Abstract
A classical problem in fluid mechanics concerns the stability and instability of different hydrodynamic patterns in various physical settings, particularly in the high Reynolds number limit of laminar flows with boundary layers. Despite extensive studies when the fluid is governed by incompressible Navier-Stokes equations, there are very few mathematical results on the compressible fluid. This paper aims to introduce a new approach to studying the compressible Navier–Stokes equations in the subsonic and high Reynolds number regime, where a subtle quasi-compressible and Stokes iteration is developed. As a byproduct, we show the spectral instability of subsonic boundary layers.
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1 Introduction
One of the most fundamental problems in fluid mechanics is understanding the physical mechanisms that lead to the stability or instability of hydrodynamic patterns. Most laminar flows are unstable at the high Reynolds number, and small perturbations will eventually cascade into turbulence. Under many circumstances, the early stage of such transition is the instability induced by viscous disturbance wave, now called Tollmien–Schlichting or T–S wave. For incompressible flow, the physical description of T–S waves can be found, for instance, in the pioneering work by Heisenberg, C.C. Lin, Tollmien and Schlichting, cf. [5, 15, 21, 33], and Wasow [38] established a formal construction of them. Until recently, the most rigorous mathematical justification was given by Grenier–Guo–Nguyen [7].
From the physical point of view, it is important to study the compressible flow with boundary layers that arises from, for instance, the flow near the airfoil. The theoretical investigation can be traced back to Lees–Lin [19], in which Rayleigh’s criterion for inviscid flow was extended to the compressible subsonic flow. Later on, the asymptotic expansion used in [19] near the critical layer was rigorously justified by Morawitz [29]. For more investigation from the physical perspective, we refer readers to [5, 20, 21, 33] and the references therein. It is worth noting that the instability mechanisms studied in the literature are inviscid in nature, while the viscous transition mechanisms still need to be investigated. This paper aims to fill this gap by rigorously justifying the presence of T–S waves in the compressible boundary layer.
1.1 Problem and Main Result
Consider the 2D compressible Navier–Stokes equations for isentropic flow in half-space \(\{(x,y)\mid x\in \mathbb {T},y\in \mathbb {R}_+\}\)
In the above equations \(\rho \), \({U}=(u,v)\) and \(P(\rho )\) stand for the density, velocity field and pressure of the fluid. The vector field \({F}\) is a given external force. The constants \(\mu >0, \lambda \geqq 0\) are rescaled shear and bulk viscosity, respectively, while \(0<\varepsilon \ll 1\) is a small parameter which is proportional to the reciprocal of the Reynolds number. For simplicity and without loss of generality, the constant \(\mu \) is set to 1 throughout paper.
A laminar boundary layer flow is defined by
It is a steady solution to (1.1) with external force \({F}=(-\partial _Y^2U_s,0)\).
In this work, to understand the (in)stability properties of the above boundary layer profile, we study the compressible Navier–Stokes system linearized around \((\rho _s, {\textbf{U}}_s)\). Denote the Mach number by \({\mathcalligra {m}}:=\frac{1}{\sqrt{P'(1)}}\). The linearization gives
To study (1.2), we use the rescaled variable
and then look for solution to the linearized compressible Navier–Stokes system in the following form:
Plugging this ansatz into (1.2) yields to the system (we replace \((\tilde{\rho },\tilde{u},\tilde{v})\) by \((\rho ,u,v)\) for simplicity of notation)
with no-slip boundary conditions
In (1.3), \(\Delta _\alpha =(\partial _Y^2-\alpha ^2)\) and \(\text {div}_\alpha (u,v)=i\alpha u+\partial _Y v\) denote the Fourier modes of Laplacian and divergence operators respectively. For convenience, we denote by \(\mathcal {L}(\rho ,u,v)\) the linear operator (1.3). If for some \(c\in \mathbb {C}\) with positive imaginary part \(\text {Im}c>0\) and wave number \(\alpha \in \mathbb {R}\), the boundary value problem (1.3) with (1.4) has a non-trivial solution, then the boundary layer profile \((\rho _s,{\textbf{U}}_s)\) is spectral unstable. Otherwise, thus is spectral stable.
In the analysis, we focus on a class of laminar boundary layer flows that satisfy the following assumptions:
-
\(U_s\in C^3(\overline{\mathbb {R}_+})\) and satisfies
$$\begin{aligned} U_s(0)=0,~U_s(Y)>0,~\lim _{Y\rightarrow +\infty }U_s(Y)=1,~\text {and } U_s'(0)=1. \end{aligned}$$(1.5) -
There exist positive constants \(s_0\), \(s_1\) and \(s_2\), such that
$$\begin{aligned} s_1e^{-s_0Y}\leqq \partial _YU_s(Y)\leqq s_2e^{-s_0Y}, ~\forall Y\geqq 0. \end{aligned}$$(1.6) -
The boundary layer flow is assumed to be uniformly subsonic, that is \({\mathcalligra {m}}\in (0,1)\). Moreover, there exists a constant \(\sigma _1=\sigma _1({\mathcalligra {m}})>0\) such that for all \(Y\geqq 0\), it holds
$$\begin{aligned} H(Y)\overset{\hbox {{def}}}{=}\frac{-\partial _Y^2U_s(1-{\mathcalligra {m}}^2U_s^2)-2{\mathcalligra {m}}^2U_s|\partial _YU_s|^2}{|\partial _Y^2U_s|+|\partial _YU_s|^2}\geqq \sigma _1. \end{aligned}$$(1.7) -
There exists a constant \(\sigma _2>0\) such that for any \(Y\geqq 0,\) it holds
$$\begin{aligned} \left| \frac{\partial _Y^3U_s}{\partial _Y^2U_s}\right| +\frac{|\partial _Y^2U_s|}{\partial _YU_s}+\frac{1-U_s}{\partial _YU_s}\leqq \sigma _2. \end{aligned}$$(1.8)
Note that this class of profiles include the exponential profile \(U_s(Y)=1-e^{-Y}\) with \(\sigma _1({\mathcalligra {m}})=\frac{1-{\mathcalligra {m}}^2}{2}\) and \(s_0=s_1=s_2=\sigma _2=1.\) Moreover, from (1.7), we have
Hence, the profiles in this class are strictly concave.
The main result in the paper can be stated as follows:
Theorem 1.1
Let the Mach number \({\mathcalligra {m}}\in (0,\frac{1}{\sqrt{3}})\). There are positive constants \(K_0>1\) and \(\varepsilon _0\in (0,1)\), such that for any \(\varepsilon \in (0,\varepsilon _0)\) and any wave number \(\alpha =K\varepsilon ^{\frac{1}{8}}\) with \(K\geqq K_0\), there exists \(c_\varepsilon \in \mathbb {C}\) with \(\alpha \text {Im}c_\varepsilon \approx \varepsilon ^{\frac{1}{4}}\), such that the linearized compressible Navier–Stokes system (1.2) admits a solution \((\rho ,u,v)\) in the form of
Here the profile \((\tilde{\rho },\tilde{u},\tilde{v}) \in H^1(\mathbb {R}_+)\times \left( H^2(\mathbb {R}_+)\cap H^1_0(\mathbb {R}_+) \right) ^2\) and satisfies the eigenvalue problem (1.3).
Remark 1.2
In what follows, we present several remarks on Theorem 1.1.
-
(a)
As shown in the proof, the bounds on the solution depend on some negative power of \(1-{\mathcalligra {m}}\) that are uniform when \({\mathcalligra {m}}\) is in a compact set of \( [0,\frac{1}{\sqrt{3}})\). Therefore, by taking the vanishing Mach number limit, we have the Tollmien–Schlichting wave solution for the incompressible flow that was analyzed in [7] by Grenier–Guo–Nguyen.
-
(b)
The restriction of Mach number \({\mathcalligra {m}}\in (0,\frac{1}{\sqrt{3}})\) should be technical and it is only used in obtaining a positive lower bound of \({\mathcalligra {w}}_0-U_s{\mathcalligra {w}}_1\) in the proof of Lemma 3.4. However, the main purpose of this paper is to introduce a new approach to the stability and instability analysis on compressible fluid with strong boundary layers. Therefore, we will not pursue how the Mach number can be close to one in this work.
-
(c)
The growing modes are supported in the frequency regime \(n=\frac{\alpha }{\sqrt{\varepsilon }}=K\varepsilon ^{-\frac{3}{8}}\) and grow like \(\exp ( K^{-\frac{2}{3}}n^{\frac{2}{3}}t)\) with \(K\gg 1\). These parameters indicate a spectral instability in Gevrey space with index equal to \(\frac{3}{2}\). How to justify the stability in Gevrey \(\frac{3}{2}\) class for the linear problem (1.2) that shares the same index as in the incompressible case studied in [12] will be discussed in our future work.
-
(d)
By constructing a suitable approximate growing mode, we can also obtain the same results as in Theorem 1.1 for the wave number \(\alpha =C\varepsilon ^{\beta }\), with \(C>0\) and \(\beta \in (1/12,1/8)\). Here we will not give the detailed analysis in this regime because we focus on the most unstable mode when \(\beta =1/8\). We refer to [22] on the incompressible MHD system for the related discussions and analysis.
1.2 Relevant Literature
Since this paper is motivated by the study of inviscid limit and Prandtl boundary layer expansion for incompressible Navier–Stokes equations, we briefly summarize some related works in this direction. Indeed, there are two main destabilizing mechanisms in the boundary layer that makes inviscid limit problem very challenging. The first one is induced by the inflexion points inside the boundary layer profile. In this case, the linearized Navier–Stokes system exhibits a strong ill-posedness below the analyticity regularity, cf. [6, 8]. Therefore, the results of inviscid limit can be obtained only when the data is analytic at least near the boundary, cf. [18, 25, 30, 32, 34]. The second one is induced by the small disturbance around a monotone and concave boundary layer profile, called Tollmien–Schlichting instability that has been extensively studied in physical literatures and was justified rigorously in Grenier–Guo–Nguyen [7] by constructing a growing mode in Gevrey 3/2 space to linearized incompressible Navier–Stokes equations. The main idea in [7] is to use the stream function and vorticity, that is, the Orr–Sommerfeld (OS) formulation, and then to solve it via an iteration based on Rayleigh and Airy equation that can be viewed as the inviscid and viscous approximation to the original OS equation. We also refer to Grenier-Nguyen [9] for a result of nonlinear instability for small data that depend on viscosity coefficient. On the other hand, this instability result was complemented by the work of Gérard-Varet-Maekawa-Masmoudi [12, 13] that establishes the Gevrey stability of Euler plus Prandtl expansion with critical Gevrey index 3/2; see also [2] for a result in \(L^\infty \)-setting. Most recently, the formation of boundary layer is studied by Maekawa [26] using the Rayleigh profile. For the Sobolev data, the boundary layer expansion is only valid under certain symmetry assumptions or for steady flows, cf. [11, 14, 16, 24, 28] and the references therein. Finally, we refer to [3, 4, 10] for the instability analysis of boundary layer profile in different settings.
For compressible flow, even though there are many interesting results on the Navier–Stokes equations at high Reynolds number in different settings, cf. [1, 23, 31, 35,36,37, 39, 40] and the references therein, to our knowledge, the stability/instability properties of strong boundary layer for compressible Navier–Stokes equations have not yet been investigated. Compared to the incompressible Navier–Stokes equations, the major difficulty comes from the fact that the Orr–Sommerfeld formulation is no longer available for the compressible case. As a result, Rayleigh–Airy iteration approach used in [7] can not be applied, either the approaches used in [12, 13]. Therefore, the novelty of this paper is to introduce a new iteration approach to study compressible flow in the subsonic and high Reynolds number regime. We believe that the analytic techniques developed in this work can be used in other related problems for subsonic flows.
In the next subsection, we present the strategy of the proof for better understanding of the detailed analysis in the follow sections.
1.3 Strategy of Proof
The instability analysis is based on several steps.
Step 1. Construction of the approximate growing mode. Similar as in the incompressible case [7], the T–S instability is driven by the interaction of inviscid and viscous perturbations. Set the approximate growing mode (its precise definition will be given in (2.32)) \({\Xi }_{\text {app}}=(\rho _{\text {app}},u_{\text {app}},v_{\text {app}})\) as
Here the slow mode \((\rho _{\text {app}}^s,u^s_{\text {app}},v_{\text {app}}^s)\) is an approximate solution to the inviscid system, the fast mode \((\rho _{\text {app}}^f,u^f_{\text {app}},v_{\text {app}}^f)\) is an approximate solution to the full system (1.3) which exhibits viscous boundary layer structure near \(Y=0\); see (2.19) and (2.30) for the precise definition of slow and fast modes respectively. Note that the approximate solutions defined in (1.11) have zero normal velocity at the boundary, that is \(v_{\text {app}}(0;c)\equiv 0.\) Then, to recover the no-slip boundary condition, inspired by [3, 4], we analyze the zero point of \(\mathcal {F}_{\text {app}}(c)\overset{\hbox {{def}}}{=}u_{\text {app}}^s(0;c)-\frac{v^s_{\text {app}}(0;c)}{v_{\text {app}}^f(0;c)}u_{\text {app}}^f(0;c)\) by applying Rouché’s Theorem. Precisely, we study the equation \(\mathcal {F}_{\text {app}}(c)=0\) in a family of \(\varepsilon \)-dependent domains \(D_0\) (see (2.34)). Then, by Rouché’s Theorem, we can show that \(\mathcal {F}_{\text {app}}(c)\) has the same number of zero points as a linear function \(\mathcal {F}_{\text {ref}}(c)\) defined by (2.41). In addition, we prove that \(|\mathcal {F}_{\text {app}}(c)|\) has a strictly positive lower bound on \(\partial D_0\).
Step 2. Stability of the approximate growing mode. Since the approximate solution \({\Xi }_{\text {app}}\) exhibits the instability already, this step is to show the existence of an exact solution near \({\Xi }_{\text {app}}\). This is the most difficult and key step. For this, we study the resolvent problem
with a given inhomogeneous source term \((f_u,f_v)\). Here, we emphasize that in (1.12) only normal velocity field v is prescribed on the boundary. Even though we relax the boundary constraint on u in (1.12), it is still difficult for existence because the presence of stretching term \(v\partial _YU_s\). This difficulty is overcome by the following three ingredients:
-
Quasi-compressible approximation. When the inhomogeneous source \((f_u,f_v)\in H^1(\mathbb {R}_+)^2\), we introduce the following quasi-compressible system
$$\begin{aligned} \left\{ \begin{aligned}&i\alpha (U_s-c)\varrho +i\alpha \mathfrak {u}+\partial _Y \mathfrak {v}=0,\\&\sqrt{\varepsilon }\Delta _\alpha \left[ \mathfrak {u}+(U_s-c)\varrho \right] -i\alpha (U_s-c)\mathfrak {u}-\mathfrak {v}\partial _YU_s-i\alpha {\mathcalligra {m}}^{-2}\varrho =f_u,\\&\sqrt{\varepsilon }\Delta _\alpha \mathfrak {v}-i\alpha (U_s-c)\mathfrak {v}-{\mathcalligra {m}}^{-2}\partial _Y\varrho =f_v,\\&\mathfrak {v}|_{Y=0}=0, \end{aligned} \right. \nonumber \\ \end{aligned}$$(1.13)which will be denoted by \(L_Q(\varrho ,\mathfrak {u},\mathfrak {v})=(0,f_u,f_v)\). Note that the inviscid part of the original linear operator \(\mathcal {L}\) is kept in \(L_Q\), while the diffusion terms are modified to be divergence free. It turns out that for Mach number \({\mathcalligra {m}}\in (0,1)\), the system (1.13) exhibits a similar stream function-vorticity structure as the incompressible Navier–Stokes equations. In fact, if we introduce the “effective stream function” \(\Psi \) associated to the modified velocity variable \((\mathfrak {u}+(U_s-c)\varrho ,\mathfrak {v})\) which satisfies
$$\begin{aligned} \partial _Y\Psi =\mathfrak {u}+(U_s-c)\varrho ,~ -i\alpha \Psi =\mathfrak {v},~\Psi |_{Y=0}=0, \end{aligned}$$then (1.13) can be reformulated in terms of \(\Psi \) as
$$\begin{aligned} \begin{aligned} \text {OS}_{\text {CNS}}(\Psi )&:=\frac{i}{n}\Lambda (\Delta _\alpha \Psi )+(U_s-c)\Lambda (\Psi )-\partial _Y(A^{-1}\partial _YU_s)\Psi \\&=f_v-\frac{1}{i\alpha }\partial _Y(A^{-1}f_u), \end{aligned} \end{aligned}$$(1.14)where \(n=\alpha /\sqrt{\varepsilon }\), \(A(Y)=1-{\mathcalligra {m}}^2(U_s-c)^2\) and \(\Lambda (\Psi )=\partial _Y(A^{-1}\partial _Y\Psi )-\alpha ^2\Psi \). Note that A(Y) is invertible at least for \({\mathcalligra {m}}\in (0,1)\) and c near the origin. When the Mach number \({\mathcalligra {m}}=0,\) we have \(\Lambda = \Delta _\alpha \) and \(A(Y)\equiv 1\). Thus \(\text {OS}_{\text {CNS}}\) in this case reduces to the classical Orr–Sommerfeld operator for incompressible Navier–Stokes system. Therefore, \(\text {OS}_{\text {CNS}}\) can be viewed as the compressible counterpart of the Orr–Sommerfeld equation, which to our best knowledge is for the first time derived in the literatures. This formulation motivates the notion “quasi-compressible” approximation. We solve (1.14) with artificial boundary conditions \(\Psi |_{Y=0}=\Lambda (\Psi )|_{Y=0}=0\) that allows us to obtain the weighted estimates on \(\Lambda (\Psi )\). One can see that when \({\mathcalligra {m}}=0\), these boundary conditions are simply the perfect-slip boundary conditions used in [2, 11, 13] for the study of incompressible Navier–Stokes equations. However, for the problem considered in this paper, the multiplier \({\mathcalligra {w}}(Y)=-\partial _Y(A^{-1}\partial _YU_s)\) is not real. Therefore both its leading and first order terms \({\mathcalligra {w}}_0\), \({\mathcalligra {w}}_1\) (see Lemma 3.3 for the precise definitions) play a role in the energy estimates. For the bound estimations, we essentially use \({\mathcalligra {m}}\in (0,\frac{1}{\sqrt{3}})\) and the new structural condition (1.7) of the profile in order to show that the function \({\mathcalligra {w}}_0-U_s{\mathcalligra {w}}_1\) has a positive lower bound, cf. (3.29) in the proof of Lemma 3.4. After we obtain \(\Psi \) which solves (1.14), the solution \((\varrho ,\mathfrak {u},\mathfrak {v})\) to (1.13) can be recovered in terms of \(\Psi \), cf. (3.11) and (3.12). Here we would like to mention that (1.13) has a regularizing effect on density. That is, formally by applying \(\text {div}_\alpha \) to the momentum equation in (1.13) and by noting that the diffusion term lies in the kernel of \(\text {div}_\alpha \), we have \(\Delta _\alpha \varrho \in L^2(\mathbb {R}_+)\). This reveals an elliptic structure for the linearized compressible Navier–Stokes equations in the subsonic regime.
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Stokes approximation. Note that \((\varrho ,\mathfrak {u},\mathfrak {v})\) is not an exact solution to (1.12) and its error is
$$\begin{aligned}&E_Q(\varrho ,\mathfrak {u},\mathfrak {v})\overset{\hbox {{def}}}{=}\mathcal {L}(\varrho ,\mathfrak {u},\mathfrak {v})-L_Q(\varrho ,\mathfrak {u},\mathfrak {v})\nonumber \\&\qquad \quad =\left( 0, -\sqrt{\varepsilon }\Delta _\alpha \left[ (U_s-c)\varrho \right] +\lambda \sqrt{\varepsilon }i\alpha \text {div}_\alpha (\mathfrak {u},\mathfrak {v}),\right. \nonumber \\&\qquad \qquad \left. \sqrt{\varepsilon }\partial _Y\text {div}_\alpha (\mathfrak {u},\mathfrak {v})\right) . \end{aligned}$$(1.15)This error term involves a small factor of \(\sqrt{\varepsilon }\) but lies only in \(L^2(\mathbb {R}_+)\). This fact prevents us from using the standard fixed point argument to solve (1.12). To recover the regularity, we introduce another operator \(L_S\) that we call Stokes approximation. It is obtained from \(\mathcal {L}\) by removing the stretching term, that is,
$$\begin{aligned} L_S(\xi ,\phi ,\psi )\overset{\hbox {{def}}}{=}\mathcal {L}(\xi ,\phi ,\psi )+(0,\psi \partial _YU_s,0). \end{aligned}$$To eliminate the error \(E_Q(\varrho ,\mathfrak {u},\mathfrak {v})\), we then take \((\xi ,\phi ,\psi )\) as the solution to
$$\begin{aligned} L_S(\xi ,\phi ,\psi )=-E_Q(\varrho ,\mathfrak {u},\mathfrak {v}),~ \partial _Y\phi |_{Y=0}=\psi |_{Y=0}=0. \end{aligned}$$By using the energy approach in the same spirit as Matsumura-Nishida [27] and Kawashima [17], we are able to show \((\xi ,\phi ,\psi )\) is in \(H^1(\mathbb {R}_+)\times H^2(\mathbb {R}_+)^2\). Thus, the error term \(E_S(\xi ,\phi ,\psi ):=(0,\psi \partial _YU_s,0)\) is in the weighted space \(H^2_w(\mathbb {R}_+)\) so that we can treat it as source term of \(L_Q\). Therefore, we can iterate the above two approximations.
-
Quasi-compressible-Stokes iteration. Recall that we have the following two decompositions of the linear operator \(\mathcal {L}:\)
$$\begin{aligned} \mathcal {L}=L_Q+E_Q=L_S+E_S. \end{aligned}$$The solvability to (1.12) can be justified via an iteration scheme that is illustrated as follows. Assume that at the N-th step we have an approximate solution in the form of \(\sum _{j=0}^N{\Xi }_j\) which satisfies
$$\begin{aligned} \mathcal {L}\left( \sum _{j=0}^N{\Xi }_j\right) =(0,f_u,f_v)+{\mathcal {E}}_N. \end{aligned}$$Here \({\mathcal {E}}_N\) is an error term at this step. Provided that \({\mathcal {E}}_N\) is smooth enough and has zero value at its first component, we can introduce a corrector
$$\begin{aligned} {\Xi }_{N+1}=-L_Q^{-1}( {\mathcal {E}}_N )+L_S^{-1}\circ E_Q\circ L_Q^{-1}({\mathcal {E}}_N ), \end{aligned}$$where \(L_Q^{-1}\) and \(L_S^{-1}\) denote respectively the solution operators to quasi-compressible and Stokes approximate systems. The approximate solution at the \(N+1\)-step is therefore defined by \(\sum _{j=0}^{N+1}{\Xi }_j\). Then we have
$$\begin{aligned} \begin{aligned} \mathcal {L}\left( \sum _{j=1}^{N+1} {\Xi }_j\right)&=(0,f_u,f_v)+{\mathcal {E}}_{N+1}\\&:=(0,f_u,f_v)+E_S\circ L_S^{-1}\circ E_Q \circ L_Q^{-1}({\mathcal {E}}_N). \end{aligned} \end{aligned}$$A combination of the smallness of \(E_Q\), the regularizing effect of \(L_S^{-1}\) and the strong decay property of \(E_S\) yields the contraction in \(H^1_w(\mathbb {R}_+)\) of truncated error operator \(E_S\circ L_S^{-1}\circ E_Q \circ L_Q^{-1}\) so that the convergence of series \(\sum _{j=1}^\infty {\Xi }_j\) in \(H^1(\mathbb {R}_+)\times H^2(\mathbb {R}_+)^2\) follows, cf. the proof of Proposition 3.1.
Step. 3. Recovery of the no-slip boundary condition. We look for solutions to the original system (1.3) with \(v|_{Y=0}=0\) in the form of \((\rho ,u,v)=(\rho _{\text {app}},u_{\text {app}},v_{\text {app}})-(\rho _{R},u_{R},v_{R})\). Here the remainder \((\rho _R,u_R,v_R)\) solves \(\mathcal {L}(\rho _R,u_R,v_R)\!=\!E, v_R|_{Y=0}\!=\!0\), where E is the error due to the approximation \((\rho _{\text {app}},u_{\text {app}},v_{\text {app}})\). In this step, we need to decompose the error into regular and smallness parts as in (2.44). The remainder is divided accordingly into \({\Xi }_R={\Xi }_{\text {re}}+{\Xi }_{\text {sm}}.\) The reason for such decomposition is that the regular part \(E_{\text {re}}\) coming from the rough approximation to Rayleigh equation has a worse bound than the smallness part so that we can compensate some extra order of \(\varepsilon \) by the favorable bounds of \({\Xi }_{\text {re}}\) due to strong decay and \(H^1\)-regularity of \(E_{\text {re}}\), cf. Proposition 3.1. Eventually, we can prove that \(|u_R(0;c)|\leqq C\varepsilon ^{\frac{1}{16}}\) on \(\partial D_0\), which is smaller than \(|u_{\text {app}}(0;c)|\). Then we conclude Theorem 1.1 by Rouché’s Theorem.
The rest of the paper is organized as follows: in the next section, we will construct the approximate growing mode. In Sect. 3, we will show the solvability of the linearized system (1.3) with zero normal velocity condition in order to resolve the remainder due to the approximation. The proof is divided into several steps. Firstly, two approximate systems, that is, Quasi-compressible and Stokes approximations will be introduced in Sects. 3.1 and 3.2 respectively. Based on these two systems, the iteration scheme will be analyzed in Sect. 3.3. The proof of Theorem 1.1 will be given in the final section. In the Appendix, we will give the proof of the invertibility of operator \(\Lambda \) that is used in the construction of solution to the equation (3.16).
In the paper, for any \(z\in \mathbb {C}\setminus \mathbb {R}_-\), we take the principle analytic branch of \(\log z\) and \(z^k, k\in (0,1)\), that is
Notations: Throughout the paper, C denotes a generic positive constant and \(C_a\) means that the generic constant depending on a. These constants may vary from line to line. \(A\lesssim B\) and \(A=O(1)B\) mean that there exists a generic constant C such that \(A\leqq CB\). And \(A\lesssim _a B\) implies that the constant C depends on a. Similar definitions hold for \(A\gtrsim B\) and \(A\gtrsim _a B\). Moreover, we use notation \(A\approx B\) if \(A\lesssim B\) and \(A\gtrsim B\). \(\Vert \cdot \Vert _{L^2}\) and \(\Vert \cdot \Vert _{L^\infty }\) denote the standard \(L^2(\mathbb {R}_+)\) and \(L^\infty (\mathbb {R}_+)\) norms respectively. For any \(\eta >0\), \(L^\infty _\eta (\mathbb {R}_+)\) denotes the weighted Lebesgue space with the norm \(\Vert f\Vert _{L^\infty _\eta }\triangleq \sup _{Y\in \mathbb {R}_+}\left| e^{\eta Y}f(Y)\right| \). And the weighted Sobolev space \(W^{k,\infty }_\eta (\mathbb {R}_+)\) \((k\in \mathbb {N})\) has the norm \(\Vert f\Vert _{W^{k,\infty }_\eta }=\sum _{j\leqq k}\Vert \partial _Y^j f\Vert _{L^\infty _\eta }\).
2 Approximate Growing Mode
In the following three subsections, we will construct the approximate growing mode that satisfies the no-slip boundary condition. Similar to the incompressible Navier–Stokes equations, this is based on the superposition of the slow mode and the fast mode that represent the interaction of the inviscid and viscous effects near the boundary.
2.1 Slow Mode
In this subsection, we will construct the slow mode to capture the inviscid behavior. For this, we consider the following system denoted by \(\mathcal {I}(\rho ,u,v)={0}\):
By introducing a new function \(\Phi =\frac{i}{\alpha }v\), from (2.1)\(_1\), we have
Substituting this into (2.1)\(_2\) yields
where
Note that for the uniformly subsonic boundary layer, that is \({\mathcalligra {m}}\in (0,1)\), when \(|c|\ll 1\), A(Y) is invertible so that we can represent \(\rho \) in terms of \(\Phi \) by
Plugging (2.4) into (2.1)\(_3\), we derive the following equation for \(\Phi \), which can be viewed as an analogy of the classical Rayleigh equation in the compressible setting:
We remark that the equation (2.5) was firstly derived by Lees-Lin in [19] for the study of stability of shear flow in inviscid fluid. Thus (2.5) is sometimes referred to as Lees-Lin equation.
The slow mode will be constructed based on an approximate solution to (2.5). Since (2.5) has similar structure of the Rayleigh equation, the construction is similar as [7, 22] for incompressible flow. In what follows we sketch the key steps to make the paper to be self-contained.
Starting from \(\alpha =0\), the equation (2.5) admits following two independent solutions
For \(\alpha >0\), to capture the decay property of the solution, we set
Then we define
Direct computation yields the following error terms:
To have a better approximate solution for (2.5) up to \(O(\alpha ^2)\), the following approximate Green’s function is needed:
Then we define a corrector
and set
Hence, by (2.8) and (2.9), we have
In summary, \(\Phi _{\text {app}}^s\) is the slow mode with properties given in the following lemma:
Lemma 2.1
Let the Mach number \({\mathcalligra {m}}\in (0,1).\) Then for each \(Y\geqq 0\), \(\Phi _{\text {app}}^s(Y;c)\) is holomophic in the upper-half complex plane \(\{c\in \mathbb {C}\mid \text {Im}c>0\}\). Moreover, there exists \(\gamma _1\in (0,1)\), such that if \(\text {Im}c>0\) and \(|c|<\gamma _1\), the boundary values of \(\Phi _{\text {app}}^s\) have the following expansions:
Proof
Since \(\text {Im}c>0\), \(U_s(Y)-c\ne 0, ~\forall Y\geqq 0\). The analyticity of \(\Phi _{\text {app}}^s\) follows from the explicit formula (2.7), (2.10) and (2.11). Now we derive the boundary values \(\Phi _{\text {app}}^s(0;c) \) and \(\partial _Y\Phi _{\text {app}}^s(0;c).\)
Firstly, note that
Then it holds that
Then by using (2.11) and the fact that \(\beta =\alpha [(1-{\mathcalligra {m}}^2)^{\frac{1}{2}}+O(1)|c|]\), one has
To estimate the boundary value \(\partial _Y\Phi _{\text {app}}^s(0;c)\), differentiating (2.15) yields that
Similar to (2.16), by using \(U_s'(0)=1\), we obtain
Here we have used (2.16) in the last identity. Consequently, it holds that
Finally, we have \(\varphi _-(0)=-1+O(1)|c\log \text {Im}c|,\) cf. Lemma 3.1 in [22]. Then by substituting this into (2.17) and (2.18), we obtain (2.13) and (2.14). The proof of Lemma 2.1 is completed. \(\quad \square \)
With \(\Phi ^s_{\text {app}}\), we can define the slow mode of fluid quantities \({\Xi }_{\text {app}}^s\overset{\hbox {{def}}}{=}(\rho _{\text {app}}^s,u_{\text {app}}^s, \)\( v_{\text {app}}^s)\) by using (2.2) and (2.4) as follows:
One can directly check that \({\Xi }_{\text {app}}^s\) satisfies
where the error \(\text {Ray}_{\text {CNS}}(\Phi _{\text {app}}^s)\) is given in (2.12). Therefore, \({\Xi }_{\text {app}}^s\) is an approximate solution to the inviscid equation (2.1) up to \(O(\alpha ^2)\).
2.2 Fast Mode
To capture the viscous effect of (1.3) in the boundary layer, we need to construct a boundary sublayer corresponding to the fast mode in the approximate solution. Let \(z\overset{\hbox {{def}}}{=}\delta ^{-1}Y\). Here \(0<\delta \ll 1\) is the scale of boundary sublayer which will be determined later. Now we scale the density and velocity fields in the sublayer by setting
This leads to the following rescaled system associated to (1.3):
Here the constant is \(n\overset{\hbox {{def}}}{=}\frac{\alpha }{\sqrt{\varepsilon }}\), which is the rescaled frequency. Recalling \(U_s'(0)=1,\) we can rewrite \(U_s(Y)-c\) as
where \(z_0\overset{\hbox {{def}}}{=}-\delta ^{-1}c\), and
In view of (2.22)–(2.26), it is natural to set
so that \(in\delta ^3=1.\) Formally, we have the expansion
Inserting this expansion into (2.22)–(2.24) and taking the leading order, we can derive the following system for \((\mathfrak {p}_0,\mathcal {U}_0,\mathcal {V}_0)\)
where the variable z lies in the segment \(e^{\frac{1}{6}\pi i}\mathbb {R}_+\). From (2.27), we observe that the leading order terms of the density and divergence of velocity field vanish in the sublayer. We also require \((\mathcal {U}_0,\mathcal {V}_0)\) to concentrate near the boundary, that is,
Differentiating (2.28), by (2.27), we obtain
Therefore, from (2.27) and (2.29) we have
Here \(\text {Ai}(1,z)\) and \(\text {Ai}(2,z)\) are respectively the first and second order primitives of the classical Airy function \(\text {Ai}(z)\) which is the solution to Airy equation
\(\text {Ai}(2,z),\) \(\text {Ai}(1,z)\) and \(\text {Ai}(z)\) all vanish at infinity along \(e^{\frac{1}{6}\pi i}\mathbb {R}_+\). They satisfy the relations \(\partial _z\text {Ai}(k,z)=\text {Ai}(k-1,z)\), \(k=1,2,\) where \(\text {Ai}(0,z)\equiv \text {Ai}(z)\). For the detailed construction of these profiles, we refer to [12].
Finally, by rescaling the leading order profile \((\mathfrak {p}_0,\mathcal {U}_0,\mathcal {V}_0)\) via (2.21), we define the fast mode as
Obviously,
2.3 Approximate Growing Mode
Based on slow and fast modes constructed in the above two subsections, we are now ready to construct an approximate growing mode to (1.3) with boundary condition (1.4). Set
where \({\Xi }_{\text {app}}^s\), \({\Xi }_{\text {app}}^f\) are defined in (2.19), (2.30) respectively, and the function \(\Phi _{\text {app}}^s(Y;c)\) is defined in (2.11) with boundary data satisfying (2.13) and (2.14). Thanks to (2.31), the normal velocity \(v_{\text {app}}\) satisfies the zero boundary condition, that is, \(v_{\text {app}}(0;c)\equiv 0.\) Therefore, the approximate solution (2.32) satisfies the full no-slip boundary condition (1.4) if and only if the following function vanishes at some point c:
To find the zero point of \(\mathcal {F}_{\text {app}}(c)\), we consider the Mach number \({\mathcalligra {m}}\in (0,1)\) and the wave number \(\alpha =K\varepsilon ^{\frac{1}{8}}\) with \(K\geqq 1\) being a large but fixed real number. Set
and define a disk centered at \(c_0\) by
with some constant \(\theta \in (0,1)\). Clearly, for any \( {\mathcalligra {m}}\in (0,1)\), there exists a positive constant \(\tau _0>0\) (\(\tau _0\rightarrow 0\) as \({\mathcalligra {m}}\rightarrow 1\)), such that for sufficiently large K, the following estimates hold for any \(c\in D_0\):
With the above preparation, we will prove the following proposition about the existence of approximate growing mode:
Proposition 2.2
Let \({\mathcalligra {m}}\in (0,1)\). There exists a positive constant \(K_0>1\), such that if \(K\geqq K_0\), then there exists \(\varepsilon _1\in (0,1)\), such that for \(\alpha =K\varepsilon ^{\frac{1}{8}}\) and \(c\in D_0\) with \(\varepsilon \in (0,\varepsilon _1)\), the function \(\mathcal {F}_{\text {app}}(c)\) has a unique zero point in \(D_0\). Moreover, on the circle \(\partial D_0,\) it holds that
Proof
The proof follows the approach used in Proposition 3.2 in the authors’ paper [22] with Liu on the incompressible MHD system. For completeness, we sketch the main steps as follows. Firstly, we take \(K_0\) sufficiently large so that (2.35) holds in the disk \(D_0\). Then for \(\alpha =K\varepsilon ^{\frac{1}{8}}\) and any \( c\in D_0,\) by (2.13), (2.14) and (2.35), we have
Thus from (2.4) the expression for \(\rho _{\text {app}}^s\), one obtains
Next, we consider the ratio \(\frac{\text {Ai}(1,z_0)}{\text {Ai}(2,z_0)}\). Recall \(z_0(c)=-\delta ^{-1}c=-e^{\frac{1}{6}\pi i}K^{\frac{1}{3}}\varepsilon ^{-\frac{1}{8}}c\). (2.35) implies that
Then using the asymptotic behavior of Airy profile (for example cf. [12]) and by (2.39), we obtain
Now we set
On one hand, there exists a unique zero point \(c_0\) in (2.33) to the mapping \(\mathcal {F}_{\text {ref}}(c)\). On the boundary \(\partial D_0\), it holds that
On the other hand, we can show that \(\mathcal {F}_{\text {ref}}(c)\) is the leading order of \(\mathcal {F}_{\text {app}}(c)\). In fact, by (2.37), (2.38) and (2.40), we have the following estimate on the difference:
Here the positive constants \(C_{\mathcalligra {m}}\) are independent of K and \(\varepsilon \) and \(C_{K,{\mathcalligra {m}}}\) depends on K and Mach number \({\mathcalligra {m}}\), but not on \(\varepsilon \). Now we take \(K_0\) larger if needed and then take \(\varepsilon _1\in (0,1)\) suitably small such that for \(\varepsilon \in (0,\varepsilon _1)\) and \(K\geqq K_0\), it holds that
Consequently, one obtains
Combining this with (2.42) yields (2.36). Moreover, since \(\text {Ai}(1,z)\) and \(\text {Ai}(2,z)\) are analytic functions and \(\text {Ai}(2,z_0)\ne 0\) due to (2.40), both \(\mathcal {F}_{\text {app}}(c)\) and \(\mathcal {F}_{\text {ref}}(c)\) are analytic in \(D_0\). Therefore, by Rouché’s Theorem, \(\mathcal {F}_{\text {app}}(c)\) and \(\mathcal {F}_{\text {ref}}(c)\) have the same number of zeros in \(D_0\). The proof of Proposition 2.2 is then completed. \(\quad \square \)
We now conclude this subsection by summarizing the relations between the parameters n (rescaled frequency), \(\delta \) (scale of sublayer), \(\alpha \) (wave number) and \(\varepsilon \) (viscosity) that will be used frequently later:
If, in particular, \(\alpha =K\varepsilon ^{\frac{1}{8}}\) and \(c\in D_0\), then
where the relations may depend on K but not on \(\varepsilon \).
2.4 Estimates on Error Terms
In this subsection, we will give the detailed estimate on the error of the approximate solution (2.32) by using a decomposition that takes the decay and regularity in Y into consideration.
\(\underline{``Regular+Smallness'' decomposition:}\) Precisely, the approximate solution \({\Xi }_{\text {app}}\) to (1.3) has the following error representation:
Here the regular part
with \(\text {Ray}_{\text {CNS}}(\Phi _{\text {app}}^s)\) defined in (2.12). Observe that \(E_{v,\text {re}}\) has strong decay in Y due to the background boundary layer profile, and the smallness part reads
where \(\eta \overset{\hbox {{def}}}{=}\delta ^{-1}\Phi _{\text {app}}^s(0;c)\). As we will see, the error terms \(E_{u,\text {sm}}\) and \(E_{v,\text {sm}}\) are of higher order in \(\varepsilon \) than \(E_{v,\text {re}}\).
The estimates on these error terms are summarized in the next proposition. Let us first define some weighted Sobolev spaces for later use:
Recall \(K_0\geqq 1\) and \(\varepsilon _1\in (0,1)\) are constants given in Proposition 2.2. For \(K\geqq K_0\) and \(\varepsilon \in (0,\varepsilon _1),\) the disk \(D_0\) is defined in (2.34). The following proposition gives the precise error bound estimates:
Proposition 2.3
Let the Mach number \({\mathcalligra {m}}\in (0,1)\). There exists \(\varepsilon _2\in (0,\varepsilon _1)\), such that for \(\varepsilon \in (0,\varepsilon _2)\), \(\alpha =K\varepsilon ^{\frac{1}{8}}\) and \(c\in D_0\), the error terms \(E_{v,\text {re}}\), \(E_{u,\text {sm}}\) and \(E_{v,\text {sm}}\) satisfy the estimates
Here the constant K is uniform in \(\varepsilon \).
The proof of Proposition 2.3 follows from a series of estimations on approximate solutions. First of all, we show some properties of corrector \(\varphi _{1,\alpha }\) and the approximate solution \(\Phi _{\text {app}}^s\) to Rayleigh operator that are defined in (2.10) and (2.11) respectively. Fix \({\mathcalligra {m}}\in (0,1)\) and set \({\beta }_1\overset{\hbox {{def}}}{=}\frac{1}{2}(1-{\mathcalligra {m}}^2)^{\frac{1}{2}}\alpha \).
Lemma 2.4
Let \(\gamma _1\) be the constant given in Lemma 2.1. There exists \(\gamma _2\in (0,\gamma _1)\), such that for any c lies in the half disk \( \{c\in \mathbb {C} \mid \text {Im}c>0,~|c|< \gamma _2 \}\) and \(\alpha \in (0,1)\), the corrector \(\varphi _{1,\alpha }\) satisfies
Moreover, if, in addition, \(\alpha =K\varepsilon ^{\frac{1}{8}}\) and \(c\in D_0\), we have the following uniform bounds:
Proof
Recall that \(\beta =\alpha \left[ (1-{\mathcalligra {m}}^2)^{\frac{1}{2}}+O(1)|c|\right] \) from (2.6) and \(A(Y)=1-{\mathcalligra {m}}^2U_s^2+O(1)|c|\) from (2.3). Taking \(\gamma _2\in (0,\gamma _1)\) suitably small, we have \(\text {Re}\beta >\beta _1\) and \(|A^{-1}|\leqq \frac{1}{2(1-{\mathcalligra {m}}^2)}\lesssim 1\) for \(|c|<\gamma _2\). Then the proof of (2.50)–(2.53) follows from an argument exactly as in Lemma 3.6 in [22] by using the explicit expression (2.15). Hence, we omit it for brevity.
For (2.54), we recall (2.11) and observe that
By (2.43), we have \(\alpha /\text {Im}c\lesssim 1\). Thus putting (2.50)–(2.53) and (2.55) together yields the desired estimate (2.54). The proof of the lemma is then completed. \(\quad \square \)
By Lemma 2.4, we can immediately obtain the following estimates on the slow mode \({\Xi }_{\text {app}}^s\) given in (2.19):
Corollary 2.5
If \(\alpha =K\varepsilon ^{\frac{1}{8}}\) and \(c\in D_0\cap \{c\in \mathbb {C}\mid |c|< \gamma _2\}\), the slow mode \({\Xi }_{\text {app}}^s\) satisfies the following estimates:
Proof
The estimate on \(v_{\text {app}}^s\) follows from (2.54) directly. For \(\rho _{\text {app}}^s\), using (2.54) gives
Differentiating (2.4) with respect to Y yields the formulas
and
Taking \(L^2\)-norm and by (2.54), we obtain
and
The velocity field \(u_{\text {app}}^s\) can be estimated in the same way so that we omit the details. And this completes the proof of the corollary. \(\quad \square \)
The next lemma gives some pointwise estimates on the fast mode \((u_{\text {app}}^f,v_{\text {app}}^f)\) defined in (2.30). The proof follows from Lemma 3.9 in [22] by using the pointwise estimate of Airy profiles. Thus, we omit the details, for brevity.
Lemma 2.6
The fast mode \((u_{\text {app}}^f,v_{\text {app}}^f)\) has the pointwise estimates
for some constant \(\tau _1>0\) which does not depend on n.
With the above estimates, we are now ready to prove Proposition 2.3 as follows:
Proof of Proposition 2.3
We start with proving (2.48) for \(E_{v,\text {re}}\). Recall the definition (2.45) and explicit formula (2.12). By taking \(\varepsilon _2>0\) suitably small, such that \(D_0\subset \{\text {Im}c>0, |c|\leqq \gamma _2\}\), we have \(\text {Re}\beta >\beta _1\) and \(|A|\gtrsim 1\). Then thanks to (1.8), (2.6), the bounds (2.50)–(2.54) and the fact that
we have
This, combined with the concavity (1.9), gives the following weighted estimate:
Moreover, differentiating (2.12) yields
With this, by the bounds (2.50)–(2.54) and the concavity (1.9), we obtain
Putting (2.60) and (2.61) together yields the estimate (2.48) on part of the error with decay.
Now we turn to estimate the part of error with smallness \((E_{u,\text {sm}},E_{v,\text {sm}})\) which is defined in (2.46). Keeping in mind the bounds of parameters (2.43) and
because of (2.13). Note that also \(|U_s(Y)-U_s'(0)Y|\lesssim Y^2, |\partial _YU_s(Y)-\partial _YU_s(0)|\lesssim Y\) and
By the bounds on \((u_{\text {app}}^s,v_{\text {app}}^s)\) and \((u_{\text {app}}^f,v_{\text {app}}^f)\) given in Corollary 2.5 and Lemma 2.6, we have
and
Estimates (2.62) and (2.63) give the bound (2.49) for \((E_{u,\text {sm}},E_{v,\text {sm}})\). Then the proof of the proposition is completed. \(\quad \square \)
3 Solvability of Remainder System
In this section, we will construct a solution to the resolvent problem
where \((f_u,f_v)\) is a given inhomogeneous source term. If \((f_u,f_v)\in H^1(\mathbb {R}_+)^2\), we define the operator
Recall (2.47) the weighted function space \(L^2_w(\mathbb {R}_+)\). The following is the main result in this section:
Proposition 3.1
(Solvability of resolvent problem) Let the Mach number \({\mathcalligra {m}}\in (0,\frac{1}{\sqrt{3}}).\) There exists \(\varepsilon _{3}\in (0,1)\), such that for any \( \varepsilon \in (0,\varepsilon _3)\), \(\alpha =K\varepsilon ^{\frac{1}{8}}\) and \(c\in D_0\), the following two statements hold.
(1) If \((f_u,f_v)\in L^2(\mathbb {R}_+)^2\), then there exists a solution \({\Xi }=(\rho ,u,v)\in H^{1}(\mathbb {R}_+)\times H^2(\mathbb {R}_+)^2\) to (3.1) which satisfies the following estimates:
(2) If in addition we have \((f_u,f_v)\in H^1(\mathbb {R}_+)^2\) with \(\Vert \Omega (f_u,f_v)\Vert _{L^2_w}<\infty ,\) then there exists a solution \({\Xi }=(\rho ,u,v) \in H^{1}(\mathbb {R}_+)\times H^2(\mathbb {R}_+)^2\) to (3.1) which satisfies the following improved estimates
Moreover, if \((f_u,f_v)(\cdot ~;c)\) is analytic in c with values in \(L^2\), then the solution \({\Xi }(\cdot ~;c)\) is analytic with values in \(H^1(\mathbb {R}_+)\times H^2(\mathbb {R}_+)^2\).
Remark 3.2
-
(a)
By Sobolev embedding \(H^1(\mathbb {R}_+)\hookrightarrow L^\infty (\mathbb {R}_+)\), the mapping \(u(0;c): D_0\mapsto \mathbb {C}\) is analytic.
-
(b)
The solutions to (3.1) are in general not unique because we do not prescribe the boundary data at \(Y=0\) for u.
-
(c)
The constants in estimates (3.3)–(3.8) are uniform for \({\mathcalligra {m}}\in (0,{\mathcalligra {m}}_0]\) with any \({\mathcalligra {m}}_0\in (0,\frac{1}{\sqrt{3}})\).
-
(d)
As one can see from the proof, the argument also works for a wider regime of parameters:
$$\begin{aligned} |\alpha |\lesssim 1, ~ |c|\ll 1,~\frac{|c|^2}{\text {Im}c}\ll 1,~ \frac{1}{n(\text {Im}c)^2}\ll 1. \end{aligned}$$(3.9)In fact, the boundedness of wave number \(\alpha \) is essentially used in the proof. In addition, we require c to satisfy (3.103) so that \(c\in \Sigma _{Q}\cap \Sigma _{S}\) where \(\Sigma _{Q}\) and \(\Sigma _{S}\) are resolvent sets of \(L_Q\) and \(L_S\) respectively. Moreover, in view of (3.114), we require smallness of \(\frac{1}{n(\text {Imc})^2}\) in order to establish the convergence of iteration. These requirements can be fulfilled by the smallness in (3.9).
As mentioned in the Introduction, the proof of Proposition 3.1 is based on the following two newly introduced decompositions, that is, quasi-compressibile approximation and the Stokes approximation.
3.1 Quasi-Compressible Approximation
Following the strategy described in the Introduction, we first consider the approximate problem
with a given inhomogeneous source term \((s_1,s_2)\).
By the continuity equation (3.10)\(_1\), we can define an “effective stream function” \(\Psi \) satisfying that
Then, by (3.10)\(_2\), we can express the density \(\rho \) in terms of \(\Psi \) as
Substituting (3.12) into (3.10)\(_3\), we derive the following equation for \(\Psi \) which can be viewed as the Orr–Sommerfeld equation in the compressible setting:
Here the modified vorticity operator \(\Lambda \) is given by
and \(\Omega \) is given in (3.2).
In order to solve (3.13), we consider the following boundary condition:
If \(\Psi \) solves problem (3.13) with boundary conditions (3.15), it is straightforward to check that \((\rho ,u,v)\) defined by (3.11) and (3.12) is a solution to (3.10).
Thus, in what follows, we consider the boundary value problem
with a given inhomogeneous source \(h\in L^2_w(\mathbb {R}_+)\). Let us first introduce the multiplier
A straightforward computation yields the properties of w stated in the following lemma:
Lemma 3.3
Let \({\mathcalligra {m}}\in (0,1)\) and \(U_s\) satisfy (1.5)–(1.8). There exists \(\gamma _3>0\), such that if \(|c|< \gamma _3\), \({\mathcalligra {w}}(Y)\) has the expansion
Here \({\mathcalligra {w}}_0\) and \({\mathcalligra {w}}_1\) are given by
where the function H(Y) is defined in (1.7). Moreover, it holds that
Set the function space
For the problem (3.16), we have
Lemma 3.4
(A priori estimates) Let \({\mathcalligra {m}}\in (0,\frac{1}{\sqrt{3}})\) and \(\Psi \in \mathbb {X}\) be a solution to (3.16). There exists \(\gamma _4\in (0,\gamma _3)\), such that for \(\alpha \in (0,1)\) and c lies in
then \(\Psi \) satisfies the following estimates
Proof
Taking inner product of (3.16) with the multiplier \(-{\mathcalligra {w}}\overline{\Lambda (\Psi )}\) leads to
Now we estimate \(J_1-J_4\) separately. Let us consider \(J_3\) first. By integrating by parts and using the boundary condition \(\Psi |_{Y=0}=0,\) we obtain
Recalling (2.3) about the definition of A(Y), we have
With this identity, the assumption \({\mathcalligra {m}}\in (0,1)\), and (1.5) for the positivity of \(U_s\), we can deduce from (3.25) that
and
where the constants may depend on \({\mathcalligra {m}}\) but not on either \(\varepsilon \) or c.
For \(J_2\), we obtain from the expansion (3.18) and bound (3.20) that
Using the explicit formula (3.19) of \({\mathcalligra {w}}_0\) and \({\mathcalligra {w}}_1\) gives
Since \({\mathcalligra {m}}\in (0,\frac{1}{\sqrt{3}})\), we have \((1-3{\mathcalligra {m}}^2U_s^2)\geqq (1-3{\mathcalligra {m}}^2)>0.\) Thus, by (1.9) and (3.29), it holds that
where the positive constants \(C_1\) and \(C_2\) are uniform in \(\varepsilon \) and c. Therefore, taking real and imaginary part of \(J_2\) respectively and using the bounds (3.20), (3.28) and (3.30) yield
and
For \(J_1\), we rewrite
where \([\Delta _\alpha ,\Lambda ](\Psi )\) is the commutator \(\Delta _\alpha \left[ \Lambda (\Psi )\right] -\Lambda \left[ \Delta _\alpha (\Psi )\right] \). By integrating by parts and using the boundary condition \(\Lambda (\Psi )|_{Y=0}=0,\) we obtain
Then by (1.8) and (3.20), we have
Thus the last integral on the right hand side of (3.34) is bounded by
By taking real and imaginary parts of \(J_{11}\) respectively and using (3.35), we deduce that
and
where we have used the fact that
and
by the expansion (3.18).
Next we estimate \(J_{12}\). Recall (3.14) the definition of \(\Lambda \). We have
We rewrite \(\partial _Y^2\Psi \) and \(\partial _Y^3\Psi \) as
By \(\alpha \in (0,1)\), it holds that
By (1.8), we have
Then applying the bounds (3.39) and (3.40) to (3.38) gives
which by (1.9) implies
Substituting (3.41) into \(J_{12}\) and using Cauchy-Schwarz inequality yield
where we have used (3.20). By (3.36), (3.37) and (3.42), we can deduce from real and imaginary parts of (3.33) that
and
Finally, for \(J_4\), we have by Cauchy-Schwarz inequality that
Thus, we have completed the estimation on \(J_1-J_4\).
By taking imaginary part of (3.24) and using previous bounds (3.27), (3.32), (3.44) and (3.45) for \(J_1-J_4\), we have
Similarly, taking real part of (3.24) and using (3.26), (3.31), (3.43) and (3.45) give
Multiplying estimate (3.47) by \(\text {Im}c\), suitably combining it with (3.46) and using Young’s inequality, we can obtain that
By taking \(\gamma _4\in (0,\gamma _3)\) suitably small such that
we can absorb the first and second terms on the right hand side of (3.48) by the left hand side. Thus,
The above two inequalities immediately imply the estimates (3.22) and (3.23). The proof of the lemma is completed. \(\quad \square \)
With the a priori estimates in Lemma 3.4, we can prove the existence, uniqueness and analytic dependence on c of the solution to the compressible Orr–Sommerfeld equation (3.16) in the following lemma:
Lemma 3.5
(Construction of the solution) Let \({\mathcalligra {m}}\in (0,\frac{1}{\sqrt{3}})\), \(\alpha \in (0,1)\) and \(c\in \Sigma _Q\). If \(\Vert h\Vert _{L^2_w}<\infty \), there exists a unique solution \(\Psi \in \mathbb {X}\) to (3.16) which satisfies estimates (3.22) and (3.23). Moreover, if \(h(\cdot ~;c)\) is analytic in c in \(L^2_w(\mathbb {R}_+)\), then \(\Psi (\cdot ~;c)\) is analytic in \(\mathbb {X}\).
Remark 3.6
By elliptic regularity, the solution \(\Psi \) is in \(H^4(\mathbb {R}_+)\cap H^1_0(\mathbb {R}_+)\).
Proof
The proof is based on a cascade of approximate process and a continuity argument. First of all, we set \(W\overset{\hbox {{def}}}{=}\Lambda (\Psi )\) and reformulate (3.16) as
Here the inverse operator \(\Lambda ^{-1}: L^2(\mathbb {R}_+)\rightarrow H^2(\mathbb {R}_+)\cap H^1_0(\mathbb {R}_+)\) is constructed in Lemma 5.1. If one can show the solvability of (3.49) in \(H^1_w(\mathbb {R}_+)\), then by Lemma 5.1, \(\Psi \overset{\hbox {{def}}}{=}\Lambda ^{-1}(W)\in \mathbb {X}\) and it solves the equation (3.16). Now we elaborate the construction of solution to (3.49) in the following three steps.
Step 1. Fix any parameter \(l>0\). We start from an auxiliary problem
We claim that there exists \(l_0>0\), such that if \(c\in \Sigma _{Q}\) and \(\Vert h\Vert _{L^2_w}<\infty \), then (3.50) admits a unique solution \(W\in H^2_w(\mathbb {R}_+)\cap H^1_0(\mathbb {R}_+)\) and the solution operator \(T_{l_0}^{-1}: L^2_w(\mathbb {R}_+)\rightarrow H^2_w(\mathbb {R}_+)\) is analytic in c. To prove this claim, we define a sequence of approximate solutions \(\{W_k\}_{k=0}^\infty \) by the following equations
where \(\text {Airy}\overset{\hbox {{def}}}{=}\frac{i}{n}\Delta _{\alpha }+U_s: H^2_w(\mathbb {R}_+)\cap H^1_0(\mathbb {R}_+)\rightarrow L^2_w(\mathbb {R}_+)\) is the Airy operator. For any \(c\in \Sigma _Q\) and \(l>0\), by direct energy method, it is straightforward to check that \(c+il\) lies in the resolvent set of Airy operator. Thus by an inductive argument, we can solve \(W_k\) and establish its analytic dependence on c from (3.51). In order to take the limit \(k\rightarrow \infty ,\) we need some uniform estimates. Applying the multiplier \((\partial _Y^2U_s)^{-1}\bar{W}_{k+1}\) to (3.51), we have
By Cauchy-Schwarz inequality, we deduce that
By using (3.20) and the bound in (5.1) for \(\Lambda ^{-1}\), we have
Integration by parts yields
By (1.8), the last integral in the above equality is bounded by
By taking the imaginary part of (3.52), and using the bounds obtained in (3.53)–(3.56) with Young’s inequality, we have
We choose \(\gamma _4>0\) smaller if needed so that \(\frac{C}{n\text {Im}c}<\frac{1}{2}\), for any \(c\in \Sigma _{Q}\). Then (3.57) gives
Now we take the difference \(W_{k+1}-W_k\). A Similar argument gives
By taking l suitably large, such that \(\frac{C}{\text {Im}c+l}\leqq \frac{C}{l}\leqq \frac{1}{2}\), \(\{W_{k}\}_{k=1}^\infty \) is a Cauchy sequence in \(H^1_w(\mathbb {R}_+)\). This implies the existence of a limit function \(W=\lim _{k\rightarrow \infty } W_k\) in \( H^1_w(\mathbb {R}_+)\) that is the solution to (3.50). By the elliptic regularity, \(W_k\) converges to W in \(H^2_w(\mathbb {R}_+).\) Moreover, by induction, each \(W_k\) is analytic in c, so is W by uniform convergence. This justifies the claim and step 1 is completed.
Step 2. (Bootstrap from \(T_{l_0}^{-1}\) to \(T_{0}^{-1}\)). Consider the equation (3.50) for any fix \(l\in [0,l_0]\) with \(W^l\) as its solution. Applying the multiplier \(-{\mathcalligra {w}}\bar{W}^l\) and using the same argument as in Lemma 3.4, we can show that \(W^l\) satisfies
where the constant C is uniform in \(l\in [0,l_0]\). Now we take \(l_1=l_0-\lambda \) for some fixed constant \(0<\lambda <2C^{-1}\gamma _4^{-1}n^{-1}\) and construct the solution \(W^{l_1}=T_{l_1}^{-1}(h)\) through the following iteration
Applying the a priori estimate (3.58) to \(W_{k+1}^{l_1}-W_{k}^{l_1}\) yields that
Hence, \(\{W_{k}^{l_1}\}_{k=0}^\infty \) is a Cauchy sequence in \(H^1_w\) and it has a limit \(W^{l_1}=\lim _{k\rightarrow \infty }W^{l_1}_k\). It is straightforward to check that \(W^{l_1}\) is in \(H^2_{w}(\mathbb {R}_+)\cap H^1_0(\mathbb {R}_+)\) and satisfies (3.50) with \(l=l_1\). Moreover, from the previous step we have already shown that each \(W_{k}^{l_1}\) is analytic in c. Thus analyticity of \(W^{l_1}\) follows from the uniform convergence. Thus we have completed the construction solution operator \(T^{-1}_{l_1}\). Noting that \(W^{l_1}\) satisfies the a priori estimate (3.58), we can take \(l_2=l_1-\lambda \) and construct the solution operator \(T_{l_2}^{-1}\) in the same way. Repeating the same procedure, we can eventually establish the existence and analytic dependence on c of the solution operator \(T_0^{-1}\).
Step 3. We now solve the original system (3.49) by using the following iteration:
By using the bounds in (3.41) and (5.1) on the commutator \([\Delta _\alpha ,\Lambda ]\) and \(\Lambda ^{-1}\) respectively, we have
Then applying the a priori bound (3.58) to \(W_{k+1}-W_{k}\), gives
By taking \(\gamma _4>0\) smaller if needed such that \(\frac{C}{n\text {Im}c}+\frac{C}{n^{\frac{1}{2}}\text {Im}c^{\frac{1}{2}}}\leqq C\gamma _4^{\frac{1}{2}}(1+\gamma _4^{\frac{1}{2}})<\frac{1}{2}\) for \(c\in \Sigma _Q,\) we show that \(\{W_k\}_{k=0}^\infty \) is a Cauchy sequence in \(H^1_w(\mathbb {R}_+)\). Let \(W:=\lim _{k\rightarrow \infty }W_k\). By the elliptic regularity and \(H^1_w\)-convergence, it is straightforward to check that \(W_k\) converges to W in \(H^2_w(\mathbb {R}_+)\) and W is a solution to (3.49). Moreover, since each \(W_k\) is analytic in c and the convergence is uniform in \(c\in \Sigma _Q\), we conclude that W is analytic in c. The uniqueness of solution follows from the a priori estimates obtained in Lemma 3.4. Then the proof of the lemma is completed. \(\quad \square \)
Now let \(\Psi \) be the solution to (3.16) with \(h=\Omega (s_1,s_2)\) and \(\Omega \) defined in (3.2). In terms of the fluid variables \((\varrho ,\mathfrak {u},\mathfrak {v})\) given in (3.11) and (3.12), we have the following proposition for the solvability of the quasi-compressible approximation system (3.10):
Proposition 3.7
(Solvability of quasi-compressible system) Under the same assumption on parameters \({\mathcalligra {m}}\), \(\alpha \) and c as in Lemma 3.5, if \({s}=(s_1,s_2)\in H^1(\mathbb {R}_+)^2\) and \(\Vert \Omega (s_1,s_2)\Vert _{L^2_w}<\infty \), there exists a solution \((\varrho ,\mathfrak {u},\mathfrak {v})\in H^2(\mathbb {R}_+)^3\) to the quasi-compressible approximation system (3.10). Moreover, \((\varrho ,\mathfrak {u},\mathfrak {v})\) satisfies the estimates
and
Furthermore, if both \({s}(\cdot ~;c)\) and \(\Omega (\cdot ~;c)\) are analytic in c in \(H^1(\mathbb {R}_+)\) and \(L^2_w(\mathbb {R}_+)\) respectively, then \((\varrho ,\mathfrak {u},\mathfrak {v})(\cdot ~;c)\) is analytic in c in \(H^2(\mathbb {R}_+)\).
Remark 3.8
If \(\text {div}_\alpha (s_1,s_2)=0,\) then by (3.11), (3.67) and regularity of \(\Psi \) it is easy to deduce that \((\varrho ,\mathfrak {u},\mathfrak {v})\in H^3(\mathbb {R}_+)^3\). This reveals the elliptic structure for linearized compressible Navier–Stokes equations around the subsonic boundary layer profile.
Proof
It is straightforward to check that \((\varrho ,\mathfrak {u},\mathfrak {v})\) satisfies (3.10). The analyticity directly follows from Lemma 3.4. It remains to show the estimates (3.59) and (3.60). Firstly, by using bounds given in (3.22), (3.23) with \(h=\Omega (s_1,s_2)\) and (3.39), we obtain that
and
where we have used \(n\text {Im}c\gtrsim 1\) for \(c\in \Sigma _Q\). Then by \(\mathfrak {v}=-i\alpha \Psi \), (3.61) and \(\alpha \in (0,1)\), it holds that
Next we estimate \(\varrho \). Recall (3.12) for its representation. Since \(\Psi |_{Y=0}=0,\) we can use Hardy inequality \(\Vert Y^{-1}\Psi \Vert _{L^2}\leqq 2\Vert \partial _Y\Psi \Vert _{L^2},\) and the bounds given in (3.61), (3.62) to obtain
For \(\partial _Y\varrho ,\) differentiating (3.12) yields that
where we have used the equation (3.13) in second identity. Taking \(L^2\)-norm in (3.65) and using bound (3.61), we can further deduce that
Now we estimate \(\partial _Y^2\varrho .\) By using (3.12) and (3.65), we have
Then taking \(L^2\) norm leads to
Here we have used (3.61) and (3.64) in the last inequality. Therefore, \(H^2\)-estimate of \(\varrho \) follows from (3.64), (3.66) and (3.68). Since \(\text {div}_\alpha (\mathfrak {u},\mathfrak {v})=-i\alpha (U_s-c)\varrho \), by using (3.64) and (3.66) we have
Finally, for \(\mathfrak {u}\), by using (3.11), (3.61), (3.62), (3.64), (3.66) and (3.68), we obtain that
Putting the estimates in (3.63), (3.64), (3.66), (3.68)–(3.70) together yields the estimate (3.59). Note that (3.60) directly follows from (3.71). Then the proof of proposition is completed. \(\quad \square \)
3.2 Stokes Approximation
In this section, we study the Stokes system with advection
with a given inhomogeneous source term \({q}=(q_0,q_1,q_2)\in H^1(\mathbb {R}_+)\times L^2(\mathbb {R}_+)^2\). Compared with original system (3.1), in (3.72) we remove the stretching term \(-\psi \partial _YU_s\) in the momentum equation. We impose the Neumann boundary condition \(\partial _Y\phi |_{Y=0}=0\) on the tangential velocity for obtaining estimates on the higher order derivatives. The following proposition gives the solvability of (3.72):
Proposition 3.9
Let \({\mathcalligra {m}}\in (0,1)\). Assume that \(\alpha \in (0,1)\) and \(\frac{1}{n}=\frac{\sqrt{\varepsilon }}{\alpha }\ll 1\). There exists \(\gamma _5\in (0,1)\), such that for any c lies in
the system (3.72) admits a unique solution \((\xi ,\phi ,\psi )\in H^1(\mathbb {R}_+)\times H^2(\mathbb {R}_+)^2\). Moreover, \((\xi ,\phi ,\psi )\) satisfies the following estimates:
Here the positive constant C does not depend on either \(\alpha \) or \(\varepsilon \). Furthermore, if \({q}(\cdot ~;c)\) is analytic in c in \(H^1(\mathbb {R}_+)\times L^2(\mathbb {R}_+)^2\), then \((\xi ,\phi ,\psi )(\cdot ~;c)\) is analytic in c in \(H^{1}(\mathbb {R}_+)\times H^2(\mathbb {R}_+)^2\).
Remark 3.10
We will use the bounds given in (3.74)–(3.77) only when \(q_0=0\) in the proof of convergence of iteration.
Remark 3.11
In view of (3.75)–(3.77) with \(q_0=0,\) the divergence part \(\text {div}_\alpha (\phi ,\psi )\) and the density \(\xi \) of the solution have better estimates than other components because there is no strong sublayer related to these two fluid components. This stronger estimate is crucial in the proof of convergence of the iteration later.
Proof
We first focus on the a priori estimates (3.74)–(3.77). By taking inner product of (3.72)\(_2\) and (3.72)\(_3\) with \(-\bar{\phi }\) and \(-\bar{\psi }\) respectively then integrating by parts, we obtain
By Cauchy-Schwarz and Young’s inequalities, it holds that
By using the continuity equation, \(\overline{\text {div}_\alpha (\phi ,\psi )}=i\alpha (U_s-\bar{c})\bar{\xi }+\bar{q}_0\), and the Cauchy-Schwarz inequality, we get
By (3.79) and (3.80), the real part of (3.78) gives that
By taking \(\gamma _5\in (0,1)\) sufficiently small so that \(\frac{C\sqrt{\varepsilon }}{a\text {Im}c}\leqq \frac{C}{n\text {Im}c}\leqq C\gamma _5\leqq \frac{1}{4},~\forall c\in \Sigma _S\), we can absorb the first term on the right hand side of (3.81) by the left hand side. Thus we get
and
This completes the proof of (3.74) and (3.75).
Next we estimate \(\Vert \partial _Y\xi \Vert _{L^2}\) and \(\Vert \text {div}_\alpha (\phi ,\psi )\Vert _{H^1}\). Define \(\omega \overset{\hbox {{def}}}{=}\partial _Y\phi -i\alpha \psi \) and denote \(\mathcal {D}:=\text {div}_\alpha (\phi ,\psi )\). Then
and \(\omega |_{Y=0}=0\) because of the boundary conditions in (3.72). Thus, we can rewrite (3.72)\(_2\) and (3.72)\(_3\) as
By taking inner product of (3.84) and (3.85) with \(-i\alpha \bar{\xi }\) and \(\partial _Y\bar{\xi }\) respectively, we deduce that
Integrating by parts and using boundary condition \(\omega |_{Y=0}=0\) yield that
For \(J_6\), by using the continuity equation (3.72)\(_1\), we have
which implies that
For last two terms on the right hand side, we obtain by Cauchy-Schwarz and Young’s inequalities that
and
Thus, taking real part of \(J_6\) gives
for \(0<\varepsilon \ll 1\) being sufficiently small. Again, by Young’s inequality, we get
where we have used \(\frac{1}{n}=\frac{\sqrt{\varepsilon }}{\alpha }\ll 1\). By (3.87), (3.89) and (3.90), the real part of (3.86) yields
Moreover, by (3.88) and (3.91), we obtain
Putting the bound (3.74) on \(\Vert (\phi ,\psi )\Vert _{L^2}\) into (3.91) and (3.92) yields the estimate
Hence, (3.76) holds.
Finally, we derive the estimate on \(\Vert (\partial _Y\omega ,\alpha \omega )\Vert _{L^2}\). By taking inner products of (3.84) and (3.85) with \(\partial _Y\bar{\omega }\) and \(i\alpha \bar{\omega }\) respectively then using the fact that
we obtain
which implies
By combining this with the bound (3.76) on \(\Vert \text {div}_\alpha (\phi ,\psi )\Vert _{H^1}\) and recalling (3.83), we have
which is (3.77). The uniqueness of solution follows from the a priori bounds (3.74)–(3.77).
As for the construction of solution, we introduce a parameter \(\eta \in [0,1]\) and study a sequence of auxiliary problems \(L_{S,\eta }(\xi ^\eta ,\phi ^\eta ,\psi ^{\eta })=(q_0,q_1,q_2)\) as follows:
When \(\eta =0,\) (3.94) reduces to the classical Stokes system for incompressible flow:
It is standard to show the existence and uniqueness of solution \((\xi ^0,\phi ^0,\psi ^0)\in H^1(\mathbb {R}_+)\times H^2(\mathbb {R}_+)^2\) for any \((q_0,q_1,q_2)\in H^1(\mathbb {R}_+)\times L^2(\mathbb {R}_+)^2.\) Moreover, by repeating previous energy estimates to (3.94) and slightly modifying the proof of bounds (3.82), (3.91), (3.92) and (3.93), one can deduce the following estimates on \((\xi ^\eta ,\phi ^\eta ,\psi ^\eta )\)
where \(\omega ^\eta =\partial _Y\phi ^\eta -i\alpha \psi ^\eta \) and the constant \(C>0\) does not depend on \(\eta \). Putting above inequalities together yields the following uniform-in-\(\eta \) estimate
where the constant \(C(\varepsilon ,\alpha )\) may depend on \(\varepsilon \) and \(\alpha \), but not on \(\eta \in [0,1]\). Thus the existence of solution to (3.72) as well as its analytic dependence on c can be established by the same bootstrap argument as in Lemma 3.5. By uniqueness, the solution obtained satisfies the bounds (3.74)–(3.77). And this completes the proof of the proposition. \(\quad \square \)
3.3 Quasi-Compressible-Stokes Iteration
In this subsection, we will construct a solution \({\Xi }=(\varrho ,\mathfrak {u},\mathfrak {v})\) to the linearized system (3.1) via an iteration scheme based on the solutions to quasi-compressible and Stokes approximations given in Propositions 3.7 and 3.9.
We first consider the case when source term \((f_u,f_v)\in L^2(\mathbb {R}_+)^2\). At zeroth step, we define \({\Xi }_0=(\xi _0,\phi _0,\psi _0)\) as the solution to Stokes approximate system
which yields an error
Because of the regularizing effect of solution operator to Stokes approximation \(L_S\), this error has higher regularity and fast decay so that \({\mathcal {E}}_0\in H^2_w(\mathbb {R}_+)\). We can then eliminate it by considering \((\varrho _1,\mathfrak {u}_1,\mathfrak {v}_1)\) as the solution to quasi-compressible approximation
Then we have
where the error operator \(E_Q\) is defined in (1.15). According to Proposition 3.7, the solution \((\varrho _1,\mathfrak {u}_1,\mathfrak {v}_1)\) is in \( H^2(\mathbb {R}_+)^3\). Thus the error term \(E_Q(\varrho _1,\mathfrak {u}_1,\mathfrak {v}_1)\) is in \(L^2(\mathbb {R}_+)\). This allows us to correct this error by using the solution \((\xi _1,\phi _1,\psi _1)\) to the Stokes approximate system again:
Now we set \({\Xi }_1=(\varrho _1,\mathfrak {u}_1,\mathfrak {v}_1)+(\xi _1,\phi _1,\psi _1)\) as the approximate solution as the first step, which together with \({\Xi }_0\) generates an error term
Now we can iterate the above process. Given the approximate solution \({\Xi }_j\) as well as the error
in the j-th (\(j\geqq 1\)) step, we define the \(j+1\)-order approximate solution \({\Xi }_{j+1}\) as
where \((\varrho _{j+1},\mathfrak {u}_{j+1},\mathfrak {v}_{j+1})\) is the solution to quasi-compressible system
and \((\xi _{j+1},\phi _{j+1},\psi _{j+1})\) solves the Stokes approximate system
Observe that for each positive integer \(N\geqq 0\), it holds that
where the error term in N-th step is \({\mathcal {E}}_N=\left( 0,-\psi _N\partial _YU_s,0\right) \). Therefore, at this point, formally the series \({\Xi }=\sum _{j=0}^\infty {\Xi }_j\) gives a solution to the original system (3.1).
If in addition \(f_u,f_v\in H^1(\mathbb {R}_+)\) and \(\Vert \Omega (f_u,f_v)\Vert _{L^2_w}<\infty \) where operator \(\Omega \) is defined in (3.2), then we introduce \((\varrho _0,\mathfrak {u}_0,\mathfrak {v}_0)\in H^2(\mathbb {R}_+)^3\) as the solution to the quasi-compressible system
which yields an error term
The new error term \(\mathcal {E}_{-1}\) is in \(L^2(\mathbb {R}_+)^3\). So we can take \({\Upsilon }=(\tilde{\rho },\tilde{u},\tilde{v})\) as the solution to original linear system (3.1) with inhomogeneous source term \(-\mathcal {E}_{-1}\), that is \(\mathcal {L}({\Upsilon })=-{\mathcal {E}}_{-1}\). Then it is clear that \({\Xi }\overset{\hbox {{def}}}{=}(\varrho _0,\mathfrak {u}_0,\mathfrak {v}_0)+{\Upsilon }\) defines a solution to (3.1).
The above iteration can be rigorously justified by proving the convergence of iteration that is given in Proposition 3.1.
Proof of Proposition 3.1
Recall the bounds on the parameters |c| and n in (2.43). We can take \(0<\varepsilon \ll 1\) suitably small such that the following bounds hold for any \(c\in \overline{D_0}:\)
Here the constants \(\gamma _4\) and \(\gamma _5\) are given in Proposition 3.7 and 3.9 respectively. Thus, we have \(\overline{D_0}\subsetneq \Sigma _{Q}\cap \Sigma _{S}\), where \(\Sigma _Q\) and \(\Sigma _S\) are resolvent sets of \(L_Q\) and \(L_S\), which are defined in (3.21) and (3.73) respectively. From (3.99), we know that \((\varrho _{j+1},\mathfrak {u}_{j+1},\mathfrak {v}_{j+1})\) is the solution to quasi-compressible approximation (3.10) with inhomogeneous source term \(s_{1,j+1}=\psi _j\partial _YU_s\), \(s_{2,j+1}=0.\) Then we have
To eliminate the singular factor \(\alpha ^{-1}\), we use the fact that \(\partial _Y\psi _j=\text {div}_\alpha (\phi _j,\psi _j)-i\alpha \phi _j\), the two bounds given in (1.9), (3.20) and Hardy inequality to obtain
Similarly, we get
Thus, by applying bounds given in (3.59) and (3.60) in Proposition 3.7 to \((\varrho _{j+1}, \)\( \mathfrak {u}_{j+1},\mathfrak {v}_{j+1})\) and using (3.104)–(3.106), we obtain
and
Here we have also used \(\alpha \in (0,1)\).
Next according to (3.100), we solve \((\xi _{j+1},\phi _{j+1},\psi _{j+1})\) from the Stokes approximation (3.72) with inhomogeneous source term \(q_{0,j+1}=0\),
By (3.107), we have
Then by applying (3.74)–(3.77) in Proposition 3.9 to \((\xi _{j+1},\phi _{j+1},\psi _{j+1})\), using (3.109) and \(\alpha =n\sqrt{\varepsilon }\), we can deduce that
Set
By the estimates (3.110) and (3.111), we have
Recall (2.43) for the bounds on c and n when \(\alpha =K\varepsilon ^{\frac{1}{8}}\) and \(c\in D_0\). By taking \(\varepsilon _3\in (0,1)\) suitably small so that \(\frac{C}{n(\text {Im}c)^2}\leqq C\varepsilon ^{\frac{1}{8}}<\frac{1}{2}\) for any \(\varepsilon \in (0,\varepsilon _3)\), we can deduce from (3.114) that
Furthermore, by using the bounds obtained in (3.107), (3.108), (3.112)–(3.115) and \(\frac{1}{n^{\frac{1}{2}}(\text {Im}c)^{\frac{3}{2}}}\lesssim 1\) for any \(c\in D_0\), we get
In view of (3.115)–(3.119), we have justified the convergence of \({\Xi }=(\rho ,u,v)=\sum _{j=0}^\infty {\Xi }_j\) in \(H^1(\mathbb {R}_+)\times H^2(\mathbb {R}_+)^2\). This gives the existence of solution. Moreover, Recall (3.95). By applying (3.74)–(3.77) to \({\Xi }_0\) with \(q_0=0\), \(q_1=f_u\) and \(q_2=f_v\), we derive the following estimates:
By summarizing the estimates (3.115)–(3.122), we have
and
Putting (3.123)–(3.126) together yields the estimates (3.3)–(3.6). The analytic dependence on c of the solution \((\rho ,u,v)\) follows from the uniformly convergence. Therefore, the proof of the first part of Proposition 3.1 is completed.
Now we assume that \(f_u,f_v\in H^1(\mathbb {R}_+)\) and \(\Vert \Omega (f_u,f_v)\Vert _{L^2_w}<\infty \). As discussed in the formal presentation of the iteration scheme, we can decompose the solution \((\rho ,u,v)\) into \((\rho ,u,v)=(\varrho _0,\mathfrak {u}_0,\mathfrak {v}_0)+{\Upsilon }=(\varrho _0,\mathfrak {u}_0,\mathfrak {v}_0)+(\tilde{\rho },\tilde{u},\tilde{v})\), where \((\varrho _0,\mathfrak {u}_0,\mathfrak {v}_0)\) is the solution to (3.101) that generates an error \({\mathcal {E}}_{-1}\) defined in (3.102), and \({\Upsilon }\) solves \(\mathcal {L}({\Upsilon })=-{\mathcal {E}}_{-1}\). By (3.59) and (3.60) in Proposition 3.7, we have
and
Then we can estimate the \(L^2\)-bound of the error \({\mathcal {E}}_{-1}\) by
By (3.129), applying (3.3)–(3.6) to \({\Upsilon }=(\tilde{\rho },\tilde{u},\tilde{v})\) leads to
By summarizing the estimates (3.127), (3.128), (3.130)–(3.133) and using the fact that \(n(\text {Im}c)^2\gtrsim 1\) and \(n^{\frac{1}{2}}(\text {Im}c)^{\frac{3}{2}}\gtrsim 1\) for \(c\in D_0\), we derive the following estimates:
Thus, the improved estimates (3.7) and (3.8) are proved. And this completes the proof of the proposition. \(\quad \square \)
4 Proof of Theorem 1.1
Finally, in this section, we prove Theorem 1.1. We construct the solution to linearized system (1.3) with no-slip boundary condition (1.4) in the following form
Here \({\Xi }_{\text {app}}\) is the approximate solution obtained in (2.32) which satisfies (2.44), \({\Xi }_{\text {sm}}=(\rho _{\text {sm}},u_{\text {sm}},v_{\text {sm}})\) and \({\Xi }_{\text {re}}=(\rho _{\text {re}},u_{\text {re}},v_{\text {re}})\) solve the remainder system
and
respectively. By Proposition 2.3 and 3.1, both \({\Xi }_{\text {sm}}\) and \({\Xi }_{\text {re}}\) are well-defined. Moreover, it is straightforward to check that \({\Xi }=(\rho ,u,v)\) satisfies
To recover the no-slip boundary condition on the tangential component, we introduce the mapping
On one hand, from Proposition 2.2, \(\mathcal {F}_{\text {app}}(c)\) is analytic and has a unique zero point in \(D_0\). On the other hand, according to Remark 3.2 (a), both \(u_{\text {sm}}(0;c)\) and \(u_{\text {de}}(0;c)\) are analytic in \(D_0\). Then by applying estimates (3.3), (3.5) to \({\Xi }_{\text {sm}}\) with \((f_u,f_v)=(E_{u,\text {sm}},E_{v,\text {sm}})\), using the bound in (2.49) and the Sobolev inequality, we deduce that
Here we have used (2.43) in the last inequality. For \(u_{\text {re}}(0;c)\), we use the bounds given in (3.7) for \(\Vert u_{\text {re}}\Vert _{H^1}\) with \((f_u,f_v)=(0,E_{v,\text {re}})\) and (2.48) to get that
Thus, by recalling the lower bound of \(|\mathcal {F}_{\text {app}}(c)|\) on the circle \(\partial D_0\) in (2.36), and by using the bounds in (4.2) and (4.3), it holds that
by taking \(\varepsilon \in (0,1)\) suitably small. Therefore, by Rouché’s Theorem, \(\mathcal {F}(c)\) and \(\mathcal {F}_{\text {app}}(c)\) have the same number of zero points in \(D_0\). This justifies the existence of a unique \(c\in D_0\) such that \({\Xi }(Y;c)\) defined in (4.1) solves the linear equation (1.3) with the no-slip boundary condition (1.4). The proof of Theorem 1.1 is completed. \(\quad \square \)
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Acknowledgements
The authors would like to thank the anonymous referees for their valuable comments for improving the writing of the paper. The research of T. Yang was supported by the General Research Fund of Hong Kong (Project No. 11303521) and the NSFC Grant 12171186.
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Appendix
Appendix
Recall (3.14) the definition of operator \(\Lambda \).
Lemma 5.1
(The operator \(\Lambda ^{-1}\)) Let \({\mathcalligra {m}}\in (0,1)\) and \(\alpha \in (0,1)\). For small \(|c|\ll 1\), if \(\Vert h\Vert _{L^2}<\infty \), there exists a unique solution \(\psi \in H^2(\mathbb {R}_+)\cap H^1_0(\mathbb {R}_+)\) to \(\Lambda (\psi )=h\) in \(\mathbb {R}_+\), \(\psi |_{Y=0}=0\) which satisfies the following estimates
Moreover, the solution mapping \(\Lambda ^{-1}(\cdot ~;c): L^2(\mathbb {R}_+)\rightarrow H^2(\mathbb {R}_+)\cap H^1_0(\mathbb {R}_+)\) is analytic in c.
Proof
We first establish the a priori estimate (5.1). Denote \(\langle \cdot ,\cdot \rangle \) the standard inner product in \(L^2(\mathbb {R}_+)\). Taking inner product with \(-\bar{\psi }\) in the equation \(\Lambda (\psi )=h\) gives that
Note that
Then for suitably small |c|, the real part of (5.2) yields
which gives \(H^1\)-estimate of \(\psi \). The estimate of \(\partial _Y^2\psi \) can be obtained by using the equation to have
If in addition, \((1+Y)h\in L^2(\mathbb {R}_+)\), we can obtain \(|\langle h,\bar{\psi } \rangle |\leqq \Vert Y^{-1}\psi \Vert _{L^2}\Vert Yh\Vert _{L^2}\leqq C\Vert \partial _Y\psi \Vert _{L^2}\Vert Yh\Vert _{L^2}\) by using the Hardy inequality for the term \(Y^{-1}\psi \). Thus real part of (5.2) yields the \(H^1\)-estimate
The estimate on \(\partial _Y^2\psi \) then follows from the equation and \(H^1\)-estimate (5.5) similar to the estimation in (5.4). Hence the estimate (5.1) holds. The uniqueness of solution follows from the a priori estimate (5.1).
Set
For \({\mathcalligra {m}}\in (0,1)\), the operator \(\Lambda _0\) is clearly uniformly elliptic and invertible in \(L^2(\mathbb {R}_+)\). Then by (5.3), the existence and analytic dependence on c of \(\Lambda ^{-1}\) follow from the standard perturbative argument. Hence, we omit the details for brevity. The proof of the lemma is completed. \(\quad \square \)
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Yang, T., Zhang, Z. Linear Instability Analysis on Compressible Navier–Stokes Equations with Strong Boundary Layer. Arch Rational Mech Anal 247, 83 (2023). https://doi.org/10.1007/s00205-023-01919-z
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DOI: https://doi.org/10.1007/s00205-023-01919-z