ| ≤R),
0 (|ξ|> R).
We define P1 by
P1 =F−1
[
1R0(ξ)
(ˆ
Π+(ξ) + ˆΠ−(ξ)
)
+1R1(ξ)Πrig
]
F, (4.10)
where R0 is the positive constant given in Lemma 4.1 and R1 is a positive
constant to be determined later. We also set
P∞=I−P1. (4.11)
It then follows that P1 is a bounded projection on Hk×Hk for all k, and
that P1 satisfies∂x1P1 =P1∂x1,
∥P1u∥Hk ≤Ck∥u∥2, (4.12)
and
P1e−tL =e−tLP1
=F−1[
1R0(ξ)
(
eλ+(ξ)tΠˆ+(ξ) +eλ−(ξ)tΠˆ−(ξ)
)
+1R1(ξ)eλrig(ξ)tΠrig
]
F.
By (4.9), a standard argument shows that theP1-part ofe−tL has the
Lemma 4.2 For any nonnegative integers k andl, P1e−tL =e−tLP1 satisfies the estimate
∥∂x1k ∂xl′P1e−tLu0∥2 ≤C(1 +t)− 1 4−
k
2∥u0∥
1 (4.13)
uniformly for t ≥0 and u0 ∈(L2×L2)∩(L1×L1).
We next investigate the asymptotic behavior of e−tLP1 as t → ∞. We
define S(t), S±(t) and Srig(t) by
S(t) = S+(t) +S−(t) +Srig(t),
Sj(t) = F−1Sˆj(t)F, j =±, rig,
ˆ
S±(t) =
1 2e
−2ν+ν′
2 ξ 2t±iξt
Π±,0,
ˆ
Srig(t) = e−νξ 2t
Πrig.
We then see that S(t) gives the asymptotic leading part of e−tLP1 in the
following sense.
Lemma 4.3 Let k = 0,1.The following estimates hold uniformly for t >0 : (i) ∥∂k
x1S±(t)P1u0∥2 ≤C(1 +t)− 1 4−
k
2∥u0∥1,
∥∂k
x1S±(t)(I−P1)u0∥2 ≤C
{
t−k2e−2ν+4ν′R2 0t∥u
0∥2+ (1 +t)− 3 4−
k
2∥u0∥1},
(ii) ∥∂k
x1Srig(t)P1u0∥2 ≤C(1 +t)− 1 4−
k
2∥u0∥1,
∥∂x1k Srig(t)(I−P1)u0∥2 ≤Ct−
k
2e−
ν
2R 2 1t∥u
0∥2,
(iii) ∥∂k x1
(
e−tL−S(t))
P1u0∥2 ≤C(1 +t)− 3 4−
k
2∥u0∥1.
Proof. The first inequality of (i) can be obtained in a standard manner. As for the second one, since ˆΠ±(ξ) = Π±,0+ ˜Π±(ξ), we see that
Π±,0(I−P1) = F−1
[
Π±,0(I −1R0) +1R0
(
I −Πˆ+(ξ)−Πˆ−(ξ)
)]
=F−1[
(1−1R0)Π±,0 −1R0Π±,0
(˜
Π+(ξ) + ˜Π−(ξ)
)] .
It then follows from Lemma 4.1 (iii) that
∥∂x1k S±(t)(I−P1)u0∥22
≤C {∫
|ξ|≥R0|
ξ|2ke−(2ν+ν′)ξ2t∥uˆ0∥2L2(D)dξ
+
∫
|ξ|≤R0|
ξ|2(k+1)e−(2ν+ν′)ξ2t∥uˆ0∥2L1(D)dξ
}
≤C{t−ke−2ν+ν
′ 2 R
2 0t∥u
0∥22+ (1 +t)− 3 2−k∥u
which is the desired inequality.
The inequalities in (ii) can be obtained similarly to those in (i). As for (iii), we see from Lemma 4.1 (iii) that
eλ±(ξ)t
−e(−2ν+ν
′ 2 ξ
2±iξ)t
=
e(−2ν+ν
′ 2 ξ
2±iξ)t
(e±iλ˜±(ξ)t
−1)
≤C|ξ|3te−2ν+ν
′ 2 ξ
2t
for |ξ| ≤R0. We thus obtain
∥∂x1k (
e−tL−S(t))
P1u0∥2 ≤C
(∫
|ξ|≤R0|
ξ|2(k+1)e−(2ν+ν′)ξ2tdξ )12
∥u0∥1
≤C(1 +t)−34−
k
2∥u0∥1.
This completes the proof. □
Remark 4.4 Let us consider the inhomogeneous problem
∂tu+Lu=F, u|t=0 =u0, (4.14)
where F = ⊤(f0,f), f = ⊤(f1,f′). Let u
1(t) = P1u(t). Applying P1 to (4.14), we have
∂tu1+Lu1 =P1F, u1|t=0 =P1u0. (4.15) Since Πˆ+(ξ) + ˆΠ−(ξ) = Π0+ Π1, we see that P1 is written as
P1 =F−1[1R0(ξ)Π0+1R0(ξ)Π1+1R1(ξ)Πrig]F. It then follows that
u1(t) =σ0(t)b0+σ1(t)b1+σrig(t)brig, where
b0 =⊤(1,0,0′), b1 =⊤(0,1,0′),
σ0 =F−1[1R0⟨ϕˆ⟩], σ1 =F−1[1R0⟨wˆ1⟩], σrig=F−1[1R1⟨wˆ′,b′rig⟩]. The problem (4.15) is then written as
∂tσ0 +∂x1σ1 =F−1[1R0⟨fˆ0⟩],
∂tσ1 −(2ν+ν′)∂x12 σ1+∂x1σ0 =F−1[1R0⟨fˆ1⟩],
∂tσrig−ν∂x12 σrig =F−1[1R1⟨fˆ′,b′rig⟩],
σ0|
t=0 =F−1[1R0⟨ϕˆ0⟩], σ1|t=0 =F−1[1R0⟨wˆ01⟩], σrig|t=0 =F−1[1R1⟨wˆ0′,b′rig⟩],
(4.16)
where u0 =⊤(ϕ0,w0), w0 =⊤(w01,w′0). Therefore, ⊤(σ0, σ1) is a solution of linearized one-dimensional compressible Navier-Stokes equations on R, and
As for the P∞-part we will use the following Poincar´e’s and Korn type
inequalities.
Lemma 4.5 If the constant R1 in the definition ofP1 is taken suitably large, then the following estimates
∥ϕ∥2 ≤C∥∇ϕ∥2,
∥w∥2 ≤C1∥∇w∥2 ≤C2∥D(w)∥2 hold for u=⊤(ϕ,w) =P
∞u with w·n|∂Ω = 0.
To prove Lemma 4.5, we prepare the following Korn’s inequality.
Lemma 4.6 There holds the estimate
∥∇w∥2 ≤C{∥D(w)∥2+∥Π′
rigw′∥2} (4.17)
for w =⊤(w1,w′)∈H1(Ω) with w·n|
∂Ω = 0.
Proof. We first assume that Π′
rigw′(x1,·) =0 for a.e. x1. By the definition
of D(w), we have
∥D(w)∥22 = (D(w),D(w)) = 2
3
∑
j,k=1
∫
Ω|
∂xjw
k
|2dx+ 2Re
3
∑
j,k=1
∫
Ω
∂xjw
k∂ xkw
jdx. (4.18)
As for the second term on the right-hand side, since w·n|∂Ω = 0,we see, by
integration by parts, that
3
∑
j,k=1
∫
Ω
∂xjw
k∂
xkwjdx
=
3
∑
k=1
∫
Ω
∂x1wk∂xkw
1dx+ ∑ j=2,3
∫
Ω
∂xjw
1∂
x1wjdx
+ ∑
j=2,3
∑
k=2,3
∫
Ω
∂xjw
k∂
xkwjdx
=
∫
Ω
divw ∂x1w1dx+
∫
Ω
∂x1w1∇′·w′dx+
∫
R
I(w′,w′)(x1)dx1,
(4.19)
where
I(w1′,w′2) = ∑
j=2,3
∑
k=2,3
∫
D
∂xjw
k 1∂xkw
j
Since,Π′
rigw′(x1,·) =0,according to the proof of [34, Lemma 4], one can
show that for any δ >0 there exists a positive constantCδ such that
ReI(w′,w′)(x1)≥ ∥∇′·w′(x1,·)∥2L2(D)−δ∥∇′w(x1,·)∥2L2(D)
−Cδ∥D′(w′(x1,·))∥2L2(D).
(4.20)
This, together with (4.19), gives
Re
3
∑
j,k=1
∫
Ω
∂xjw
k∂ xkw
jdx≥ ∥divw∥2
2 −δ∥∇′w′∥22−Cδ∥D′(w′)∥22. (4.21)
It then follows from (4.18) and (4.21) that
∥D(w)∥22 ≥2∥∇w∥22+ 2∥divw∥22−2δ∥∇′w′∥22−2Cδ∥D′(w′)∥22. (4.22)
Taking δ= 12, we have
∥∇w∥22+∥divw∥22 ≤ ∥D(w)∥22+ 2C∥D′(w′)∥22 ≤C∥D(w)∥22. (4.23) This shows the desired inequality forwsatisfyingw·n|∂Ω = 0 andΠ′rigw′(x1,·) = 0 for a.e. x1.
For general w with w·n|∂Ω = 0,we decompose w′ in (4.20) as
w′ =v′+Π′
rigw′, (4.24)
where v′ = (I−Π′
rig)w′.Then Π′rigv′ = 0 and
I(w′,w′) =I(v′,v′) +I(v′,Π′
rigw′) +I(Π′rigw′,v′) +I(Π′rigw′,Π′rigw′).
Since ∇′ ·(Π′
rigw′) = 0 and D′(Π′rigw′) = O′, we have ∇′·v′ =∇′ ·w′ and D′(v′) =D′(w′). Then, applying (4.20) with w′ replaced byv′, we have
Re I(v′,v′)≥ ∥∇′ ·v′∥2L2(D)−δ∥∇′v′∥2L2(D)−Cδ∥D′(v′)∥2L2(D)
=∥∇′·w′∥2L2(D)−δ∥∇′v′∥2L2(D)−Cδ∥D′(w′)∥2L2(D)
≥ ∥∇′ ·w′∥2L2(D)−δ∥∇′w′∥2L2(D)−Cδ∥Πrigw′∥2L2(D)
−Cδ∥D′(w′)∥2L2(D).
Furthermore,
|I(v′,Π′rigw′) +I(Π′rigw′,v′)| ≤δ∥∇′v′∥2L2(D)+Cδ∥Π′rigw′∥2L2(D)
≤δ∥∇′w′∥2L2(D)+Cδ∥Π′rigw′∥2L2(D),
Therefore, instead of (4.20), we obtain
ReI(w′,w′)(x1)≥ ∥∇′·w′(x1,·)∥2L2(D)−2δ∥∇′w′(x1,·)∥2L2(D)
−Cδ∥D′(w′(x1,·))∥2L2(D)−Cδ∥Π′rigw′(x1,·)∥2L2(D).
This, together with (4.19), then gives
Re
3
∑
j,k=1
∫
Ω
∂xjw
k∂
xkwjdx
≥ ∥divw∥22 −2δ∥∇′w′∥22−Cδ∥D′(w′)∥22−Cδ∥Π′rigw′∥22,
and hence
∥D(w)∥22 ≥2∥∇w∥22+ 2∥divw∥22 −4δ∥∇′w′∥22
−2Cδ∥D′(w′)∥22−2Cδ∥Π′rigw′∥22.
Taking δ = 14, we obtain the desired inequality. This completes the proof. □
We now give a proof of Lemma 4.5.
Proof of Lemma 4.5. Let u = ⊤(ϕ,w) satisfy u = P
∞u. Since ˜Π+(ξ) +
˜
Π−(ξ) = Π0+ Π1, we see that
ϕ=F−1[ ˆϕ−1R0⟨ϕˆ⟩] =F−1
[
(1−1R0) ˆϕ+1R0( ˆϕ− ⟨ϕˆ⟩)
]
, (4.25)
w1 =F−1[ ˆw1−1R0⟨wˆ1⟩] = F−1
[
(1−1R0) ˆw1+1R0( ˆw1− ⟨wˆ1⟩)
]
. (4.26)
By the Poincar´e’s inequality, we have
∥F−1[
1R0( ˆϕ− ⟨ϕˆ⟩)
]
∥2 ≤C∥∇′ϕ∥2 ≤C∥∇ϕ∥2. (4.27)
Since
1−1R0(ξ) =
{
0 (|ξ| ≤R0),
1 (|ξ|> R0),
we have
∥F−1[(1−1R0) ˆϕ]∥2 ≤
C R0∥
ξϕˆ∥2 =
√
2πC
R0 ∥
∂x1ϕ∥2 ≤
C R0∥∇
ϕ∥2. (4.28) We thus obtain
∥ϕ∥2 ≤C∥∇ϕ∥2. (4.29)
Similarly we have
We next show that∥∇w∥2 ≤C∥D(w)∥2ifR1 >0 is taken suitably large.
We first observe that
w=F−1[(I−1R0Π1 −1R1Πrig) ˆw]
=F−1[(1R0(I −(Π1+Πrig)) + (I−1R1) + (1R1 −1R0)(I−Πrig)) ˆw]
=:w1 +w2+w3.
As for w1, since Π′rigwj′ =0′ (j = 1,3),we see from Lemma 4.6 that
∥∇wj∥2 ≤C∥D(wj)∥2, j = 1,3. (4.31)
Lemma 4.6 also shows that
∥∇w2∥2 ≤C{∥D(w2)∥2+∥Π′rigw2′∥2}
≤C{∥D(w2)∥2+∥w′2∥2}
≤C{∥D(w2)∥2+
1
R1∥
∂x1w2′∥2}.
Set R2 = max{R0,2C1 }. Then
∥∇w2∥2 ≤2C∥D(w2)∥2.
We thus obtain
∥∇w∥2 ≤ ∥∇w1∥2+∥∇w2∥2+∥∇w3∥2
≤C{∥D(w1)∥2+∥D(w2)∥2+∥D(w3)∥2}.
Noting that supp ˆw1 ⊂ {|ξ| ≤R0}, supp ˆw2 ⊂ {|ξ| ≥R1} and supp ˆw3 ⊂
{R0 ≤ |ξ| ≤R1},we see that
2π∥D(w)∥22 =
3
∑
j=1
∥Dˆ( ˆwj)∥22+ 2Re ( ˆD( ˆw1),Dˆ( ˆw2))
+ 2Re ( ˆD( ˆw2),Dˆ( ˆw3)) + 2Re ( ˆD( ˆw3),Dˆ( ˆw1))
=
3
∑
j=1
∥Dˆ( ˆwj)∥22 = 2π
3
∑
j=1
∥D(wj)∥22.
Therefore, we have
We finally prove that ∥w∥2 ≤ C∥∇w∥2. In view of (4.30), it suffices to prove that ∥w′∥2 ≤ C∥∇′w′∥2. But, since w′ · n′|∂D = 0, we have
∥w′(x1,·)∥L2(D) ≤ C∥∇′w′(x1,·)∥L2(D), which gives the desired estimate.
This completes the proof. □
Hereafter we fix the constant R1 in the definition of P1 so that Lemma
4.5 holds true.
5
Reformulation of problem
In this section we reformulate the problem. The problem (2.5)-(2.8) is written as
{
∂tu+Lu=F(u),
u|t=0 =u0.
(5.1)
Here u=⊤(ϕ,w)∈D(L) and F(u) =⊤(f0(ϕ,w),f(ϕ,w)).
One can prove the local solvability for (5.1) as in [14]. See also [28].
Proposition 5.1 Assume that u0 = ⊤(ϕ0,w0) ∈ H2×H∗2 and ∥ϕ0∥∞ ≤ 12.
Then there exists a positive number T0 depending on ∥u0∥H2 such that the problem (5.1) has a unique solution u = ⊤(ϕ,w) on [0, T
0] satisfying u ∈
C([0, T0];H2×H∗2)∩C1([0, T0];L2×L2) with w∈ ∩1j=0Hj(0, T0;H3−2j) and
∥ϕ0(t)∥∞≤ 43 for t ∈[0, T0]. Furthermore, the inequality
sup
t∈[0,T0]{∥
u(t)∥H2+∥∂tu(t)∥2}+
∫ T0
0 ∥
w∥2H3dt≤C{1+∥u0∥2H2}β∥u0∥2H2 (5.2) holds with some positive constants C and β.
The global existence ofu(t) follows in a standard manner from Proposition 5.1 and the a priori bound ∥u(t)∥H2 ≤C∥u0∥H2∩L1 for u0 sufficiently small
in H2∩L1. The a priori bound will be given in Proposition 5.7 below.
To solve the problem (5.1), we decompose (5.1) into a problem for a low frequency partu1(t) ofu(t) and a one for a high frequency partu∞(t) ofu(t).
We decomposeu=⊤(ϕ,w) into
u=u1+u∞,
where
u∞ =P∞u=⊤(ϕ∞,w∞), w∞=⊤(w1∞,w′∞).
By applying the operators P1 and P∞ to (5.1), we obtain the following
proposition.
Proposition 5.2 LetT be a given positive number and let u(t)be a solution of (5.1) on [0, T]. Assume that u ∈C([0, T];H2×H∗2)∩C1([0, T];L2×L2)
with w ∈ ∩1j=0Hj(0, T;H3−2j). Then
u1 =⊤(ϕ1,w1)∈C1
(
[0, T];Hl×[Hl∩H∗2])
for l = 0,1,2,· · ·, and
u∞=⊤(ϕ∞,w∞)∈C([0, T];H2×H∗2)∩C1([0, T];L2×L2)
with w∞ ∈ ∩1j=0Hj(0, T;H3−2j). Furthermore, u1 and u∞ satisfy
u1 =e−tL0P1u0+
∫ t
0
P1e−(t−τ)L0F1(u(τ))dτ, (5.3)
∂tu∞+Lu∞ =F∞(u), u∞|t=0 =P∞u0. (5.4) Here F1(u) =P1F(u1+u∞), F∞(u) =P∞F(u1+u∞).
We define M(t)≥0 by
M(t) =M1(t) +M∞(t) (t∈[0, T]), (5.5)
where M1(t) and M∞(t) are defined by
M1(t) = sup 0≤τ≤t
{ 1
∑
k=0
(1 +τ)14+
k
2∥∂k
x1u1(τ)∥2+ (1 +τ) 3
4∥∂tu1(τ)∥ 2
} ,
M∞(t) =
(
sup
0≤τ≤t
(1 +τ)32{∥u∞(τ)∥2
H2 +∥∂tu∞(τ)∥22}
)12 .
By the definition of P1,we see that
P1u=F−1[1R0⟨ϕˆ⟩]b0+F−1[1R0⟨wˆ1⟩]b1+F−1[1R1⟨wˆ′,b′rig⟩]brig,
and F−1[1
R0⟨ϕˆ⟩], F−1[1R0⟨wˆ1⟩] and F−1[1R1⟨wˆ′,b′rig⟩] are functions of x1
only. Therefore, by the Gagliardo-Nirenberg-Sobolev inequality, we have
∥u1(t)∥∞≤C∥u1(t)∥ 1 2
2∥∂x1u1(t)∥ 1 2
2 ≤C(1 +t)− 1 2M
As for u∞(t),we have
∥u∞(t)∥∞ ≤C∥u∞(t)∥H2 ≤C(1 +t)− 3 4M
∞(t).
We also introduce the quantitiesE∞(t) andD∞(t) foru∞(t) = ⊤(ϕ∞(t),w∞(t)) :
E∞(t) =∥u∞(t)∥2H2 +∥∂tu∞(t)∥22,
D∞(t) =∥∇ϕ∞(t)∥2H1 +∥∇w∞(t)∥2H2 +∥∂tu∞(t)∥2H1.
Theorem 3.1 is an immediate consequence of the following proposition.
Proposition 5.3 If ∥u0∥H2∩L1 is sufficiently small, then
M(t)≤C∥u0∥H2∩L1. (5.6)
To prove Proposition 5.3, we reformulate (5.3)-(5.4) as follows. We will make use of a momentum formulation for the low frequency part which is useful to derive the decay estimate. Let u = u1 +u∞, where u1 and u∞
satisfy (5.3)-(5.4). Then, usatisfies (2.5)-(2.8). If we write uasu=⊤(ϕ,w),
then ϕ and w satisfy
∂tϕ+ divw =g0(ϕ,w), (5.7)
∂tw−νdivD(w)−ν′∇divw+∇ϕ=g(ϕ,w), (5.8)
where
g0(ϕ,w) =−div(ϕw),
g(ϕ,w) =−ϕ∂tw−(1 +ϕ)w· ∇w−
(
∇p(ρ)− ∇ϕ)
=−ϕ∂tw−(1 +ϕ)w· ∇w−
p′′(1)
2 ∇(ϕ
2)
− 12∇(
p(3)(ϕ)ϕ3) .
Here
p(3)(ϕ) =
∫ 1
0
(1−θ)2p′′′(1 +θϕ)dθ.
From the system (5.7)-(5.8) we derive a momentum formulation for the low frequency part. As in [36], we introduce a dimensionless momentum m:
m= (1 +ϕ)w, (5.9)
and define its low-frequency part m1 by
where w=w1+w∞ and the operator P1 is defined by P1w=F−1
(
1R0Π1wˆ +1R1Πrigwˆ
) .
Note that P1 is a bounded projection from L2 to Hk for any nonnegative
integer k and it holds that ∂x1P1 =P1∂x1 and
∥P1w∥Hk ≤Ck∥w∥2. (5.11)
Before proceeding further, we show that w1 is uniquely determined by m1, ϕ and w∞ through the relation (5.10), i.e.,
w1 =m1−P1(
ϕ(w1+w∞))
. (5.12)
Lemma 5.4 There exists a positive constant δ0 such that the following as-sertion holds true. Let
m1 ∈C1([0, T];L2), w∞ ∈ ∩1j=0Cj([0, T];H2−2j), ϕ∈ ∩1j=0Cj([0, T];H2−2j).
If∥ϕ∥C([0,T];H2)+∥∂tϕ∥C([0,T];L2) ≤δ0,then there uniquely existsw1 ∈C1([0, T];L2) that satisfies (5.12).
Furthermore, there hold the estimates
∥w1∥C([0,T];L2) ≤C
(
∥m1∥C([0,T];H2)+∥ϕ∥C([0,T];H2)∥w∞∥C([0,T];H2)
)
, (5.13)
∥∂tw1∥C([0,T];L2)
≤C(∥∂tm1∥C([0,T];H2)+∥∂tϕ∥C([0,T];L2)∥w∞∥C([0,T];H2) (5.14)
+∥ϕ∥C([0,T];H2)∥∂tw∞∥C([0,T];L2)
) ,
where C is a positive constant independent of T.
Proof. We first observe that, by the Sobolev inequality and (5.11),
∥w1∥∞ ≤C∥w1∥H2 =C∥P1w1∥H2 ≤C∥w1∥2.
Let ∥ϕ∥C([0,T];H2) +∥∂tϕ∥C([0,T];L2) ≤ δ0. We set Γ(w1) = m1 −P1
(
ϕ(w1+
w∞)
)
. We claim that Γ is a map on C1([0, T];L2). In fact, by the Sobolev inequality and (5.11), we have
∥Γ(w1)∥C([0,T];L2)
≤ ∥m1∥C([0,T];L2)+C∥ϕ(w1+w∞)∥C([0,T];L2)
≤C{∥m1∥C([0,T];L2)+∥ϕ∥C([0,T];L∞)∥w1+w∞∥C([0,T];L2)
}
(5.15)
≤C{∥m1∥C([0,T];L2)+∥ϕ∥C([0,T];H2)
(
∥w1∥C([0,T];L2)+∥w∞∥C([0,T];L2)
and
∥∂tΓ(w1)∥C([0,T];L2)
≤ ∥∂tm1∥C([0,T];L2)+C∥∂tϕ(w1+w∞)∥C([0,T];L2)
+C∥ϕ(∂tw1+∂tw∞)∥C([0,T];L2)
≤C{∥∂tm1∥C([0,T];L2)+C∥∂tϕ∥C([0,T];L2)
(
∥w1∥C([0,T];L∞)+∥w∞∥C([0,T];L∞)
)
+C∥ϕ∥C([0,T];L∞)
(
∥∂tw1∥C([0,T];L2)+∥∂tw∞∥C([0,T];L2)
)}
(5.16)
≤C{∥∂tm1∥C([0,T];L2)+∥∂tϕ∥C([0,T];L2)
(
∥w1∥C([0,T];L2)+∥w∞∥C([0,T];H2)
)
+∥ϕ∥C([0,T];H2)
(
∥∂tw1∥C([0,T];L2)+∥∂tw∞∥C([0,T];L2)
)} .
Therefore, Γ is a map on C1([0, T];L2).
We next show that Γ is a contraction map. By the Sobolev inequality and (5.11), we have
∥Γ(w(1)1 )−Γ(w(2)1 )∥C([0,T];L2) ≤C∥ϕ∥C([0,T];L∞)∥w1(1)−w(2)1 ∥C([0,T];L2)
≤C∥ϕ∥C([0,T];H2)∥w(1)1 −w1(2)∥C([0,T];L2),
and
∂t
[
Γ(w1(1))−Γ(w1(2))]
C([0,T];L2)
≤C{∥ϕ∥C([0,T];L∞)∥∂tw1(1)−∂tw1(2)∥C([0,T];L2)
+∥∂tϕ∥C([0,T];L2)∥w1(1)−w(2)1 ∥C([0,T];L∞)
}
≤C{
∥ϕ∥C([0,T];H2)+∥∂tϕ∥C([0,T];L2)
}
∥w1(1)−w(2)1 ∥C1([0,T];L2).
We thus obtain
∥Γ(w1(1))−Γ(w1(2))∥C1([0,T];L2) ≤Cδ0∥w1(1)−w1(2)∥C1([0,T];L2).
Therefore, if δ0 > 0 is sufficiently small, then Γ is a contraction. By the
contraction mapping principle, we thus see that there exists a unique w1 ∈
C1([0, T];L2) such that w1 = Γ(w1). This shows the unique existence of w1
satisfying (5.12). The estimates (5.13) and (5.14) now follow from (5.15) and
(5.16). This completes the proof. □
Remark 5.5 We see from (5.11) that w1 in Lemma 5.4 satisfies w1 ∈
C1([0, T];Hk) for any k and
By (5.7), we have
(1 +ϕ)w· ∇w= div(
(1 +ϕ)w⊗w)−wdiv(
(1 +ϕ)w)
= div(
(1 +ϕ)w⊗w)+w∂tϕ.
(5.18)
Therefore, applying the operator P1 to (5.7)-(5.8), we obtain
∂tϕ1 + divm1 = 0,
and
P1(∂tw−νdivD(w)−ν′∇divw) +∇ϕ1
=P1(
−ϕ∂tw−(1 +ϕ)w· ∇w− p ′′(1)
2 ∇(ϕ 2)
−12∇(p
(3)(ϕ)ϕ3))
=P1
(
−ϕ∂tw−div((1 +ϕ)w⊗w) +wdiv((1 +ϕ)w)
−p′′2(1)∇(ϕ 2)
− 12∇(p
(3)(ϕ)ϕ3))
=P1
(
−ϕ∂tw−div((1 +ϕ)w⊗w)−w∂tϕ− p ′′(1)
2 ∇(ϕ 2)
− 12∇(p
(3)(ϕ)ϕ3))
=P1
(
−∂t(ϕw)−div((1 +ϕ)w⊗w)
)
+P1
(
− p′′2(1)∇(ϕ 2)
− 12∇(p
(3)(ϕ)ϕ3)) .
The latter equation is written as
P1(
∂t((1 +ϕ)w)−νdivD(w)−ν′∇divw
)
+∇ϕ1
=−P1
(
div((1 +ϕ)w⊗w))
+P1
(
− p′′2(1)∇(ϕ 2)
− 12∇(p
(3)(ϕ)ϕ3)) .
We thus obtain
∂tϕ1+ div m1 = 0, (5.19)
∂tm1−νdiv D(m1)−ν′∇div m1+∇ϕ1
=−νdivD(P1(ϕw))−ν′∇divP1(ϕw) (5.20)
−P1(div((1 +ϕ)w⊗w)−p′′2(1)∇(ϕ2)− 1 2∇(p
(3)(ϕ)ϕ3)) .
Setting
u1∗ =⊤(ϕ1,m1), (5.21)
we arrive at
∂tu1∗ +Lu1∗ =P1G1(u) +P1G2(u), m1 =w1+P1(ϕw),
u1∗|t=0 =P1u∗0,
(5.22)
where u=⊤(ϕ,w), w =w
1+w∞, and
G1(u) =
(
−div(ϕw)
−div((1 +ϕ)w⊗w)−p′′2(1)∇(ϕ2)− 1 2∇(p
(3)(ϕ)ϕ3)
)
,
G2(u) =L
(
0
P1(ϕw)
) .
Since ˆΠ+(ξ) + ˆΠ−(ξ) = Π0+ Π1,we see that
P1
(
G1(u) +G2(u)
)
=P1∂x1G(u) +P1∂x1G˜(u), (5.23)
where
G(u) = −
0 (w1)2+p′′(1)
2 ϕ 2
w1w′
, (5.24)
˜
G(u) = −
0
ϕ(w1)2+1
2p(3)(ϕ)ϕ3+ (2ν+ν′)∂x1(ϕw1)
ϕw1w′+ν∂x1(ϕw′)
. (5.25)
In terms ofu1∗(t) and u∞(t), Proposition 5.2 is restated as follows.
Proposition 5.6 Let u(t) be a solution of (5.1) on [0, T]. Assume that
u ∈ C([0, T];H2×H∗2)∩C1([0, T];L2×L2) with w ∈ ∩1j=0Hj(0, T;H3−2j).
Then
u1∗ =⊤(ϕ1,m1)∈C1([0, T];Hl×Hl) for l = 0,1,2,· · ·, and
u∞=⊤(ϕ∞,w∞)∈C([0, T];H2×H∗2)∩C1([0, T];L2×L2)
with w∞ ∈ ∩1j=0Hj(0, T;H3−2j). Furthermore, u1∗ and u∞ satisfy
u1∗(t) = e−tLP1u∗0+
∫ t
0
e−(t−τ)LP1∂x1
(
G(u) + ˜G(u))
(τ)dτ, (5.26)
m1 =w1+P1(ϕw), (5.27)
∂tu∞+Lu∞ =F∞(u), u∞|t=0 =P∞u0, (5.28)
where u = ⊤(ϕ,w), w = w
1 +w∞; u∗0 = ⊤(ϕ0,m0), m0 = w0 +ϕ0w0;
G(u)andG˜(u)are the ones given by (5.24) and (5.25); F∞(u) =P∞F(u) =: ⊤(f0
∞(u),f∞(u)), f∞(u) =:⊤(f∞1 (u),f∞′ (u)).
We define M∗(t)≥0 by
M∗(t) = M1∗(t) +M∞(t) (t∈[0, T]). (5.29)
Here M1∗(t) is defined by
M1∗(t) = sup
0≤τ≤t
{ 1
∑
k=0
(1 +τ)14+
k
2∥∂k
x1u1∗(τ)∥2+ (1 +τ) 3
4∥∂tu1∗(τ)∥ 2
}
,
and M∞(t) is defined as before.
We see from Lemma 5.4 (and its proof) thatM(t) is equivalent toM∗(t)
if ∥ϕ∥C([0,t];H2) +∥∂tϕ∥C([0,t];L2) ≤ δ0. We also note that, by the
Gagliardo-Nirenberg-Sobolev inequality,
∥u1∗(t)∥∞ ≤C∥u1∗(t)∥
1 2
2∥∂x1u1∗(t)∥
1 2
2 ≤C(1 +t)− 1
2M1∗(t).
As forM1∗(t) and M∞(t), we will prove the following estimates.
Proposition 5.7 Let u(t) be a solution of (5.1) on [0, T]. Then there exists a positive constant ε1 such that if∥u(t)∥H2 ≤ε1 andM∗(t)≤1fort ∈[0, T], the estimates
M1∗(t)≤C{∥u∗0∥1+M∗(t)2} (5.30)
and
E∞(t) +
∫ t
0
e−d(t−τ)D∞(τ)dτ
≤C {
e−dtE∞(0) + (1 +t)−
3
2M∗(t)4+
∫ t
0
e−d(t−τ)R(τ)dτ
} (5.31)
hold uniformly for t ∈ [0, T] with a positive constant C independent of T. Here d=d(ν, ν′) is a positive constant, and R(t) is a quantity that satisfies
R(t)≤C{(1 +t)−32M
∗(t)3 +M∗(t)D∞(t)}. (5.32)
The estimate (5.30) will be proved in Section 6, and the estimates (5.31) and (5.32) will be proved in Sections 7 and 8, respectively.
From Propositions 5.1 and 5.7, one can show the following uniform esti-mate of M∗(t) as in [12].
Proposition 5.8 If ∥u0∥H2∩L1 is sufficiently small, then
M∗(t)≤C∥u0∥H2∩L1. (5.33)
6
Estimates for
u
1∗(t)
In this section, we estimate the low-frequency part u1∗ = ⊤(ϕ1,m1) and
prove the estimate (5.30) in Proposition 5.7.
As for the nonlinearities, by using (4.12), it is not difficult to show the following estimates.
Lemma 6.1 Let k= 0,1. Then
∂x1k P1
(
G(u) + ˜G(u))
1 ≤C(1 +t)
−1 2−
k
2M
∗(t)2(1 +M∗(t)), (6.1)
P1∂x1
(
G(u) + ˜G(u))
2 ≤C(1 +t)
−3 4M
∗(t)2(1 +M∗(t)). (6.2)
We now give a proof of (5.30).
Proof of (5.30). By (5.26) and Lemma 4.2, we have
∥∂x1k u1∗(t)∥2
≤ ∥∂x1k e−tLP1u∗0∥2+
∫ t
0 ∥
∂x1k e−(t−τ)LP1∂x1(G(u) + ˜G(u))(τ)∥2dτ
≤C(1 +t)−1 4−
k
2∥u
∗0∥1+
∫ t
0 ∥
∂x1k e−(t−τ)LP1∂x1(G(u) + ˜G(u))(τ)∥2dτ
for k = 0,1. As for the second term on the right-hand side, we write it as
∫ t
0 ∥
∂x1k e−(t−τ)LP1∂x1(G(u) + ˜G(u))(τ)∥2dτ
=
( ∫ 2t
0
+
∫ t
t
2
)
∥∂x1k e−(t−τ)LP1∂x1(G(u) + ˜G(u))(τ)∥2dτ
=:I1(t) +I2(t).
As forI1(t),since∂x1P =P ∂x1,we write∂x1k e−(t−τ)LP1∂x1 =∂x1k+1e−(t−τ)LP1,
and applying Lemmas 4.2 and 6.1 to obtain
I1(t)≤C
∫ 2t
0
(1 +t−τ)−34−
k
2(1 +τ)− 1 2dτ M
∗(t)2 ≤C(1 +t)−
1 4−
k
2M
∗(t)2.
As forI2(t), applying Lemmas 4.2 and 6.1, we have
I2(t)≤C
∫ t
t
2
(1 +t−τ)−14−
k
2(1 +τ)−1dτ M
∗(t)2 ≤C(1 +t)−
1 4−
k
2M
We thus obtain
∥∂x1k u1∗(t)∥2 ≤C(1 +t)− 1 4−
k
2{∥u∗0∥1+M∗(t)2} (6.3)
for k = 0,1.
Let us estimate the time derivative. By Remark 4.4, we have
Lu1∗ =
0 ∂x1 ⊤0′
∂x1 −(2ν+ν′)∂x12 ⊤0′
0′ 0′ −ν∂2
x1I′
u1∗, I′ = (
1 0 0 1
) .
This, together with (4.12), implies that
∥ −Lu1∗(t)∥2 ≤C{∥∂x12 m1(t)∥2+∥∂x1u1∗(t)∥2}
≤C(1 +t)−3 4{∥u
∗0∥1 +M∗(t)2}.
Since ∂tu1∗ =−Lu1∗+P1∂x1
(
G(u) + ˜G(u))
,applying Lemma 6.1, we have
∥∂tu1∗(t)∥2 ≤C{∥Lu1∗(t)∥2+∥P1(G(u) + ˜G(u))(t)∥2}
≤C(1 +t)−34{∥u
∗0∥1+M∗(t)2}.
(6.4)
By (6.3) and (6.4), we deduce the desired estimate (5.30). This completes
the proof. □
7
Estimates for
u
∞(
t
)
In this section, we estimate the high-frequency part u∞ = P∞u by using
the Matsumura-Nishida energy method ([27]) and prove estimate (5.31) with
R(t) satisfying (5.32) in Proposition 5.7. We define the operator P∞ by
P∞=I−P1.
Throughout this section we setν0 = min
{2
3ν, 2 3ν+ν
′}
. As was shown in [32], since
∥div w∞∥22 ≤3 3
∑
j=1
∥∂xjw
j
∞∥22 ≤
3
4∥D(w∞)∥
2
2, (7.1)
we see that
ν
2∥D(w∞)∥
2
2+ν′∥div w∞∥22 ≥
3
4ν0∥D(w∞)∥
2
In fact, (7.2) clearly holds if ν′ ≥0.If ν′ <0, by (7.1)
ν
2∥D(w∞)∥
2
2+ν′∥div w∞∥22 ≥
( ν
2 + 3 4ν
′
)
∥D(w∞)∥2 2
= 3
4
(
2 3ν+ν
′
)
∥D(w∞)∥2 2.
We thus obtain (7.2).
To prove (5.31), we prepare some basic estimates.
Proposition 7.1 Letkandj be nonnegative integers satisfying0≤2k+j ≤
2. Then
1 2
d dt∥∂
k
t∂x1j u∞∥22+
3
8cKν0∥∇∂
k
t∂x1j w∞∥22+
3ν0
4 ∥∂
k
t∂x1j ϕ˙∞∥22 ≤CR (1)
j,k, (7.3)
where cK = C 2 1 C2
2 with C1 and C2 being the constants in Lemma 4.5,
˙
ϕ∞=∂tϕ∞+w· ∇ϕ∞,
R(1)j,k = 1
2(divw,|∂
k
t∂x1j ϕ∞|2)−([∂tk∂x1j ,w]· ∇ϕ∞, ∂tk∂x1j ϕ∞)
+ (∂tk∂x1j f˜∞0 , ∂tk∂x1j ϕ∞) + (∂tk∂x1j f∞, ∂tk∂x1j w∞) +
3ν0
2 ∥∂
k
t∂x1j f˜∞0 ∥22,
˜
f∞0 = ˜f∞0 (u) =F−1[1R0⟨w\· ∇ϕ∞⟩]−(w· ∇ϕ1+ϕ divw)
+F−1[1R0⟨w\· ∇ϕ1+ϕ\divw⟩].
Here and in what follows we abbreviate f∞0(u), f˜∞0(u), f∞(u) and F∞(u) as
f0
∞, f˜∞0 , f∞ and F∞, respectively.
Proof. Equation (5.28) is written as
∂tϕ∞+w· ∇ϕ∞+ divw∞= ˜f∞0 , (7.4)
∂tw∞−νdivD(w∞)−ν′∇divw∞+∇ϕ∞ =f∞. (7.5)
We compute (∂x1j (7.4), ∂x1j ϕ∞) + (∂x1j (7.5), ∂x1j w∞) to obtain
1 2
d dt∥∂
j
x1u∞∥22+
ν
2∥∂
j
x1D(w∞)∥22+ν′∥∂x1j divw∞∥22
=−(∂x1j (w· ∇ϕ∞), ∂x1j ϕ∞) + (∂x1j f˜∞0, ∂x1j ϕ∞) + (∂x1j f∞, ∂x1j w∞)
=−(w· ∇∂x1j ϕ∞, ∂x1j ϕ∞)−([∂x1j ,w]· ∇ϕ∞, ∂x1j ϕ∞)
+ (∂x1j f˜∞0 , ∂x1j ϕ∞) + (∂x1j f∞, ∂x1j w∞)
= 1
2(divw,|∂
j
x1ϕ∞|2)−([∂x1j ,w]· ∇ϕ∞, ∂x1j ϕ∞) + (∂x1j f˜∞0 , ∂x1j ϕ∞)
+ (∂x1j f∞, ∂x1j w∞).
We set ˙ϕ:=∂tϕ+w·∇ϕ∞.From (7.4), we have∂x1j ϕ˙∞=−div∂x1j w∞+∂x1j f˜∞0,
and hence, by (7.1)
∥∂x1j ϕ˙∞∥22 ≤2(∥div∂x1j w∞∥22+∥∂x1j f˜∞0∥22)
≤ 32∥D(∂x1j w∞)∥22+ 2∥∂x1j f˜∞0 ∥22.
We thus obtain
3 4ν0∥∂
j
x1ϕ˙∞∥22 ≤
3
8ν0∥D(∂
j
x1w∞)∥22+
3 2ν0∥∂
j
x1f˜∞0 ∥22. (7.7)
It then follows from (4.17), (7.6) and (7.7) that
1 2
d dt∥∂
j
x1u∞∥22+
3
8cKν0∥∇∂
j
x1w∞∥22+
3ν0
4 ∥∂
j
x1ϕ˙∞∥22 ≤CR (1)
j,0. (7.8)
Replacing ∂x1j by∂t, we also have
1 2
d
dt∥∂tu∞∥
2 2+
3
8cKν0∥∇∂tw∞∥
2 2+
3ν0
4 ∥∂tϕ˙∞∥
2
2 ≤CR (1)
0,1. (7.9)
This completes the proof. □
Proposition 7.2 It holds that
1 2
d dt
{ν
2∥D(w∞)∥
2
2+ν′∥divw∞∥22−2(ϕ∞,divw∞)
}
+1
2∥∂tw∞∥
2 2
≤4∥divw∞∥22+∥w· ∇ϕ∞∥22+∥f˜∞0 ∥22+∥f∞∥22.
(7.10)
Proof. We compute ((7.4), ∂tϕ∞) + ((7.5), ∂tw∞) to obtain
∥∂tu∞∥22+
ν
4 d
dt∥D(w∞)∥
2 2+
ν′
2 d
dt∥divw∞∥
2 2
+{(divw∞, ∂tϕ∞) + (∇ϕ∞, ∂tw∞)}
= ( ˜f∞0 , ∂tϕ∞) + (f∞, ∂tw∞)−(w· ∇ϕ∞, ∂tϕ∞).
Since
(∇ϕ∞, ∂tw∞) = −(ϕ∞, ∂tdivw∞)
=−d
we have
∥∂tu∞∥22+
ν
4 d
dt∥D(w∞)∥
2 2+
ν′
2 d
dt∥divw∞∥
2 2−
d
dt(ϕ∞,divw∞)
= ( ˜f∞0 , ∂tϕ∞) + (f∞, ∂tw∞)−2(∂tϕ∞,divw∞)−(w· ∇ϕ∞, ∂tϕ∞)
≤ 1
4∥∂tu∞∥
2
2+C{∥w· ∇ϕ∞∥22+∥f˜∞0 ∥22+∥f∞∥22}.
(7.11)
Adding −2(divw∞, ∂tϕ∞) to both sides of (7.11), we obtain
∥∂tu∞∥22 +
ν
4 d
dt∥D(w∞)∥
2 2+
ν′
2 d
dt∥divw∞∥
2 2−
d
dt(ϕ∞,divw∞)
≤ −2(divw∞, ∂tϕ∞) +
1
4∥∂tu∞∥
2
2+C{∥w· ∇ϕ∞∥22+∥f˜∞0 ∥22+∥f∞∥22}
≤ 12∥∂tu∞∥22 +C{∥divw∞∥22+∥w· ∇ϕ∞∥22+∥f˜∞0 ∥22+∥f∞∥22},
which gives the desired estimate. This completes the proof. □
We next establish the interior estimates. Letζ ∈C∞(D). It follows from
(7.4) and (7.5) that
∂t(ζϕ∞) +w· ∇(ζϕ∞) + div(ζw∞) = g∞0 +w∞· ∇ζ, (7.12)
∂t(ζw∞)−νdivD(ζw∞)−ν′∇div(ζw∞) +∇(ζϕ∞) = g∞, (7.13)
where
g0∞=ζf˜∞0 + (w· ∇ζ)ϕ∞,
g∞=ζf∞+ν[ζ,divD]w∞+ν′[ζ,∇div]w∞+ϕ∞∇ζ.
We introduce ζin(x′) = ζin(|x′|)∈Cc∞(D) satisfying
ζin(x′) = 1 for |x′| ≤
3
4, ζin(x
′) = 0 for
|x′| ≥ 7
8.
Setting ζ =ζin in (7.12)-(7.13), we obtain the following interior estimates.
Proposition 7.3 Let 1≤ |α| ≤2. Then
1 2
d dt∥∂
α
x(ζinu∞)∥22+
3
8cKν0∥∇∂
α
x(ζinw∞)∥22+
3ν0
4 ∥∂
α
x(ζinϕ˙∞)∥22
≤C{R(2)α +ν0∥w∞∥2H|α|}+ε∥ϕ∞∥2H|α| +ε∥w∞∥2H|α|+1+
C ε∥w∞∥
2 H|α|,
where
R(2)α = 1
2(divw,|∂
α
x(ζinϕ∞)|2)−([∂xα,w]· ∇(ζinϕ∞), ∂xα(ζinϕ∞))
+ (∂xαg0∞,in, ∂xα(ζinϕ∞)) + (∂xα(ζinf∞), ∂xα(ζinw∞)) +
3ν0
2 ∥∂
α
x(ζing0∞)∥22. Here g∞0 ,in and g∞,in are the functions obtained by replacing ζ in g∞0 and g∞
with ζin, respectively.
Proof. Applying ∂xα to (7.12) and (7.13), we see that
∂t∂xα(ζϕ∞) +w· ∇∂xα(ζϕ∞) + div∂αx(ζw∞) (7.15)
=∂xαg∞0 +∂xα(w∞· ∇ζ)−[∂xα,w]· ∇(ζϕ∞),
∂t∂xα(ζw∞)−νdivD(∂xα(ζw∞))−ν′∇div∂xα(ζw∞) +∇∂xα(ζϕ∞) (7.16)
=∂xαg∞.
Since ∂xα(ζinw∞) = 0 for |x′| ≥ 78,one can obtain the desired estimates as in
the proof of Proposition 7.1. This completes the proof. □
We next consider the estimates near the boundary∂D. For this purpose we rewrite (7.12)-(7.13) in the cylindrical coordinates (x1, r, θ) with x2 =
rcosθ, x3 =rsinθ. The velocity field w is written as
w=Ew˜, w˜ =⊤( ˜w1,w˜2,w˜3) = ⊤(w1, wr, wθ),
where
E = (e1,er,eθ) =
1 0 0
0 cosθ −sinθ
0 sinθ cosθ
.
The gradient ∇x =⊤(∂x1, ∂x2, ∂x3) is rewritten as
∇x = ˜E∇y, ∇y =⊤(∂y1, ∂y2, ∂y3) =⊤(∂x1, ∂r, ∂θ),
where
˜
E = (e1,er,
1
reθ) =
1 0 0
0 cosθ −1rsinθ
0 sinθ 1rcosθ
.
For an integer m satisfying m≥1, we definewm by
wm =Ew˜m,
˜
