On the spectral properties for the linearized problem around space-time periodic states of the compressible Navier-Stokes equations

This paper studies the linearized problem for the compressible Navier-Stokes equation around space-time periodic state in an infinite layer of $\mathbb{R}^n$ ($n=2,3$), and the spectral properties of the linearized evolution operator is investigated. It is shown that if the Reynolds and Mach numbers are sufficiently small, then the asymptotic expansions of the Floquet exponents near the imaginary axis for the Bloch transformed linearized problem are obtained for small Bloch parameters, which would give the asymptotic leading part of the linearized solution operator as $t\rightarrow\infty$.

If the external forceG takes the formG =G para = ⊤ (g 1 (x n ,t), 0, · · · , 0), then the system (1.1)-(1.2) with the boundary condition (1.5) has a timeperiodic parallel flow, i.e., a time periodic solution of the formũ para = ⊤ (ρ * ,ṽ para ) withṽ para = ⊤ (ṽ 1 para (x n ,t), 0, · · · , 0). The stability of parallel flows has been widely studied in the hydrodynamic stability theory. As for the mathematical study of the stability of time periodic parallel flows of (1.1)-(1.2), the nonlinear dynamics of solutions around time periodic parallel flows was investigated by Brezina [2]. (See also [3,4] for the linearized analysis.) It was proved in [2] that if the Reynolds and the Mach numbers are sufficiently small, then time periodic parallel flows are asymptotically stable under perturbations small in some Sobolev space on the layerΩ. Furthermore, it was shown that the asymptotic leading part of the perturbation is given by a product of a time periodic function and a solution of an n − 1 dimensional linear heat equation in the case n = 3, and by a product of a time periodic function and a solution of a one-dimensional viscous Burgers equation in the case n = 2; the hyperbolic aspect of the perturbation decays faster. (See [2] and references therein for the mathematical analysis of the stability of parallel flows in compressible fluids.) On the other hand, in reality, the external forceG para = ⊤ (g 1 (x n ,t), 0, · · · , 0) often undergoes a perturbation inx ′ variable. Under such a situation, the external force depends not only (x n ,t) but alsox ′ , so the time periodic parallel flow is no longer a solution of (1.1)-(1.2) sinceG depends onx ′ . In this paper, we thus consider the situation where the external forceG periodically depends onx ′ variable as described in (1.4). Under such a situation, as was mentioned above, ifG is sufficiently small, the system (1.1)-(1.2) with (1.5) has a spacetime periodic stateũ p = ⊤ (ρ p ,ṽ p ). We shall establish the results on the spectral properties of the linearized evolution operator aroundũ p which suggest that the asymptotic leading part of the perturbation ofũ p exhibits diffusive behaviors similar to those in the case of parallel flows in [2] if the Reynolds and the Mach numbers are sufficiently small.
Our main results are summarized as follows. If the external force G and Bloch parameter η ′ are sufficiently small, then (1.13) Here β 0 is a positive constant; and λ η ′ ,k is a simple eigenvalue of −B η ′ satisfying with some constants a j , a jk ∈ R, where (a jk ) 1≤j,k≤n−1 is positive definite. It follows from (1.13) that the spectrum of U η ′ (1, 0) with |η ′ | ≪ 1 satisfies where e λ η ′ ,0 is a simple eigenvalue of U η ′ (1, 0). This yields the asymptotic behavior U η ′ (m, 0) ∼ e mλ η ′ ,0 as m → ∞, which, together with (1.14), would imply that U(t, 0) would behave diffusively as t → ∞. We also establish the boundedness of the eigenprojection for the eigenvalue λ η ′ ,0 with |η ′ | ≪ 1 which is needed in the analysis of the nonlinear problem. Our results, in fact, will yield the following diffusive behavior of a part of the solution operator U(t, s) as t − s → ∞ that includes the space-time periodic nature of the problem. In a similar manner to [2,3,4], based on (1.13), (1.14) and the Floquet theory, one can show that there exist a bounded projection P(t) on L 2 (Ω) × L 2 (Ω) such that P(t + 1) = P(t) and the following estimates hold: Here u (0) = u (0) (x ′ , x n , t) is some function 2π αi -periodic in x i (i = 1, · · · , n − 1) and 1-periodic in t and H(t)σ 0 is a solution of the linear heat equation The main difference in the analysis of this paper to the case of the parallel flow in [3,4] is as follows. In the case of the parallel flow, by the Fourier transform in x ′ ∈ R n−1 , the spectral analysis for the linearized problem is reduced to the one for a one-dimensional problem on the interval (0, 1) with a parameter of the Fourier variable ξ ′ ∈ R n−1 ; and the one-dimensional aspect of the reduced problem was essentially used in the analysis in [3,4], e.g., to obtain a regularity estimate of time-periodic eigenfunctions for the Floquet exponents of the operator corresponding to −B η ′ with |η ′ | ≪ 1. On the other hand, the Bloch transformed problem (1.12) is a multi-dimensional problem, i.e., a problem on Ω per , which requires approaches different to those in [3,4], e.g., we specify the eigenspace for the eigenvalue 0 of −B 0 and construct time periodic eigenfunctions of −B η ′ with |η ′ | ≪ 1 in a higher order Sobolev space, based on the energy methods in [9,13] and the argument to construct time periodic solutions in [16].
We also mention that dissipative systems on infinite layers and cylindrical domains often provide space-time periodic patterns (cf., [5,15]). The analysis of this paper is thus a preparatory study of the dynamics around space-time periodic patterns of the viscous compressible system (1.1)-(1.2). This paper is organized as follows. In Section 2, we transform the equations (1.1)-(1.2) into a non-dimensional form and introduce basic notation that is used throughout the paper. In Section 3, we first state the existence of a space-time periodic state and then state the main results of this paper. Section 4 is devoted to the proof of the main results. In Section 5, we give a proof of the existence of a space-time periodic state.

Preliminaries
In this section, we transform (1.1)-(1.2) into a non-dimensional form and introduce some function spaces and notations which are used throughout the paper.
We rewrite the problem into the non-dimensional form. We introduce the following non-dimensional variables: HereΩ per = Π n−1 j=1 T 2π α j × (0, d); and · H m (Ωper ) denotes the usual H m -norm overΩ per (whose definition is given below). Under this change of variables, the domainΩ is transformed into The equations (1.1)-(1.2) are rewritten as Here ν,ν, γ and S are non-dimensional parameters defined by We note that p ′ (1) = 1 and [G] 3,1,Ωper = 1.
Furthermore, due to the assumption (1.3), we have The boundary and initial conditions (1.5)-(1.6) are transformed into (2.4) We next introduce notation used throughout this paper. Let D be a domain. We denote by L p (D) (1 ≤ p ≤ ∞) the usual Lebesgue space on D and its norm is denoted by · L p (D) . Let m be a nonnegative integer. We simply write the set of all vector fields w = ⊤ (w 1 , · · · , w n ) on D as w j ∈ L p (D) (resp., H m (D)) and its norm is denoted by · L p (D) (resp., · H m (D) ).
We set where [k] is the largest integer smaller than or equal to k. The inner product of L 2 is defined as for f, g ∈ L 2 (Ω per ). Here, g denotes the complex conjugate of g. Moreover, the mean value of f = f (x) and g = g(x, t) over Ω per and Ω per × T 1 is written as respectively. We next introduce a weighted inner product: Here ρ p denotes the density of the space-time periodic state given in Proposition 3.1 below. By Proposition 3.1, we see that ρ p ≥ ρ on Ω per × T 1 for a positive constant ρ and that |ρ p (x, t) − 1| ≤ 1 2 and |p ′ (ρ p (x, t)) − 1| ≤ 1 2 for all (x, t) ∈ Ω per × T 1 . Therefore, u 1 , u 2 defines an inner product.
In terms of the Bogovskii operator B, we introduce the following inner product on L 2 * (Ω per ) × L 2 (Ω per ). For each t ∈ T 1 , we define ((u 1 , u 2 )) t by where δ is a positive constant. One can see that there exists a positive constant C such that if 0 < δ < 1 2Cγ , then ((·, ·)) t defines an inner product satisfying 1 2 We next introduce the Bloch transform. Let S (R n−1 ) be the Schwartz space on R n−1 . We define the Bloch transform T by for ϕ ∈ S (R n−1 ), whereφ denotes the Fourier transform of ϕ: The operators T and S have the following properties. See, e.g., [14,15] for the details.
(iii) S is the inverse operator of T .

Main Results
In this section, we state the main results of this paper. We first state the existence of the space-time periodic state of (2.1)-(2.3). We consider the time periodic problem for in Ω per under the boundary condition γ 2 , then the following assertions hold. There exist a space-time periodic solution where C is a positive constant independent of ν,ν, γ and S.
The proof of Proposition 3.1 is essentially the same as that given in [16]. Since we solve the time periodic problem in H 4 (Ω per ) × H 5 (Ω per ) and we need to know the dependence of the estimates on the parameters ν,ν and γ, we will give an outline of the proof of Proposition 3.1 in Section 5 below.
Our main result is concerned with the spectrum of the linearized solution operator U(t, s) around the space-time periodic solution u p = ⊤ (φ p , v p ).
As was mentioned in the introduction, we apply Bloch transform to (1.11). By Proposition 2.2, we then obtain (1.12) and consider the spectrum of −B η ′ to obtain the Floquet exponents of (1.12) for |η ′ | ≪ 1.
γ 2 , then the following assertions hold.
It then follows that are eigenfunctions for B η ′ and B * η ′ associated with eigenvalue λ η ′ ,0 and λ η ′ ,0 , respectively. Note that eigenfunctions for eigenvalues λ η ′ ,k are given by e 2πikt u η ′ and the same holds for the adjoint eigenfunctions.
We have the following estimates for the eigenfunctions for u η ′ and u * η ′ .
Theorem 3.4. Under the same assumptions of Theorem 3.3 the following estimates hold uniformly for |η ′ | ≤ r 0 and t ∈ T 1 : Theorems 3.3 and 3.4 will be proved in Section 4.

Proof of Theorems 3.3 and 3.4
In this section, we prove Theorems 3.3 and 3.4. To do so, we consider the resolvent problem Here We set We begin with investigating the spectral properties of B 0 . For this purpose, we first consider the unique solvability for the time periodic problem Proposition 4.1. There exists positive constants ν 0 , γ 0 , ε 0 and a such that γ 2 , then the following assertions hold true. For any F ∈ L 2 (T 1 ; where C is a positive constant independent of ν,ν, γ and S. To prove Proposition 4.1, we prepare the following lemma about the estimate of solution of the initial value problem for (4.2) under the initial condition Lemma 4.2. There exists positive constants ν 0 , γ 0 , ε 0 and a such that if (Ω per )), one can prove the existence of a solution to (4.2) and (4.4) with u 0 ∈ L 2 * (Ω per ) × H 1 0 (Ω per ) in a standard way by combining the method of characteristics and the parabolic theory.
We prove the estimate (4.5). We employ the energy method by Iooss-Padula [9]. We compute Re((∂ t u+L(t)u, u)) t = Re((F, u)) t . In a similar way to the proof of [7, Lemma 4.3], we see from Lemma 2.1 and Proposition 3.1 that there exist positive constants ν 0 , γ 0 , ε 0 and a such that γ 2 , then the following estimate holds: On the other hand, we have It then follows Multiplying this by e β0t and integrating the resulting inequality over [0, t], we have (4.6) We apply Lemma 4.3 below to the right hand side of (4.6) and obtain This completes the proof.
Therefore, we obtain This completes the proof.
We are in a position to prove Proposition 4.1.
Proof of Proposition 4.1. We denote by u ♯ the solution of (4.2) and (4.4) with u 0 = 0. We then see from Lemma 4.2 that u ♯ satisfies for t ≥ 0. Let m, n ∈ N with m > n. Since F is periodic in t of period 1, the function u ♯ (t + (m − n)) − u ♯ (t) is the solution of (4.2) and (4.4) with We set t = n in this inequality. It then follows from (4.7) that We thus obtain (4.8) We then see from the argument by Valli that the solution u of (4.2) and (4.4) with u 0 =ũ ♯ is a time periodic solution of (4.2). Furthermore, applying (4.5) and (4.8), we obtain This completes the proof.
The following proposition shows that 0 is an eigenvalue of −B 0 . We also give the estimates of an eigenfunction for the eigenvalue 0.
Since v p ∈ ∩ 2 j=0 C j (T 1 ; H 4−2j (Ω per )), one can prove the existence of solution u to (4.2) and (4.4) with u 0 ∈ H 2 * (Ω per ) × (H 2 (Ω per ) ∩ H 1 0 (Ω per )) in a standard way by combining the method of characteristics and the parabolic theory. The estimate of the solution u is obtained in a similar manner to the proof of Proposition 5.5 below. We here give an outline of the proof of the estimate.
We rewrite L(t) as 1+φp(t) (ν∆ +ν∇div) Here p (1) As in the proof of Proposition 5.7, applying Proposition 5.6 with m = 2 for HereẼ 2 (u) and D 2 (u) are the same functionals as those given in Propositions 5.7 and 5.3, respectively; andÑ 2 is a functional satisfying It then follows that there exist positive constants ν 0 , γ 0 , ε 0 and a such that if (4.14) Applying now the argument of the proof of Proposition 5.5 below, we have ds .
Here E 2 (u) is the same functional given in Proposition 5.3 below; and C E is the positive constant given in Lemma 5.12 below. This, together with Lemma 4.3, implies that ds .
The desired estimate follows from this inequality by applying Proposition 3.1 to the second term of the right-hand side. This completes the proof.
We are in a position to prove Proposition 4.4.
To prove that 0 is a simple eigenvalue of −B 0 , we prepare the following lemma.
Then the following assertions hold.
We now prove the simplicity of the eigenvalue of −B 0 . Let X 0 and X 1 be defined by X 0 = Π (0) X and X 1 = (I − Π (0) )X.
As for X 0 and X 1 , it holds the following assertions.
This implies that f = 0 and hence F ∈ X 1 . We thus obtain Ran(B 0 ) ⊂ X 1 .
We next prove X 1 ⊂ Ran(B 0 ). Let F = ⊤ (f, g) ∈ X 1 . We will show that there exists a unique solution u ∈ D(B 0 ) ∩ X 1 to  Hence, there exists a unique solution u = ⊤ (φ, w) ∈ D(B 0 ) ∩ X 1 to (4.15). This shows X 1 ⊂ Ran(B 0 ). Therefore X 1 = Ran(B 0 ). Let us show X 0 = Ker(B 0 ). We assume that u is the solution of B 0 u = 0. Decomposing u into u = u 0 + u 1 with u j ∈ X j for j = 0, 1, we have B 0 u j = 0. It follows from the previous argument that u 1 is a unique solution to B 0 u 1 = 0, and we see from (4.19) with F = 0 that u 1 = 0. Consequently, it holds that u = u 0 ∈ X 0 and Ker(B 0 ) ⊂ X 0 . Therefore, Ker(B 0 ) = X 0 . This completes the proof.
Remark 4.9. One can show that, for each k ∈ Z, 2πik is a simple eigenvalue of −B 0 .
Proposition 4.11. There exist positive constants ν 0 , γ 0 , ε 0 and a such that if We are now in a position to prove Theorem 3.3.
Proof of Theorem 3.3. We first observe that Let Σ be the set given in Theorem 3.3. We see from Proposition 4.11 that if λ ∈ Σ, then This, together with (4.26) and (4.27), implies that It then follows that there exists positive constant r 0 = r 0 (ν,ν, γ, ν 0 , γ 0 ) such that if |η ′ | ≤ r 0 , then We thus find that if |η ′ | ≤ r 0 , then Σ ⊂ ρ(−B η ′ ) and for λ ∈ Σ This proves the assertion (i). As for the assertion (ii), it suffices to show that if |η ′ | ≤ r 0 , then with some constants a j , a jk ∈ R and λ η ′ ,0 satisfies In view of Proposition 4.8, Proposition 4.11, (4.26) and (4.27), we can apply the analytic perturbation theory ( [12]) to see that the set consists of a simple eigenvalue, say λ η ′ ,0 , for sufficiently small η ′ , and that λ η ′ ,0 is expanded as We set u 1 is a solution of In fact, there exists a solution u 1 ) to (4.28), and λ (2) jk is written as We thus estimate u (k) 1 to prove the estimate (3.6).
and |η ′ | ≤ r 0 . Then the following estimate holds: (4.29) Proof. As in Proposition 4.1, we have , it follows from Propositions 3.1 and 4.4 that and we have the desired estimate.
To derive the estimate (3.6), we next introduceũ , which is a unique stationary solution of the Stokes system where We use the following lemma ( [11,Theorem 4.7]).
Then there exists a constant κ 0 > 0 independent of ν,ν and γ such that for all η ′ ∈ R n−1 .
By using Lemma 4.13, we have the following estimate.
and |η ′ | ≤ r 0 , then the following estimate holds: Proof. We consider It follows from the estimate for the Stokes problem (see, e.g., [8]) that ∂ xφ This completes the proof.
We next prove Theorem 3.4. To do so, we establish the estimate for (λ + B 0 ) −1 F in a higher order Sobolev space.
We are now in a position to prove Theorem 3.4.
Consequently, we have Similarly, we obtain This completes the proof.

Proof of Proposition 3.1
In this section we give a proof of Proposition 3.1. We set The system (2.1)-(2.2) is then written as We consider (5.1)-(5.2) on Ω per under the conditions Under some smallness assumption on the size of S, we have the following result on the existence of a time periodic solution of (5.1)-(5.4).
where C is independent of ν,ν, γ and S.

Global solution
In this subsection, we prove the global existence of solution to (5.1)-(5.5) with energy estimate that is stated in the following proposition.
Proposition 5.3. Let u 0 and G satisfy (5.6) and the compatibility conditions (5.7) and (5.8). There exist positive constants ν 0 , γ 0 , a, C 0 , C 1 and C E such that , then there exists a unique global solution u to where E m (u) and D m (u) (m = 2, 4) are the quantities satisfying the following inequalities with some positive constant C : As in [6], we can prove Proposition 5.3 by combining local existence and the a priori estimates. The local existence is proved by applying the local solvability result in [10,16]. In fact, we can show that the following assertion. The global existence is proved by combining Proposition 5.4 and the following a priori estimates.
Proposition 5.5. Let T be a positive number and assume that u is a solution of (5.1)- (5.5) ). There exist positive constants ν 0 , γ 0 , a, C 0 , C 1 and C E independent of T , ν,ν, γ and S such that the following assertion holds. If for t ∈ [0, T ].
Proposition 5.5 is proved by the energy estimate and nonlinear estimates. In what follows, we denote We also denoteφ := ∂ t φ + w · ∇φ. It then follows thaṫ We have the following basic estimates in a similar manner to [7, Section 5].
Combining the basic estimates given in Proposition 5.6, we have the following H m -energy estimate (m = 2, 4) in a similar argument to that in [7, Section 5]. for m = 2, 4. Here where C is independent of ν,ν, γ and S.
We next estimate N 4 (u).
Proposition 5.8 is an immediate consequence of the following Proposition 5.9.
Here C is some positive constant independent of ν,ν, γ and S.
To prove Proposition 5.9, we use the following Sobolev inequalities.
In fact, Proposition 5.13 is an immediate consequence of the following Proposition 5.14. Here we also use the assumption (1.3) as in the proof of Proposition 5.9.
Proposition 5.14 can be proved in a similar manner to Proposition 5.9. We now establish the H 2 -energy estimate forũ.