Operators
For our convenience, let us introduce the following operators We denote by Orr the Orr- Sommerfeld operator
Orr(φ) := (U ưc)(∂ z 2 ưα 2 )φưU ′′ φưε(∂ z 2 ưα 2 ) 2 φ, (2.1) by Rayα the Rayleigh operator
Ray α (φ) := (Uưc)(∂ z 2 ưα 2 )φưU ′′ φ, (2.2) by Dif f the diffusive part of the Orr-Sommerfeld operator,
Dif f(φ) :=−ε(∂ z 2 −α 2 ) 2 φ, (2.3) by Airythe modified Airy equation
Airy(φ) :=ε∂ z 4 φ−(U −c+ 2εα 2 )∂ z 2 φ, (2.4) and finally, byReg the regular zeroth order part of the Orr-Sommerfeld operator
Orr=Ray α +Dif f=−Airy+Reg (2.6)
Asymptotic behavior as z → + ∞
To construct independent solutions of the equation, we analyze their behavior as \( z \) approaches infinity It is observed that solutions must resemble those of the constant-coefficient limiting equation: \( \varepsilon \partial_z^4 \phi = (U^+ - c + 2\varepsilon \alpha^2) \partial_z^2 \phi - \alpha^2 (\varepsilon \alpha^2 + U^+ - c) \phi \), where \( U^+ = U(+\infty) \) The solutions to this limiting equation take the form \( Ce^{\lambda z} \), with \( \lambda \) being either \( \pm \lambda_s \) or \( \pm \lambda_f \), where \( \lambda_s = \pm \alpha + O(\alpha^2 \sqrt{\varepsilon}) \) and \( \lambda_f = \pm 1 \).
In the analysis of the system, we identify two slow-behavior solutions, φ₁ and φ₂, characterized by eigenvalues λ approximately equal to ±α, with one solution decaying and the other growing Additionally, we discover two fast-behavior solutions, φ₃ and φ₄, where the eigenvalues λ are on the order of ±1/√ε, again with one decaying and the other growing Notably, the slow solutions φ₁ and φ₂ serve as perturbations of the eigenfunctions derived from the Rayleigh equation, while the fast solutions φ₃ and φ₄ are specifically associated with the Orr-Sommerfeld equation and relate to the classical Airy equation Thus, we aim to construct four independent solutions of the given equation.
• φ 1 and φ 2 which are perturbations of the decreasing/increasing eigenvector of the Rayleigh equation To leading order in small α and ε, φ 1 and φ 2 behave at infinity, respectively, like (U(z)−c) exp(−αz) and (U(z)−c) exp(αz).
The perturbations φ3 and φ4 of the second primitive Airy equation exhibit behavior characterized by the order exp(±|Z| 3/2) as |Z| approaches infinity In this context, Z is defined as η(z)/ǫ 1/3, representing a fast variable near the critical layer, with a size proportional to ǫ 1/3 Additionally, η(z) corresponds to Langer’s variable, which asymptotically behaves like z 2/3 as z tends toward infinity.
A solution to the equations (1.9)–(1.11) is expressed as a linear combination of the functions φ 1, φ 2, φ 3, and φ 4, which must satisfy the specified boundary conditions Notably, considering the asymptotic behavior of φ 2 and φ 4, we find that any bounded solution to these equations can be represented solely as a combination of φ 1 and φ 3 Consequently, our analysis will focus exclusively on φ 1 and φ 3.
Outline of the construction
We now present the idea of the iterative construction We start from the Rayleigh solution φRay so that
The error term Dif f(φ Ray ) = ǫ(∂ z 2 − α 2 ) 2 φ Ray, represented by O 1 (z), is of order O(ǫ) in L ∞ It's important to highlight that the operator ∂ z 2 − α 2 and so Dif f(ã) eliminate the slow decay term O(e − αz ) in φ Ray In proximity to the critical layer, the Rayleigh solution typically features a singular solution of the form (z−z c ) log(z−z c ), resulting in φRay exhibiting a similar singularity at z = zc Consequently, Dif f(φRay) encompasses singularities of orders log(z−z c ) and (z−z c ) − k, for k = 1, 2, 3 To address these singularities, we employ the Airy operator, which effectively smoothes out the singularity within the critical layer However, in terms of spatial decay at infinity, the inverse of the Airy(ã) operator introduces linear growth in the spatial variable, hindering the convergence of our iteration Therefore, we introduce a modified Airy operator Aa(ã) to resolve this issue.
In our construction, we define φ₁ as φ_Ray + Airy - 1(A_s) + ∂z - 2A - a₁(I₀), where A_s = χ_Diff(φ_Ray) represents the singular component, and I₀ = (1 - χ)Diff(φ_Ray) indicates the regular component Additionally, ∂z - 1 is defined as -R ∞ z The function χ(z) is a smooth cut-off function that equals 1 on the interval [0, 1] and transitions to 0 for values greater than or equal to 2.
Our main technical task is to show that O 1 is indeed in the next vanishing order, when ǫ→0, or precisely the iteration operator
Ray α − 1 (2.10) demonstrates contractive properties within appropriate function spaces Our methodology circumvents the need for inner and outer expansions, focusing instead on a thorough examination of singularities and precise estimates of the resolvent solutions.
Function spaces
Throughout the paper, z c is some complex number and will be fixed, depending only onc,through U(z c ) =c.
We will use the function spaces Xp η , for p ≥ 0, to denote the spaces consisting of measurable functionsf =f(z) such that the norm kfkX p η := sup
|e ηz ∂ z k f(z)| is bounded In case p= 0, we simply writeXη,k ã k η in places of X 0 η ,k ã kX 0 η , respectively.
We also introduce the function spaces Y p η ⊂X p η , p≥0, such that for any f ∈Y p η , the functionf additionally satisfies
|f(z)| ≤C, |∂ z f(z)| ≤C(1 +|log(z−z c )|), |∂ k z f(z)| ≤C(1 +|z−z c | 1 − k ) for all |z−z c | ≤1 and for 2≤k≤p The best constantC in the previous bounds defines the normkfkY p η
This paper presents key estimates, highlighting the ability to construct an inverse operator Ray − 1 for Ray α through nearly explicit computations Notably, if Ray α (φ) equals f, then the relationship between the operators can be established.
If \( U - c \) does not vanish, particularly when \( c \) is complex, classical elliptic regularity indicates that if \( f \in C^k \), then \( \phi \in C^{k+2} \), resulting in a gain of two derivatives However, the quality of these derivative estimates diminishes as \( z - z_c \) decreases The key factor is that the weight \( (z - z_c)^l \) effectively manages this singularity Additionally, by differentiating equation (2.11) \( l \) times, we find that \( \partial_z^{2+l} \phi \) is bounded by \( C/(z - z_c)^{l+1} \) if \( f \in X_{\eta,k} \), thus introducing an additional \( z - z_c \) factor in the derivative estimates between \( f \) and \( \phi \) Moreover, since \( e^{\pm \alpha z} \) is in the kernel of \( \partial_z^2 - \alpha^2 \), if \( f \) decays like \( e^{-\eta z} \), we can expect \( \phi \) to decay at most as \( e^{-\alpha z} \) at infinity Consequently, if \( f \) belongs to \( X_{k,\eta} \), then \( \phi \) will be in \( Y_{k+2,\alpha} \), achieving a gain of two derivatives and an additional \( z - z_c \) weight, albeit with a loss of rapid decay at infinity We will also construct an inverse \( Ray^{-1} \) that is continuous from \( X_{k,\eta} \) to \( Y_{k+2,\alpha} \) for any \( k \).
By utilizing Airy functions, their double primitives, and the Langer transformation, we can nearly explicitly construct the inverse Airy operator, denoted as Airy − 1 Next, we analyze the composition of Airy − 1 with the differential operator, referred to as Airy − 1 ◦Dif f Although this operator is formally of order 0, it is singular, necessitating the use of two derivatives for control To minimize its impact, we require a factor of az−z c in the norms Through extensive calculations involving nearly explicit Green functions, we demonstrate that Airy − 1 ◦Dif f maintains a small norm as an operator from Y k+2 α to X k η.
Reg is constrained within the bounds of X k η due to the simple multiplication by a bounded function By integrating these estimates, we can derive exact solutions to the Orr-Sommerfeld equations, originating from the Rayleigh or Airy equations This process results in the formulation of four independent solutions, each represented as a convergent series that provides its expansion Ultimately, we combine the various terms from all these solutions to establish the dispersion relation of the Orr-Sommerfeld equations, with a thorough analysis revealing our instability result.
The plan of the paper follows the previous lines.
In this part, we shall construct an exact inverse for the Rayleigh operator Ray α for small α and so find the complete solution to
To achieve our goal, we begin by inverting the Rayleigh operator Ray 0 at α = 0 by presenting an explicit Green function Next, we utilize this inverse to develop an approximate inverse for the Ray α operator, achieved through the creation of an approximate Green function Ultimately, we derive the exact inverse of Ray α using an iterative procedure.
Precisely, we will prove in this section the following proposition.
Proposition 3.1 Let pbe in{0,1,2} andη >0 Assume that Im c6= 0 andα|logImc|is sufficiently small Then, there exists an operatorRaySolverα, ∞(ã)fromXp η toY p+2 α (defined by (3.20)) so that
In addition, there holds kRaySolver α, ∞ (f)kY p+2 α ≤CkfkX η p(1 +|log(Im c)|),for allf ∈Xp η
Case α = 0
As mentioned, we begin with the Rayleigh operator Ray0 when α = 0 We will find the inverse of Ray 0 More precisely, we will construct the Green function of Ray 0 and solve
We recall that z c is defined by solving the equation U(z c ) = c We first prove the following lemma.
Lemma 3.2 Assume that Im c 6= 0 There are two independent solutions φ 1,0 , φ 2,0 of Ray 0 (φ) = 0 with the Wronskian determinant
Furthermore, there are analytic functions P 1 (z), P 2 (z), Q(z) with P 1 (z c ) =P 2 (z c ) = 1 and Q(zc)6= 0 so that the asymptotic descriptions φ1,0(z) = (z−zc)P1(z), φ2,0(z) =P2(z) +Q(z)(z−zc) log(z−zc) (3.4) hold for z near zc, and
V+| ≤Cze − η 0 | z | , (3.5) as |z| → ∞, for some positive constants C, η0 and forV+=U+−c Here when z−zc is on the negative real axis, we take the value of log(z−z c ) to belog|z−z c | −iπ.
The function φ 1,0 (z) = U(z) - c serves as an exact solution to the equation Ray 0 (φ) = 0, demonstrating that the asymptotic expansion for φ 1,0 is valid in proximity to z c, where U(z c) equals c Subsequently, we develop a second particular solution, φ 2,0, by enforcing the Wronskian determinant to equal one.
From this, the variation-of-constant method φ 2,0 (z) =C(z)φ 1,0 (z) then yields φ 1,0 C∂ z φ 1,0 + φ 2 1,0 ∂ z C−∂ z φ 1,0 Cφ 1,0 = 1.
This gives∂zC(z) = 1/φ 2 1,0 (z) and therefore φ 2,0 (z) = (U(z)−c)
Note thatφ 2,0 is well defined if the denominator does not vanishes, hence if Imc6= 0 or if
As φ 2,0 is not properly defined for z < z c when z c ∈ R + , it is coherent to choose the determination of the logarithm which is defined onC−R −
The logarithm choice ensures that φ2,0 is holomorphic in C− {zc + R−} Specifically, if ℑzc = 0, φ2,0 remains holomorphic in z, except along the half-line zc + R− For real values of z, φ2,0 is holomorphic as a function of c, except when z − zc is real and negative, which occurs when z < zc Additionally, for a fixed z, φ2,0 is holomorphic in c as long as zc does not cross.
R + , and provided z−zc does not cross R− The Lemma then follows from the explicit expression (3.7) of φ 2,0
Let φ 1,0 , φ 2,0 be constructed as in Lemma 3.2 Then the Green function G R,0 (x, z) of theRay 0 operator can be defined by
In this discussion, we observe that the complex variable \( c \) has a non-zero imaginary part, specifically \( \text{Im} \, c \neq 0 \) Consequently, the Green function \( G_{R,0}(x, z) \) is a well-defined and continuous function for the variables \( (x, z) \), maintaining continuity at the point \( x = z \) However, it is important to note that the first derivative of this function experiences a discontinuity at the same point We will now proceed to define the inverse of \( Ray_0 \).
The following lemma asserts that the operatorRaySolver0(ã) is in fact well-defined from
X 0 η to Y 2 0 , which in particular shows that RaySolver 0 (ã) gains two derivatives, but losses the fast decay at infinity.
Lemma 3.3 Assume that Im c 6= 0 For any f ∈ X 0 η , the function RaySolver 0 (f) is a solution to the Rayleigh problem (3.3) In addition, RaySolver 0 (f)∈Y 2 0 , and there holds kRaySolver 0 (f)kY 2 0 ≤C(1 +|logImc|)kfkX 0 η , for some universal constant C.
Proof As long as it is well-defined, the functionRaySolver0(f)(z) solves the equation (3.3) at once by a direct calculation, upon noting that
Next, by scaling, we assume thatkfkX 0 η = 1 By Lemma 3.2, it is clear thatφ 1,0 (z) and φ 2,0 (z)/(1 +z) are uniformly bounded Thus, by direct computations, we have
That is,G R,0 (x, z) grows linearly in x for large xand has a singularity of order |x−z c | − 1 whenx is nearz c , for arbitraryz≥0 Since|f(z)| ≤e − ηz , the integral (3.8) is well-defined and satisfies
0 e − ηx max{(1 +x),|x−z c | − 1 }dx≤C(1 +|log Imc|), in which we used the fact that Imzc ≈Imc.
Finally, as for derivatives, we need to check the order of singularities for z nearz c We note that |∂ z φ 2,0 | ≤C(1 +|log(z−z c )|), and hence
Thus,∂ z RaySolver 0 (f)(z) behaves as 1 +|log(z−z c )|near the critical layer In addition, from the Ray0 equation, we have
This proves thatRaySolver 0 (f)∈Y 2 0 by definition of the function spaceY 2 0
Lemma 3.4 Assume that Im c6= 0 Letp be in {0,1,2} For any f ∈Xp η , we have kRaySolver 0 (f)kY p+2 0 ≤CkfkX p η (1 +|log(Imc)|)
Proof This is Lemma 3.3 when p = 0 When p= 1 or 2, the lemma follows directly from the identity (3.10).
Case α 6 = 0: an approximate Green function
In this article, we examine the two solutions, φ 1,0 and φ 2,0, of the Rayleigh equation Ray 0 (φ) = 0, as established in Lemma 3.2 It is observed that the solutions to Ray 0 (φ) = f approach a constant value as z approaches +∞, specifically φ 1,0 converging to U + −c Furthermore, we proceed to develop normal mode solutions to the Rayleigh equation for the case where α is not equal to zero.
The spatially asymptotic limit of the Rayleigh equation reveals two normal mode solutions, characterized by their behaviors as e ± αz at infinity To analyze the mode exhibiting the behavior of e − αz, we define φ1,α=φ1,0e − αz and φ2,α=φ2,0e − αz.
A direct calculation shows that the Wronskian determinant
W[φ1,α, φ2,α] =∂zφ2,αφ1,α−φ2,α∂zφ1,α=e − 2αz is non zero In addition, we can check that
We are then led to introduce an approximate Green function G R,α (x, z) defined by
The Green function GR,α(x, z) exhibits singular behavior near z = zc, characterized by two distinct sources of singularities: one originating from the term 1/(U(x)−c) when x is close to zc, and the other stemming from the (z−zc) log(z−zc) singularity of φ2,0(z) This can be further understood through the perspective provided in equation (3.13).
Ray α (G R,α (x, z)) =δ x −2α(U −c)E R,α (x, z), (3.14) for each fixedx Here the error term E R,α (x, z) is defined by
We then introduce an approximate inverse of the operator Ray α defined by
G R,α (x, z)f(x)dx (3.15) and the error remainder
Lemma 3.5 Assume that Imc6= 0, and let p be 0,1, or 2 For any f ∈X p η , with η > α, the function RaySolver α (f) is well-defined in Y p+2 α , satisfying
Furthermore, there hold kRaySolverα(f)k Y p+2 α ≤C(1 +|logIm c|)kfkX p η , (3.17) and kErr R,α (f)kY p η ≤Cα(1 +|log(Im c)|)kfkX p η , (3.18) for some universal constant C.
The proof is analogous to that of Lemmas 3.3 and 3.4, demonstrating that the order of singularities near the critical layer can be established in the same manner as these previous lemmas.
Let us check the right behavior at infinity Consider the casep= 0 and assumekfkX 0 η 1 Similarly to the estimate (3.9), Lemma 3.2 and the definition of G R,α yield
|GR,α(x, z)| ≤Ce − α(z − x) max{(1 +x),|x−zc| − 1 }. Hence, by definition,
0 e αx e − ηx max{(1 +x),|x−z c | − 1 }dx which is clearly bounded byC(1 +|log Imc|)e − αz This proves the right exponential decay of RaySolver α (f)(z) at infinity, for allf ∈X 0 η
Next, by definition, we have
The exponential decay of Err R,α (f)(z) at infinity is directly derived from the integral representation of f(z) and its associated derivative ∂ z φ 1,0 (z) To fully understand this behavior, it is essential to analyze the order of singularity in proximity to the critical layer, particularly for bounded values of z.
The inequality |E R,α (x, z)| is bounded by the expression C(1 + |log(z−z c)|)e αx max{1, |x−z| − 1} This lemma can be established by utilizing the additional factor of U−c in front of the integral (3.16) to effectively control the log(z−z c) component Furthermore, the estimates for the derivatives can be derived in a similar manner.
Case α 6 = 0: the exact solver for Rayleigh
We now construct the exact solver for the Rayleigh operator by iteration Let us denote
It then follows thatRay α (S 0 )(z) =f(z) +E 0 (z) Inductively, we define
S n (z) :=−RaySolver α (E n − 1 )(z), E n (z) :=−Err R,α (E n − 1 )(z), forn≥1 It is then clear that for all n≥1,
This leads us to introduce the exact solver for Rayleigh defined by
The proof of Proposition 3.1 demonstrates that as \( n \) approaches infinity, \( E_n \) converges to 0 in \( X_\eta \) provided that \( \alpha \log \text{Im} c \) remains sufficiently small Specifically, the inequality \( \| E_n \|_\eta \leq C n \alpha^n (1 + |\log(\text{Im} c)|)^n \| f \|_\eta \) supports this convergence Furthermore, the analysis in (3.17) indicates that \( \| \text{RaySolver}_\alpha(E_n) \|_{Y^2_\alpha} \) is bounded by \( C n \alpha^n (1 + |\log(\text{Im} c)|)^n \| f \|_\eta \), reinforcing the behavior of the series \( \sum_{n \geq 0} \).
(−1) n RaySolver α (E n )(z) converges inY 2 α , assuming that αlog Imc is small.
Taking the limit as n approaches infinity in equation (3.19) confirms that equation (3.2) is valid, at least in the distributional sense The estimates for values of z close to zc can be derived from analogous estimates on RaySolver α (ã), as demonstrated in Lemma 3.5 Consequently, the proof of Proposition 3.1 is complete.
Exact Rayleigh solutions
We shall construct two independent exact Rayleigh solutions by iteration, starting from the approximate Rayleigh solutionsφ j,α defined as in (3.12).
Lemma 3.6 For α small enough so that α|logImc| ≪ 1, there exist two independent functions φ Ray, ± ∈e ± αz L ∞ such that
Furthermore, we have the following expansions in L ∞ : φ Ray, − (z) =e − αz
. φ Ray,+ (z) =e αz O(1), as z→ ∞ At z= 0, there hold φ Ray, − (0) =U 0 −c+α(U + −U 0 ) 2 φ 2,0 (0) +O(α(α+|z c |)) φRay,+(0) =αφ2,0(0) +O(α 2 ) withφ 2,0 (0) = U 1 ′ c − 2U U c ′ c ′′ 2z c logz c +O(z c ).
Proof Let us start with the decaying solution φRay, −, which is now constructed by induc- tion Let us introduce ψ 0 =e − αz (U−c), e 0 =−2α(U −c)U ′ e − αz , and inductively for k≥1, ψ k =−RaySolver α (e k − 1 ), e k =−Err R,α (e k − 1 ).
By definition, it follows that
We observe that ke0k η+α ≤Cα and kψ0k α ≤C Inductively for k≥1, by the estimate (3.18), we have ke k k η+α ≤Cα(1 +|log(Imc)|)ke k − 1 k η+α ≤Cα(Cα(1 +|log(Imc)|)) k − 1 , and by Lemma 3.5, kψ k kα ≤C(1 +|log(Im c)|)ke k − 1 kη+α≤(Cα(1 +|log(Im c)|)) k
Thus, for sufficiently small α, the series φ N converges inX α and the error terme N →0 in
X η+α This proves the existence of the exact decaying Rayleigh solution φ Ray, − inX α , or ine − αz L ∞
As for the growing solution, we simply define φ Ray,+ =αφ Ray, − (z)
By definition, φRay,+ solves the Rayleigh equation identically Next, since φRay, −(z) tends to e − αz (U + −c+O(α)), φ Ray,+ is of order e αz asz→ ∞.
Finally, at z= 0, we have ψ 1 (0) =−RaySolver α (e 0 )(0) =−φ 2,α (0)
From the definition, we haveφ Ray, − (0) =U 0 −c+ψ 1 (0) +O(α 2 ) This proves the lemma,upon using that U 0 −c=O(z c ).
Our ultimate goal is to inverse the Airy operator defined as in (2.4), and thus we wish to construct the Green function for the primitive Airy equation
Classical Airy equations
Classical Airy functions are essential for analyzing phenomena near the critical layer This section will highlight key properties of these functions, which are governed by the classical Airy equation.
In connection with the Orr-Somerfeld equation with ǫbeing complex, we are interested in the Airy functions with argument z=e iπ/6 x, x∈R.
Let us state precisely what we will be needed These classical results can be found in [9, 15]; see also [1, Appendix].
Lemma 4.1 The classical Airy equation (4.2) has two independent solutions Ai(z) and Ci(z) so that the Wronskian determinant of Ai and Ciequals
W(Ai, Ci) =Ai(z)Ci ′ (z)−Ai ′ (z)Ci(z) = 1 (4.3)
In addition,Ai(e iπ/6 x)andCi(e iπ/6 x)converge to0asx→ ±∞(xbeing real), respectively. Furthermore, there hold asymptotic bounds:
2 | x | x/3, k∈Z, x∈R, (4.5) in whichAi(0, z) =Ai(z), Ai(k, z) =∂ z − k Ai(z) for k≤0, and Ai(k, z) is the k th primitives of Ai(z) for k≥0 and is defined by the inductive path integrals
Ai(k−1, w)dw so that the integration path is contained in the sector with |arg(z)| < π/3 The Airy functions Ci(k, z) for k6= 0 are defined similarly.
The following lemma whose proof can again be found in the mentioned physical refer- ences will be of use in the latter analysis.
Lemma 4.2 Let S1 be the sector in the complex plane such that the argument is between 2π/3 and 4π/3 There hold expansions
2√ πz 5/4 e − 2 3 z 3/2 (1 +O(|z| − 3/2 )) for all large z in S 1 In addition, at z= 0, there holds
Ai(k,0) = (−1) k 3 − (k+2)/3 Γ( k+2 3 ) , k∈Z, in which Γ(ã) is the Gamma function defined by Γ(z) =R ∞
Langer transformation
To analyze the profileU's dependence effectively, we apply a change of variables known as Langer's transformation, which is commonly referenced in physical literature This transformation involves redefining the variables from (z, φ) to (η, Φ), with η being defined as η(z) = h3 By utilizing this transformation, we can revert to the classical Airy equations discussed in the previous section.
(4.6) and Φ = Φ(η) defined by the relation
∂ z 2 φ(z) = ˙z 1/2 Φ(η), (4.7) in which z˙= dz(η) dη and z=z(η) is the inverse of the map η=η(z).
Direct calculation gives a useful fact (U −c) ˙z 2 =U c ′ η Next, using thatc=U(z c ), one observes that forz nearzc, we have η(z) =h3
In particular, we have η ′ (z) = 1 +O(|z−z c |), (4.9) and thus the inverse z=z(η) is locally well-defined and locally increasing near z=z c In addition, ˙ z= 1 η ′ (z) = 1 +O(|z−z c |).
Next, we note that η ′ (z) 2 = U U ′ − c c η(z), which is nonzero away from z =zc Thus, the inverse of η=η(z) exists for all z≥0.
In addition, by a view of the definition (4.6) and the fact that (U−c) ˙z 2 =U c ′ η, we have
The following lemma links (4.1) with the classical Airy equation.
Lemma 4.4 Let (z, φ) 7→ (η,Φ) be the Langer’s transformation defined as in Definition 4.3 Assume that Φ(η) solves ǫ∂ η 2 ΦưU c ′ ηΦ =f(η).
Airy(φ) = ˙z − 3/2 f(η(z)) +ǫ[∂ z 2 z˙ 1/2 z˙ − 1/2 −2α 2 ]∂ 2 z φ(z) Proof Derivatives of the identity ∂ z 2 φ(z) = ˙z 1/2 Φ(η) are
∂ z 2 ( ˙z 1/2 Φ(η)) = ˙z − 3/2 ∂ η 2 Φ(η) +∂ z 2 z˙ 1/2 Φ(η) (4.12) Putting these together and using the fact that (U−c) ˙z 2 =U c ′ η, we get ε∂ z 4 φ−(U(z)−c)∂ z 2 φ=ǫz˙ − 3/2 ∂ η 2 Φ−(U −c) ˙z 1/2 Φ +ǫ∂ z 2 z˙ 1/2 Φ
Resolution of the modified Airy equation
In this section we will construct the Green function for the Airy equation:
=e − iπ/6 (αRU c ′ ) − 1/3 , and introduce the notation X = δ − 1 η(x) and Z = δ − 1 η(z), where η(z) is the Langer’s variable defined as in (4.6) We define an approximate Green function for the Airy equation:
Ai(Z)Ci(X), if x < z, (4.14) with ˙x= ˙z(η(x)) It follows that Ga(x, z) satisfies the jump conditions acrossx=z:
By definition, we have ε∂ z 2 G a (x, z)−(U −c)G a (x, z) =δ x (z) +E a (x, z), (4.15) withE a (x, z) =iπη ′′ (z) ˙xAi(X)Ci ′ (Z).
To prepare for the upcoming sections, we will outline some estimates on G a, specifically considering the scenario where x is less than z Utilizing the estimates on the Airy functions derived from Lemma 4.1, we can draw significant conclusions.
For the case where \( x \) and \( z \) are bounded away from zero and \( \text{fork} \geq 0 \), similar bounds can be established when \( x > z \) Notably, the polynomial growth in \( x \) within the estimate can be substituted with the growth in \( z \), allowing for an exponentially decaying term.
We obtain the following lemma.
Lemma 4.5 Let G a (x, z) be the approximate Green function defined as in (4.14), and
E a (x, z) as defined in (4.15) Also let X =η(x)/δ and Z =η(z)/δ For k, ℓ = 0,1, there hold pointwise estimates
Proof The lemma follows directly from (4.16), upon noting that he pre-factor in terms of the lower case zis due to the Langer’s change of variables.
Let us next give a few convolution estimates.
Lemma 4.6 Let G a (x, z) be the approximate Green function defined as in (4.14), and
E a (x, z)as defined in (4.15) Also letf ∈X η , for some η >0 Then there is some constant
Proof Without loss of generality, we assumekfk η = 1 Fork= 0,1, using the bounds from Lemma 4.5 and noting thatZ =η(z)/δ≈(1 +|z|) 2/3 /δ asz becomes large, we obtain
Here, we have used the change of variable dx = δz˙ − 1 dX with ˙z ≈ (1 +|x|) 1/3 Similar estimates hold forE a (x, z) This completes the proof of the lemma.
An approximate solution Φ of (4.13) is given by the convolution
Aa(A − a 1 (f)) =f+Err a (f), with the error term defined by
The convolution lemma (Lemma 4.6) establishes that the error term kErr a (f)kη is bounded by Cδkfkη for all f in X η, indicating that Err a (f) is of order O(δ) in X η Consequently, we can iteratively define an exact solver for the Airy operator Aa(ã) Starting with a fixed f in X η, we define φ n as -A − a 1 (E n − 1).
E n =−Err a (E n − 1 ) (4.22) for all n≥1, with E 0 =f Let us also denote
It follows by induction that
Aa(S n ) =f+E n , for all n≥1 Now by (4.21), we have kE n kη ≤CδkE n − 1 kη ≤(Cδ) n kfkη.
As n approaches infinity, it is demonstrated that E_n converges to zero in the space X_η, given that δ is small According to Lemma 4.6, the inequality kφ_n k_η ≤ Cδ^(-1) kE_n - 1 k_η ≤ Cδ^(-1) (Cδ)^(n - 1) kf k_η holds true This indicates that φ_n also converges to zero in X_η as n approaches infinity, thereby confirming the convergence of the series.
S n →S ∞ inX η asn→ ∞, for someS ∞ ∈X η We then denote A − a, 1 ∞(f) =S ∞ , for eachf ∈X η In addition, we have A a (S ∞ ) = f,that is, A − a, 1 ∞(f) is the exact solver for the modified Airy operator A similar estimate follows for derivatives.
To summarize, we have proved the following proposition.
Proposition 4.7 Assume thatδ is sufficiently small There exists an exact solverA − a, 1 ∞(ã) as a well-defined operator from X η toX η , for arbitrary fixed η >0, so that
In addition, there holds kA − a, 1 ∞(f)kX k η ≤Cδ − 1 − k kfk η , k= 0,1,for some positive constant C.
An approximate Green function of primitive Airy equation
In this section we will construct an approximate Green function for (4.1) By a view of the Langer’s transformation, let us introduce an auxiliary Green function
Ai(X)Ci(Z), if ξ > η, Ai(Z)Ci(X), if ξ < η.
By definition, we have ε∂ η 2 G aux (X, Z)−U c ′ ηG aux (X, Z) =δ ξ (η) (4.23)
Next, let us take ξ = η(x) and η = η(z), where η(ã) is the Langer’s transformation and denote ˙x = 1/η ′ (x) and ˙z= 1/η ′ (z) By a view of (4.7), we define the function G(x, z) so that
∂ z 2 G(x, z) = ˙x 3/2 z˙ 1/2 G aux (δ − 1 η(x), δ − 1 η(z)), (4.24) in which the factor ˙x 3/2 was added simply to normalize the jump ofG(x, z) It then follows from Lemma 4.4 together withδ η(x) (η(z)) =δ x (z) that
The function G(x, z) serves as an approximate Green function for the primitive Airy operator ǫ∂ z 4 −(U −c)∂ z 2, with a minor error term of order ǫ∂ z 2 G=O(δ) The next step is to solve equation (4.24) for G(x, z), while ensuring that the jump conditions on G(x, z) are maintained across x=z.
In view of primitive Airy functions, let us denote
Thus, together with our convention that the Green functionG(x, z) should vanish aszgoes to +∞ for each fixedx, we are led to introduce
(4.26) in whicha 1 (x), a 2 (x) are chosen so that the jump conditions (see below) hold Clearly, by definition, G(x, z) solves (4.24), and hence (4.25) Here the jump conditions on the Green function read:
From the analysis of equation (4.24) and the jump conditions on G aux (X, Z) at the boundary X = Z, we can easily derive the corresponding jump conditions for ∂²z G and ∂z³ G To establish the jump conditions for G(x, z) and ∂z G(x, z), we define a₁(x) as Ci(δ - 1 η(x))Ai(1, x)e^(-Ai(δ - 1 η(x))Ci(1, x)) and a₂(x) as Ci(δ - 1 η(x))Ai(2, x)e^(-Ai(δ - 1 η(x))Ci(2, x)).
We obtain the following lemma.
Lemma 4.8 Let G(x, z) be defined as in (4.26) Then G(x, z) is an approximate Green function of the Airy operator (4.1) Precisely, there holds
Airy(G(x, z)) =δ x (z) +Err A (x, z) (4.30) for each fixed x, where Err A (x, z) denotes the error kernel defined by
It appears convenient to denote by G(x, z) ande E(x, z) the localized and non-localized part of the Green function, respectively Precisely, we denote
( Ai(δ − 1 η(x))Ci(2, z),e if x > z, Ci(δ − 1 η(x))Ai(2, z),e if x < z, and
Let us give some bounds on the Green function, using the known bounds on Ai(ã) and Ci(ã) We have the following lemma.
Lemma 4.9 Let G(x, z) = G(x, z) +e E(x, z) be the Green function defined as in (4.26), and let X=η(x)/δ and Z =η(z)/δ There hold pointwise estimates
√| Z || X − Z | (4.32) Similarly, for the non-localized term, we have
Proof We recall that forznearz c , we can write ˙z(η(z)) = 1+O(|z−z c |),which in particular yields that 1 2 ≤z(η(z))˙ ≤ 3 2 forzsufficiently nearz c In addition, by a view of the definition (4.6), η(z) grows like (1 +|z|) 2/3 as z→ ∞; see (4.10).
Let us assume that z ≥ 1 It suffices to give estimates on Ai(k, z),e C i(k, z) Withe notation Y =η(y)/|δ|, we have
For values of z less than or equal to 1, the previous estimates continue to hold true Specifically, when z is greater than or equal to Re z c, the relationship |Y| ≥ |Z| is maintained for y greater than or equal to z, allowing the estimates on Ai(k, z) to be derived similarly as previously discussed Additionally, the scenario where z is less than or equal to Re z c also requires consideration.
In this case, we have
That is, like Ai(η(z)/δ), the functions Ai(k, z) grow exponentially fast ase z tends to zero and is away from the critical layer Similarly, we also have
|Ci(1, z)e | ≤C(1 +|Z|) − 3/4 e − √ 2 | Z | 3/2 /3 , |Ci(2, z)e | ≤C(1 +|Z|) − 5/4 e − √ 2 | Z | 3/2 /3 , for z ≤ Re z c The estimates become significant when the critical layer is away from the boundary layer, that is whenδ ≪ |z c |.
By integrating the established bounds on Ai(ã) and Ci(ã), we can easily derive the proposed bounds on G(x, z) Additionally, we achieve derivative bounds through a similar methodology Ultimately, leveraging the previously mentioned bounds on Ai(k, z) and Ci(k, z) leads us to our final results.
|∂ x k a 2 (x)| ≤Cδ − k (1 +x) 5/6 − k/3 (1 +|X|) k/2 − 3/2 , (4.34) upon noting that the exponents in Ai(ã) and Ci(ã) are cancelled out identically.
This completes the proof of the lemma.
Similarly, we also obtain the following simple lemma.
Lemma 4.10 Let Err A (x, z) be the error kernel defined as in (4.31), and letX =η(x)/δ and Z =η(z)/δ There hold
Thus, the lemma follows directly from the estimates on ∂ z 2 G(x, z).
Convolution estimates
In this section, we establish the following convolution estimates.
Lemma 4.11 Let G(x, z) = G(x, z) +e E(x, z) be the approximate Green function of the primitive Airy equation constructed as in Lemma 4.8, and letf ∈X η , η >0 Then there is some constant C so that
∂ z k E(x,ã)f(x)dx η ′ ≤ C η−η ′ kfkη, (4.37) for k= 0,1,2 and for η ′ < η.
Proof Without loss of generality, we assume kfkη = 1 First, consider the case |z| ≤ 1. Using the pointwise bounds obtained in Lemma 4.9, we have
√| Z || X − Z |e − ηx dx≤Cδ, upon noting that dx = δz˙ − 1 (η(x))dX with ˙z(η(x)) ≈ (1 +|x|) 1/3 Here the growth of ˙ z(η(x)) inx is clearly controlled by e − ηx Similarly, since |E(x, z)| ≤C(1 +x) 4/3 , we have
(1 +x) 4/3 e − ηx dx≤C, which proves the estimates for |z| ≤1.
Next, consider the casez≥1, and k= 0,1,2 Again using the bounds from Lemma 4.9 and noting thatZ =η(z)/δ ≈(1 +|z|) 2/3 /δ asz becomes large, we obtain
Here again we have used the change of variable dx=δz˙ − 1 dX with ˙z≈(1 +|x|) 1/3
Let us now consider the nonlocal termE(x, z), which is nonzero forx > z, and consider the case z≥1 We recall that
|E(x, z)| ≤C(1 +x) 4/3 (1 +|X|) − 3/2 +C(1 +x) 1/3 |x−z|.Let us give estimate on the integrals involving the last term in E(x, z); the first term inE(x, z) can be treated easily We consider two cases: |x−z| ≤ M and |x−z| ≥ M for
M = η 1 log(1 +z) In the former case, we have
≤C(1 +z) 4/3 e − ηM e − η | z | =C(1 +z) 1/3 e − η | z | This completes the proof of the lemma.
Similarly, we also obtain the following convolution estimate for the error kernelErrA(x, z).
Lemma 4.12 Let ErrA(x, z) be the error kernel of the primitive Airy equation defined as in Lemma 4.8, and letf ∈X η for some η >0 Then there is some constantC so that
Err A (x,ã)f(x)dx η ≤Cδkfkη (4.38) for all z≥0.
Assuming kfkη = 1, we find that the error estimate |Err A (x, z)| is bounded by Cδ, particularly when |z| ≤ 1 For the case where z ≥ 1, we can derive a similar estimate to that of G(x, z).
Resolution of modified Airy equation
In this section, we shall introduce the approximate inverse of the Airyoperator We recall thatAiry(φ) =ε∂ z 4 φ−(U −c+ 2ǫα 2 )∂ z 2 φ Let us study the inhomogeneous Airy equation
Airy(φ) =f(z), (4.39) for some sourcef(z) We introduce the approximate solution to this equation by defining
Then, since the Green function G(x, z) does not solve exactly the modified Airy equation (see (4.30)), the solution AirySolver(f) does not solve it exactly either However, there holds
Airy(AirySolver(f)) =f +AiryErr(f) (4.41) where the error operator AiryErr(ã) is defined by
The error kernel of the Airy operator, denoted as Err A (x, z), is defined in Lemma 4.8 According to Lemma 4.12, the estimate for the Airy error is given by kAiryErr(f)kη ≤ Cδkfkη for all functions f in the space X η This indicates that the Airy error, AiryErr(f), is of order O(δ) within the context of X η.
For the above mentioned reason, we may now define by iteration an exact solver for the modified Airy operator Let us start with a fixed f ∈X η Let us define φ n =−AirySolver(E n − 1 )
En=−AiryErr(En − 1) (4.43) for all n≥1, with E 0 =f Let us also denote
It follows by induction that
Airy(Sn) =f +En, for all n≥1 Now by (4.42), we have kE n kη ≤CδkE n − 1 kη ≤(Cδ) n kfkη.
This proves thatE n →0 inX η asn→ ∞sinceδ is small In addition, by a view of Lemma 4.11, we have kφ n kη ′ ≤CkE n − 1 kη ≤C(Cδ) n − 1
This shows that φ n converges to zero in X η ′ for arbitrary fixed η ′ < η as n → ∞, and furthermore the series
Sn→S ∞ in X η ′ as n→ ∞, for some S ∞ ∈ X η ′ We then denote AirySolver ∞ (f) = S ∞ , for each f ∈ X η In addition, we have Airy(S ∞ ) = f,that is, AirySolver ∞ (f) is the exact solver for the modified Airy operator.
To summarize, we have proved the following proposition.
Proposition 4.13 Let η ′ < η be positive numbers Assume that δ is sufficiently small. There exists an exact solver AirySolver ∞ (ã) as a well-defined operator from X η to X η ′ so that
In addition, there holds kAirySolver ∞ (f)kη ′ ≤ C η−η ′ kfkη,for some positive constant C.
In this section, we study the smoothing effect of the modified Airy function Precisely, let us consider the Airy equation with a singular source:
Airy(φ) =ε∂ z 4 φ−(U −c)∂ z 2 φ=ǫ∂ z 4 f(z) (5.1) in which f ∈Y 4 η , that is f(z) satisfies
|∂ z k f(z)| ≤Ce − ηz , k= 0,ã ã ã ,4, (5.2) for z away from z c , and f(z) behaves as (z−z c ) log(z−z c ) for z near z c Precisely, we assume that
|f(z)| ≤C, |∂ z f(z)| ≤C(1 +|log(z−z c )|), |∂ z k f(z)| ≤C(1 +|z−z c | 1 − k ), (5.3) forz nearz c and for k= 2,3,4, for some constantC.
This article focuses on the convolution of the Green function associated with the Airy equation, specifically addressing the highly singular term ∂z⁴f(z) It examines the inverse of the Airy operator, which plays a crucial role in mitigating the singularities present in the source term ǫ∂z⁴f.
We then obtain the following crucial proposition.
Proposition 5.1 Assume that z c , δ α Let AirySolver ∞ (ã) be the exact Airy solver of the Airy(ã) operator constructed as in Proposition 4.13 and let f ∈ Y 4 η There holds the estimate:
To prove the proposition, we begin by deriving similar estimates for AirySolver(ǫ∂ z 4 f) and AiryError(ǫ∂ z 4 f) The estimate for the exact solver can be established using the methods outlined in Section 4.6 Consequently, it is sufficient to demonstrate the validity of the following two lemmas.
Lemma 5.2 Assume that z c , δ α Let G(x, z) be the approximated Green function to the modified Airy equation constructed as in Lemma 4.8 and let f ∈ Y 4 η There holds a convolution estimate:
Similarly, we also have the following.
Lemma 5.3 Assume thatzc, δ α Let ErrA(x, z) be the error defined as in Lemma 4.8 and let f ∈Y 4 η There holds the convolution estimate for Err A (x, z)
≤CkfkY 4 η e − ηz δ 2 (1 +|logδ|) (5.6) for allz≥0, and for k= 0,1,2.
Proof of Lemma 5.2 with k= 0 Let us assume that kfkY 4 η = 1 To begin our estimates, let us recall the decomposition of G(x, z) into the localized and non-localized part as
G(x, z) =G(x, z) +e E(x, z), whereG(x, z) ande E(x, z) satisfy the pointwise bounds in Lemma 4.9 In addition, we recall thatǫ∂ x j G 2,a (X, Z) and soǫ∂ x j G(x, z) are continuous acrossx=z forj= 0,1,2 Using the continuity, we can integrate by parts to get φ(z) =−ǫ
Here, I ℓ (z) and I e (z) denote the corresponding integral that involves G(x, z) ande E(x, z) respectively, andB 0 (z) is introduced to collect the boundary terms at x= 0 and is defined by
By a view of the definition ofE(x, z), we further denote
Estimate for the integral I ℓ (z) Using the bound (4.32) on the localized part of the Green function, we can give bounds on the integral term I ℓ in (5.7) Consider the case
|z−z c | ≤δ In this case, we note that η ′ (z)≈ z(η(z))˙ ≈1 By splitting the integral into two cases according to the estimate (4.32), we get
{| x − z c |≥ δ }|∂ x 3 G(x, z)∂e x f(x)|dx, in which since ǫ∂ x 3 G(x, z) is uniformly bounded, the first integral on the right is boundede by
For the second integral on the right, we note that in this case sinceX andZ are away from each other, there holds e −
≤Cδ(1 +|logδ|), in which the second-to-last inequality was due to the crucial change of variableX=δ − 1 η(x) and sodx=δz(η(x))dX˙ with|z(η(x))˙ | ≤C(1 +|x|) 1/3
In the scenario where |z−z_c| ≥ δ, we observe that as z approaches infinity, Z = δ − 1/η(z) also tends toward infinity, given that |η(z)| is approximately (1 + |z|)^(2/3) for sufficiently large z We proceed by dividing the integral over x into two segments: |x−z_c| ≤ δ and |x−z_c| ≥ δ For the integral where |x−z_c| ≤ δ, similar to previous analyses, we find that X and Z remain distinct, leading to the conclusion of ǫ.
Here the exponential decay in z was due to the decay terme − 1 6 | Z | 3/2 with Z ≈(1 +z) 2/3 Next, for the integral over {|x−zc| ≥ δ}, we use the bound (4.32) and the assumption
Ifz≤1, the above is clearly bounded byC(1 +|logδ|)δ Consider the casez≥1 We note that |Z| & |z| 2/3 /δ This implies that (1 +z) 1/3 |Z| − 1/2 1 and so the above integral is again bounded byC(1 +|logδ|)δe − ηz
Therefore in all cases, we have |I ℓ (z)| ≤Ce − ηz δ(1 +|logδ|) or equivalently,
In our analysis, we evaluate various scenarios based on the magnitude of z When z is sufficiently distant from the critical point and boundary layer, specifically when z is greater than or equal to the sum of the critical value |z c | and a small positive offset δ, we utilize integration by parts to derive our results.
Here for convenience, we recall the bound (4.34) ona 2 (x):
Now by using this bound and the fact that |Z| & |z| 2/3 /δ, the boundary term is clearly bounded by
≤Ce − ηz δ(1 +|logδ|)(1 +δ 1/2 |z| 1/3 ) whereas the integral term is estimated by δ 3 π
Next, for z≤ |z c |+δ, we write the integral I e,2 (z) into δ 3
∂ x 3 ( ˙x 3/2 a 2 (x))∂ x f(x)dx, where the first integral can be estimated similarly as done in (5.11) For the last integral, using (5.10) for boundedX yields δ 3
Thus, we have shown that
Estimate for I e,1 Following the above estimates, we can now consider the integral
Let us recall the bound (4.34) on a 1 (x):
To estimate the integral I e,1 (z), we analyze different scenarios, starting with the case where z is greater than or equal to |zc| plus δ In this situation, x remains distant from the critical layer, allowing us to utilize integration by parts three times effectively.
|x=z in which the boundary terms are bounded bye − η | z | δ(1 +|logδ|) times
Similarly, we consider the integral term in I e,1 Let M = 1 η log(1 +z) By (5.13), we have δ 2 π
Hence, we obtain the desired uniform boundI e,1 (z) for z≥ |z c |+δ.
Next, consider the case |z−z c | ≤δ in which Z is bounded We write
The first integral on the right can be estimated similarly as above, using integration by parts For the second integral, we use the bound (5.13) for boundedX to get δ 2
In the scenario where \(0 \leq z \leq |z_c| - \delta\) and \(\delta \ll |z_c|\), the critical layer is positioned away from the boundary layer Under these conditions, the linear growth in \(Z\) becomes pronounced, expressed as \(|Z|(1 + |z_c|/\delta)\) Consequently, this analysis leads to significant findings regarding the behavior of the system.
Ie,1(z)≤C(1 +|zc/δ|)δ(1 +|logδ|) (5.14) The estimate for Ie,1(z) thus follows for allz≥0.
Estimate for the boundary term B0(z) It remains to give estimates on
We note that there is no linear term E(x, z) at the boundaryx= 0 sincez≥0 Using the bound (4.32) for x= 0, we get
|ǫ∂ x 2 G(x, z)∂e x f(x)|x=0 ≤Cδ(1 +|logz c |)e − 2 3 | Z | 3 / 2 This together with the assumption thatδ z c then yields
|B0(z)| ≤Cδ(1 +|logδ|)e − ηz (5.15) Combining all the estimates above yields the lemma for k= 0.
Proof of Lemma 5.2 with k >0 We now prove the lemma for the casek= 2; the casek= 1 follows similarly We consider the integral ǫ
The equation (U(z)−c)² ∂z² G(x, z) ∂e x⁴ f(x) dx = I₁(z) + I₂(z) represents the integration of two distinct regions, where I₁(z) is defined for {|x−zc| ≤ δ} and I₂(z) for {|x−zc| ≥ δ} It is important to note that (U(z)−c) ˙z² can be expressed as U′(zc)η(z), with Z defined as η(z)/δ For the second integral, I₂(z), utilizing equations (5.2) and (5.3) along with the established bounds on the Green function for values of x that are both distant from and close to z, leads to straightforward conclusions.
Using |x−zc| ≥δ in these integrals and making a change of variable X =η(x)/δ to gain an extra factor of δ, we obtain
The integral I1(z) is estimated within the region defined by |x - zc| ≤ δ, where the function is clearly bounded by Cδ(1 + z)e^(-ηz) To derive this estimate, we apply integration by parts three times, temporarily setting aside the boundary terms for consideration.
The second derivative introduces a significant factor of δ − 2, which combines with (U−c)², resulting in a term of order |Z|² Additionally, the minor factor of ǫ cancels with δ − 3 from the third derivative, allowing for a straightforward bounding of the integral.
Finally, the boundary terms can be treated, following the above analysis and that done in the case k= 0; see (5.15) This completes the proof of the lemma.
Proof of Lemma 5.3 The proof follows similarly, but more straightforwardly, the above proof for the localized part of the Green function, upon recalling that
6 Construction of slow Orr-Sommerfeld modes
In this section, we systematically derive two precise solutions, φ1 and φ2, that exhibit slow decay and growth We present a proposition, which will be proven later, that establishes an exact solution to the Orr-Sommerfeld equations, originating from the exact solution of the Rayleigh equation.
Proposition 6.1 states that for an exact solution φ Ray within the space X α of the Rayleigh equation Ray α (φ Ray ) = f, where f is an element of X η and η is greater than 0, there exists an exact solution φ s (z) in X α that addresses the Orr-Sommerfeld equations when α and ǫ are sufficiently small.
Orr(φs) =f, so that φ s is close toφ Ray in X 2 η Precisely, we have kφs−φRaykX 2 η ≤Cδ(1 +|logδ|)(1 +|zc/δ|), for some positive constant C independent of α, ǫ.
Starting with the precise Rayleigh solutions φ Ray, ± derived from Lemma 3.6, Proposition 6.1 demonstrates the existence of two exact solutions φ s, ± for the homogeneous Orr-Sommerfeld equation.
Next, we obtain the following lemma.
Lemma 6.2 The slow modes φ s constructed in Proposition 6.1 depend analytically in c, for Im c >0.
Proof The proof is straightforward since the only “singularities” are of the forms: log(U−c), 1/(U−c), 1/(U−c) 2 , and 1/(U−c) 3 , which are of course analytic incwhen Imc >0.
Remark 6.3 It can be shown that the approximated solution φ N can be extended C γ -H¨older continuously on the axis {Imc= 0}, for 0≤γ 0, primarily because the cut-off function χ becomes zero for z ≥ 2 This leads to a singularity characterized by an order of (z − z_c) − 3, attributed to the z log z singularity present in RaySolver α, ∞ (ã).
By a view of Proposition 3.1, we have kχ(z)RaySolver α, ∞ (g)k Y 4 η1 ≤CkgkX 2 η (6.9)
In addition, by applying Proposition 5.1, we get
By takingη 1 = 1 +η, this together with (6.9) yields
It is now straightforward to conclude Lemma 6.5 simply by combining (6.8) and (6.10),upon recalling thatReg(φ) :=−(εα 4 +U ′′ +α 2 (U−c))φ.
First order expansion of the slow-decaying mode φ s
In this section, we calculate the boundary contribution of the initial terms in the expansion of slow Orr-Sommerfeld modes The leading term is expressed as φ 1 (z;c) = φ Ray (z;c) + AirySolver ∞ (A s )(z) + ∂ z − 2 A − a, 1 ∞(I 0 )(z), where A s is defined as χDif f(φ Ray ) and I 0 as (1−χ)Dif f(φ Ray ) Additionally, φ Ray (z;c) is represented as φ Ray, − (z) based on Lemma 3.6, with φRay, −(z) taking the form e − αz (U−c+O(α)).
Thus, together with Proposition 4.7, k∂ z − 2 A − a, 1 ∞(I 0 )kη ≤CkA − a, 1 ∞(I 0 )kη ≤Cδ − 1 kI 0 kη ≤Cδ 2 Next, with A s =χDif f(φ Ray ), we can write
The expression A s is defined as A s = χDif f(e − αz (U−c)) + χDif f(O(α)), where the first term is free of singularities and has an order of O(ǫ) To analyze A s, we only need to apply the smoothing-singularity lemma to the last term Consequently, Propositions 5.1 and 4.13 lead to the conclusion that the norm kAirySolver ∞ (A s )kη is bounded by Cǫ + Cαδ(1 + |logδ|)(1 + |z c /δ|).
This proves that kφ 1 (ã;c)−φ Ray, − kη ≤Cδ 2 +Cαδ(1 +|logδ|)(1 +|z c /δ|) (6.12)
In this section, we will prove the following lemma.
Lemma 6.6 Let φ 1 be defined as in (6.11), and let U 0 ′ 6= 0 For small z c , α, δ, such that δ.α and z c ≈α, there hold φ 1 (0;c)
(6.13) Here, O(ã) is to denote the bound in L ∞ norm.
The proof of the lemma follows directly from Lemma 3.6, together with the estimate (6.12) Indeed, let us recall φ Ray, − (0) =U 0 −c+α(U + −U 0 ) 2 φ 2,0 (0) +O(α(α+|z c |)) and ∂ z φ Ray, − (0) =U 0 ′ +O(αlogz c ).
7 Construction of fast Orr-Sommerfeld modes φ f
In this section, we present a construction similar to that in Proposition 6.4, starting with a rapidly decaying solution that connects to Airy solutions We define φ 3,0 (z) as γ 0 Ai(2, δ − 1 η(z)), where γ 0 = Ai(2, δ − 1 η(0)) - 1 serves as the normalization constant to ensure φ 3,0 remains bounded, specifically with φ 3,0 (0) equal to 1 Here, Ai(2, ã) represents the second primitive of the Airy solution Ai(ã), and we set δ equal to ε.
We recall that as Z tends to infinities, Ai(2, e iπ/6 Z) asymptotically behaves ase ∓
In the equation Z = η(z)/δ ≈ (1 + z)^(2/3)/δ, we find that Ai(2, e^(iπ/6)Z) asymptotically behaves like e^(±|z/√ǫ|), which is indicative of fast-decaying modes This implies that φ3,0(z) is well-defined for z ≥ 0 and exhibits exponential decay as z approaches infinity It is important to note that the critical layer is centered at z = zc and has a typical size of δ, with the Airy function playing a significant role within this critical layer.
Proposition 7.1 For α, δ sufficiently small, there is an exact solution φ 3 (z) in X η/ √ ǫ solving the Orr-Sommerfeld equation
Orr(φ 3 ) = 0 so that φ 3 (z) is approximately close to φ 3,0 (z) in the sense that
|φ 3 (z)−φ 3,0 (z)| ≤Cγ 0 δe − η | z/ √ ǫ | , (7.3) for some fixed constants η, C In particular, at the boundary z= 0, φ 3 (0) = 1 +O(δ), ∂ z φ 3 (0) =δ − 1 Ai(1, δ − 1 η(0))
When the critical layer \( z_c \) is sufficiently distant from the boundary, indicated by a large ratio of \( z_c/\delta \), the solution \( \phi_{3,0}(z) \) exhibits characteristics of an exponential boundary layer As \( z \) approaches zero, the relationship \( \eta(z) \sim z - z_c \) and \( Z = \eta(z)/\delta \sim (z - z_c)/\delta \) becomes evident, leading to this behavior.
From the construction, we also obtain the following lemma.
Lemma 7.3 The fast-decaying modeφ 3 constructed in Proposition 7.1 depends analytically in c with Im c6= 0.
Proof This is simply due to the fact that both Airy function and the Langer transformation(7.2) are analytic in their arguments.
Iterative construction of the Airy mode
Let us prove Proposition 7.1 in this section.
Proof of Proposition 7.1 We start with φ 3,0 (z) =γ 0 Ai(2, δ − 1 η(z)) We note thatφ 3,0 and
∂zφ3,0 are both bounded on z ≥ 0, and so are ε∂ z 4 φ3,0 and (U −c)∂ z 2 φ3,0 We shall show indeed thatφ 3,0 approximately solves the Orr-Sommerfeld equation In what follows, let us assume that γ 0 = 1 Direct calculations yield
Airy(φ 3,0 ) := εδ − 1 η (4) Ai(1, Z) + 4εδ − 2 η ′ η (3) Ai(Z) + 3εδ − 2 (η ′′ ) 2 Ai(Z) +εδ − 4 (η ′ ) 4 Ai ′′ (Z)
, with Z = δ − 1 η(z) Let us first look at the leading terms with a factor of εδ − 4 and of (U −c)δ − 2 Using the facts thatη ′ = 1/z,˙ δ 3 =ε/U c ′ , and (U −c) ˙z 2 =U c ′ η(z), we have εδ − 4 (η ′ ) 4 Ai ′′ (Z)−δ − 2 (η ′ ) 2 (U −c)Ai(Z)
The next terms inAiry(φ 3,0 ) are
, which is bounded forz≥0 The rest is of order O(ε 1/3 ) or smaller That is, we obtain
Here we note that the right-hand side I(z) is very localized and depends primarily on the fast variable Z asAi(ã) does Precisely, we have
|I(z)| ≤C(1 +z) − 2 (1 +|Z|) 1/4 e − √ 2 | Z | 3/2 /3 (7.4) for some constant C Let us then denote ψ(z) :=−AirySolver ∞ (I)(z), the exact Airy solver of I(z) It follows thatAiry(φ 3,0 +ψ) = 0 and there holds the bound
Next, since Airy(φ3,0+ψ) = 0, the identity (2.6) yields
Clearly, I 1 ∈ X η for some η ≈ 1/√ǫ and I 1 = O(δ), upon recalling that Z = η(z)/δ ≈
(1 +z) 2/3 /δ From this, we can apply the Iter operator constructed previously in Section 6 to improve the error estimate The proposition thus follows.
First order expansion of φ 3
By construction in Proposition 7.1, we obtain the following first order expansion of φ 3 at the boundary φ 3 (0;c) = 1 +O(δ), ∂ z φ 3 (0;c) =δ − 1 Ai(1, δ − 1 η(0))
In the study of the linear dispersion relation, we are interested in the ratio ∂ z φ 3 /φ 3 , on which the above yields φ 3 (0;c)
Ai(1, Y) (7.6) The following lemma is crucial later on to determine instability.
Lemma 7.4 Let φ 3 be the Orr-Sommerfeld solution constructed in Proposition 7.1 There holds φ 3 (0;c)
∂zφ3(0;c) =−e πi/4 |δ||z c /δ| − 1/2 (1 +O(|z c /δ| − 3/2 )) (7.7) as long as z c /δ is sufficiently large In particular, the imaginary part of φ 3 /∂ z φ 3 becomes negative when zc/δ is large In addition, when zc/δ= 0, φ 3 (0;c)
∂ z φ 3 (0;c) = 3 1/3 Γ(4/3)|δ|e 5iπ/6 , (7.8) for Γ(ã) the usual Gamma function.
Here, we recall that δ =e − iπ/6 (αRU c ′ ) − 1/3 , and from the estimate (4.8), η(0) = −z c +
We focus on the ratio CAi(Y) for the complex variable Y = -e^(iπ/6)y, where y is situated in a small neighborhood of the positive real numbers (R+) For simplicity, we will assume that y belongs to R+ The conclusion of Lemma 7.4 can be derived directly from the subsequent lemma.
Lemma 7.5 Let C Ai (ã) be defined as above Then,C Ai (ã) is uniformly bounded on the ray
Y =e 7iπ/6 y for y∈R + In addition, there holds
C Ai (−e iπ/6 y) =−e 5iπ/12 y − 1/2 (1 +O(y − 3/2 )) for all large y∈R + At y= 0, we have
Proof We notice that Y = −e iπ/6 y belongs to the sector S1 defined as in Lemma 4.2 for y∈R + Thus, Lemma 4.2 yields
CAi(Y) =−Y − 1/2 (1 +O(|Y| − 3/2 )) for largeY This proves the estimate for largey The value at y= 0 is easily obtained from those ofAi(k,0) given in Lemma 4.2 This completes the proof of the lemma.
8 Study of the dispersion relation
Linear dispersion relation
The solution to the Orr-Sommerfeld equation is expressed as a linear combination of the slow-decaying solution φ₁ and the fast-decaying solution φ₃, represented by φ = Aφ₁ + Bφ₃, where A and B are bounded functions of α, ε, and c Both φ₁ and φ₃ are defined in Propositions 6.4 and 7.1, respectively This formulation ensures that φ(z) is an exact solution that meets the boundary condition at z = +∞ Additionally, the boundary condition at z = 0 leads to specific dispersion relations, which are crucial for understanding the behavior of the system.
We shall show that for some ranges of (α, ǫ), the dispersion relation yields the existence of unstable eigenvalues c.
Ranges of α
When ε = 0, the Orr-Sommerfeld equation reduces to the Rayleigh equation, indicating that c(α,0) = U(0) + O(α) and the critical layer z c (α,0) is approximately α when U ′ (0) is not equal to zero For ε > 0, it is anticipated that (c(α, ε), z c (α, ε)) approaches (U(0), 0) as (α, ε) tend towards zero Additionally, for instability analysis, we focus on α values within the range (α low (R), α up (R)), where α low (R) is approximately R − 1/4 and α up (R) is around R − 1/6 for large R These values of α j (R) delineate the lower and upper branches of the marginal (in)stability curve for the boundary layer U Specifically, we will demonstrate that there exists a critical constant Ac1 such that when α low (R) = A1R − 1/4, the imaginary part of c transitions from negative (indicating stability) to positive (indicating instability) as the parameter A 1 increases.
A 1 = A c1 Similarly, there exists an A c2 so that with α = A 2 R − 1/6 , Im c turns from positive to negative asA2 increases acrossA2 =Ac2 In particular, we obtain instability in the intermediate zone: α≈R − β for 1/6< β