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 (1.9), it is essential to analyze their behavior as z approaches infinity As z tends to +∞, the solutions must resemble those of the limiting equation with constant coefficients: ε∂ z 4 φ= (U+−c+ 2εα 2 )∂ z 2 φ−α 2 (εα 2 +U+−c)φ, where U + = U(+∞) The solutions to this equation take the form Ce λz, with λ values being either ±λ s or ±λ f, where λ s =±α+O(α 2 √ ε) and λ f =± 1.
The analysis reveals two slow-behavior solutions, φ 1 and φ 2, characterized by λ approximately equal to ±α, with one solution decaying and the other growing In contrast, two fast-behavior solutions, φ 3 and φ 4, exhibit λ on the order of ±1/√ε, also featuring one decaying and the other growing The slow-behavior solutions φ 1 and φ 2 are perturbations of the eigenfunctions of the Rayleigh equation, while the fast-behavior solutions φ 3 and φ 4 are specific to the Orr-Sommerfeld equation and relate to the classical Airy equation Thus, four independent solutions to the equation can be constructed.
• 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 solutions to the second primitive Airy equation exhibit an exponential behavior of order exp(±|Z| 3/2) as |Z| approaches infinity In this context, Z is defined as η(z)/ǫ 1/3, representing the rapid variable close to 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 φ1, φ2, φ3, and φ4, which must satisfy the specified boundary conditions Notably, considering the asymptotic behavior of φ2 and φ4, we find that any bounded solution to these equations is effectively a combination of only φ1 and φ3 Consequently, our focus will be limited to the analysis of φ1 and φ3.
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 on the right-hand side, Dif f(φ Ray) = ǫ(∂ z 2 − α 2)² φ Ray, is of order O(ǫ) in L ∞ It's important to note that the operator ∂ z 2 − α 2 and so Dif f(ã) eliminate the slow decay term O(e − αz) in φ Ray Near the critical layer, the Rayleigh solution typically includes a singular solution of the form (z−z c) log(z−z c), which means φ Ray exhibits the same singularity at z = zc Consequently, Dif f(φ Ray) contains 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 smoothes out the singularity within the critical layer However, the inverse of the Airy(ã) operator introduces linear growth in the spatial variable, hindering the convergence of our iteration Therefore, we propose a modified Airy operator Aa(ã) to resolve this issue.
In our construction, we define φ1 as φRay + Airy - 1(A s) + ∂z - 2A - a1(I0), where As represents the singular part χDiff(φRay) and I0 denotes the regular part (1 - χ)Diff(φRay) Additionally, ∂z - 1 is expressed as -R∞ z The function χ(z) is a smooth cut-off function, equal to 1 for values in the range [0, 1] and zero 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 behavior within appropriate function spaces Our method circumvents the need for inner and outer expansions, focusing instead on a thorough analysis 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 It is important to note that if Ray α (φ) equals f, then this relationship is established.
When provided that U−c remains non-zero (as is the case when c is complex), classical elliptic regularity indicates that if f belongs to C^k, then φ will belong to C^(k+2), resulting in an increase of two derivatives However, the quality of derivative estimates diminishes as z−z_c approaches zero, with the weight (z−z_c)^l sufficiently managing this singularity Additionally, upon deriving l times, it can be shown that ∂_z^(2+l)φ is bounded by C/(z−z_c)^(l+1) if f is in X_(η,k) This leads to an additional factor of z−z_c in the derivative estimates relating f and φ Furthermore, since e^(±αz) is part of the kernel of ∂_z^2−α^2, if f decays like e^(−ηz), φ can be expected to decay at most as e^(−αz) at infinity In summary, if f is in X_(η,k), then φ is in Y_(α,k+2), achieving an increase of two derivatives along with an additional z−z_c weight, albeit with a loss of rapid decay at infinity Ultimately, we will construct an inverse Ray−1 that is continuous from X_(η,k) to Y_(α,k+2) for any k.
By employing Airy functions, their double primitives, and the Langer transformation, we can derive an almost explicit inverse Airy operator, denoted as Airy − 1 We then analyze the composition of Airy − 1 with the differential operator, Dif f Although this composition is formally of order 0, it exhibits singular behavior, necessitating the use of two derivatives for effective control To minimize its impact, we introduce a factor of az−z c in the norms Through extensive computations involving nearly explicit Green functions, we demonstrate that the operator Airy − 1 ◦Dif f maintains a small norm when mapped from Y k+2 α to X k η.
Reg is constrained within the bounds of X k η due to its nature as a simple multiplication by a bounded function By integrating these estimates, we can derive precise solutions to the Orr-Sommerfeld equations, utilizing solutions from either the Rayleigh or Airy equations This process results in the formation of four independent solutions, each represented as a convergent series that provides its expansion Ultimately, we combine the various terms from these solutions to derive the dispersion relation of Orr-Sommerfeld, and a thorough analysis of this relation yields our instability findings.
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 using a specific Green function Next, we utilize this inverse to develop an approximate inverse of the Ray α operator by creating an approximate Green function Ultimately, we derive the exact inverse of Ray α through an iterative process.
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 Ray 0 (φ) = 0, demonstrating that the asymptotic expansion for φ 1,0 is valid for values of z close to z c, where U(z c) equals c Furthermore, we develop a second particular solution, φ 2,0, by ensuring that the Wronskian determinant is set to 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 makes φ2,0 holomorphic in C−{zc + R−} Specifically, when ℑzc = 0, φ2,0 remains holomorphic in z, with the exception of 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 context, we observe that the complex variable \( c \) has a non-zero imaginary part, leading to the Green function \( G_{R,0}(x, z) \) being a well-defined and continuous function for the variables \( x \) and \( z \) Notably, there is a discontinuity in the first derivative of this function at the point where \( x = z \) 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 explore 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 important to note that as z approaches positive infinity, the solutions to Ray 0 (φ) = f converge to a constant value, specifically φ 1,0 approaching U + −c Additionally, we proceed to construct normal mode solutions to the Rayleigh equation under the condition that α is not equal to zero.
Analyzing the spatially asymptotic limit of the Rayleigh equation reveals two normal mode solutions characterized by behaviors of e ± αz at infinity To investigate 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 the point z = zc, characterized by two distinct sources of singularities: the first is due to the term 1/(U(x)−c) when x is close to zc, and the second originates from the (z−zc) log(z−zc) singularity associated with φ2,0(z) This behavior is further clarified through the analysis presented 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 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 ∂ z φ 1,0 (z) It is essential to examine the order of singularity in proximity to the critical layer, noting that for bounded values of z, the behavior remains consistent.
The inequality |E R,α (x, z)| is bounded by C(1 + |log(z−z c )|)e αx max{1, |x−z| − 1} This lemma is derived by utilizing the additional factor of U−c in front of the integral (3.16) to effectively limit the log(z−z c ) term Furthermore, similar methods apply to obtain estimates for the derivatives.
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 the norm \( kE_n k_{\eta} \) is bounded by \( C n \alpha^n (1 + | \log(\text{Im } c)|)^n kf k_{\eta} \), indicating that \( E_n \) converges to 0 in \( X_{\eta} \) as \( n \) approaches infinity, provided \( \alpha \log(\text{Im } c) \) remains sufficiently small Additionally, it follows from equation (3.17) that the norm \( kRaySolver_{\alpha}(E_n) k_{Y^2_{\alpha}} \) is also bounded by \( C n \alpha^n (1 + | \log(\text{Im } c)|)^n kf k_{\eta} \), confirming the convergence 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 a distributional sense The estimates for values of z close to zc are derived from analogous estimates on RaySolver α (ã), as outlined in Lemma 3.5 Therefore, the proof of Proposition 3.1 is concluded.
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 aims to highlight key properties of these functions, which are derived from 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 connect the profileU to classical Airy equations, we apply a well-known technique in physics known as Langer's transformation This transformation involves a change of variables from (z, φ) to (η, Φ), where η is defined as η(z) = h3.
(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).
In this section, we will provide estimates on the Airy functions relevant to the case where x is less than z, as outlined in Lemma 4.1 These estimates will serve as a foundation for the subsequent discussions.
For values where fork is greater than or equal to zero and both x and z are sufficiently distanced from zero, we can derive similar bounds for the scenario where x exceeds z It is noteworthy that the polynomial growth in x within the aforementioned estimate can be substituted with growth in z, with the exception of a term that decays exponentially.
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 function 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 Referring to Lemma 4.6, we establish that the norm of φ_n in X_η is bounded by Cδ^(-1) times the norm of E_n minus one in X_η, which is also constrained by Cδ^(-1) multiplied by (Cδ)^(n-1) times the norm of f in X_η This indicates that φ_n converges to zero in X_η as n approaches infinity, and additionally, the series converges accordingly.
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 involves solving 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 equation (4.24) and the jump conditions on G aux (X, Z) at X = Z, it is evident that the jump conditions for ∂²z G and ∂z³ G are easily derived 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 previously mentioned estimates continue to hold true Specifically, when z is greater than or equal to the real part of z c, the relationship |Y| ≥ |Z| is maintained for y values greater than or equal to z, allowing the estimates on Ai(k, z) to be derived similarly as before Additionally, we will examine the scenario where z is less than or equal to the real part of z c.
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 limits on Ai(ã) and Ci(ã), we can easily derive the proposed bounds on G(x, z) Additionally, we obtain derivative bounds through a similar process Ultimately, leveraging the previously mentioned bounds on Ai(k, z) and Ci(k, z) leads to significant 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 derive that the error |Err A (x, z)| is bounded by Cδ, particularly when |z| ≤ 1 For the case where z ≥ 1, we can apply a similar estimation approach as used for 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, represented as kAiryErr(f)kη, is bounded by Cδkfkη for all functions f in the space X η This indicates that the Airy error term, 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 related to the Airy equation, specifically addressing the most singular term ∂z^4 f(z) It examines the inverse of the Airy operator, which serves to smooth the singularities present in the source term ǫ∂z^4 f.
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 first need to establish a similar estimate for both AirySolver(ǫ∂ z 4 f) and AiryError(ǫ∂ z 4 f) The estimate for the exact solver can be derived using the methods outlined in Section 4.6 Therefore, it is sufficient to demonstrate the two lemmas presented below.
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
When examining the case where |z−z c | ≥ δ, we observe that as z approaches infinity, Z = δ − 1 η(z) also approaches infinity, given that |η(z)| is approximately (1 + |z|) 2/3 for sufficiently large z We then divide the integral 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 with X and Z being sufficiently distanced from one another, we arrive at the result 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 the estimate for I e,2 by examining various scenarios based on the value of z Specifically, when z is sufficiently distant from the critical and boundary layers, defined as z ≥ |z c | + 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, beginning with the case where z is greater than or equal to |zc| + δ In this situation, x is sufficiently 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 Here, the linear growth in \(Z\) becomes substantial, expressed as \(|Z|(1 + |z_c|/\delta)\) Consequently, this analysis leads us to derive important results 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 \( (U(z)−c) 2 ∂ z 2 G(x, z)∂e x 4 f(x)dx=I 1 (z) +I 2 (z) \) represents a relationship where \( I_1(z) \) and \( I_2(z) \) are integrals defined over specific regions around \( z_c \) It is important to note that \( (U(z)−c) \dot{z}^2 \) can be expressed as \( U' (z_c) \eta(z) \), with \( Z \) defined as \( \eta(z)/δ \) For the second integral \( I_2(z) \), applying equations (5.2) and (5.3), along with the established bounds on the Green function, allows for straightforward conclusions regarding the behavior of the integral as \( x \) varies in relation to \( z \).
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 region defined by |x - zc| ≤ δ is crucial for estimating I1(z), where the bounds are clearly established by Cδ(1 + z)e^(-ηz) To analyze this integral, we will apply integration by parts three times, temporarily setting aside the boundary terms for further 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 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 develop two precise solutions, φ₁ and φ₂, that exhibit slow decay and growth We present a proposition, which will be proven later, that provides an exact solution to the Orr-Sommerfeld equations derived from the exact solution of the Rayleigh equation.
Proposition 6.1 states that if φ Ray is an exact solution to the Rayleigh equation Ray α (φ Ray ) = f, where f belongs to the space X η for η > 0, then for sufficiently small values of α and ǫ, an exact solution φ s (z) exists in the space X α that satisfies the Orr-Sommerfeld equations.
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 results in a singularity that is 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 slow Orr-Sommerfeld mode expansion The primary term is expressed as φ 1 (z;c) = φ Ray (z;c) + AirySolver ∞ (A s )(z) + ∂ z − 2 A − a, 1 ∞(I 0 )(z) Here, A s is defined as χDif f(φ Ray ), while I 0 represents (1−χ)Dif f(φ Ray ) Additionally, φ Ray (z;c) is derived from φ Ray, − (z) as outlined in Lemma 3.6, where φRay, −(z) is given by 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 =χDif f(e − αz (U−c)) +χDif f(O(α)) includes a first term that is free of singularities and is of order O(ǫ) To analyze this, we apply the smoothing-singularity lemma specifically to the last term in A s From Propositions 5.1 and 4.13, we can establish that the norm kAirySolver ∞ (A s )kη is bounded by Cǫ + Cαδ(1 + |logδ|)(1 + |z c /δ|), demonstrating the relationship between the components of the expression and their respective constraints.
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
This section presents a construction similar to that in Proposition 6.4, starting with a fast-decaying solution that connects to Airy solutions The solution is defined as φ 3,0 (z) := γ 0 Ai(2, δ − 1 η(z)), where γ 0 = Ai(2, δ − 1 η(0)) - 1 serves as the normalized constant to ensure φ 3,0 remains bounded, specifically with φ 3,0 (0) = 1 Here, Ai(2, ã) represents the second primitive of the Airy solution Ai(ã), and δ is set 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 observe that Ai(2, e^(iπ/6)Z) asymptotically behaves like e^(±|z/√ǫ|), indicating the presence of fast-decaying modes This leads to the conclusion 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 characteristic size of δ, where the Airy function is significant within this region.
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 \( z_c/\delta \), the solution \( \phi_{3,0}(z) \) exhibits characteristics of an exponential boundary layer As \( z \) approaches zero, the relationship \( \eta(z) \sim z - z_c \) leads to \( Z = \eta(z)/\delta \sim (z - z_c)/\delta \).
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 within a small neighborhood of the positive real line, R+ For simplicity, we will assume that y belongs to R+ 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, as discussed in the introduction, can be expressed as a linear combination of the slow-decaying solution φ₁ and the fast-decaying solution φ₃ We define an exact solution in the form φ = Aφ₁ + Bφ₃, where A and B are bounded functions dependent on parameters α, ε, and c The functions φ₁ and φ₃ are derived from Propositions 6.4 and 7.1, respectively This formulation ensures that φ(z) is a valid solution to the Orr-Sommerfeld equation and meets the boundary condition at z = +∞ Additionally, applying the boundary condition at z = 0 leads to dispersion relations that involve A and B, establishing a relationship between the solutions at the specified boundaries.
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 simplifies to the Rayleigh equation, demonstrating that c(α,0) = U(0) + O(α) and the critical layer z c (α,0) is approximately α, provided U ′ (0) ≠ 0 In scenarios where U ′ (0) = 0, a different rate of convergence may apply For ε > 0, it is anticipated that (c(α, ε), z c (α, ε)) converges to (U(0), 0) as (α, ε) approaches 0 Furthermore, physical results indicate that for instability, α should lie within the range (α low (R), α up (R)), where α low (R) ≈ R − 1/4 and α up (R) ≈ R − 1/6 for sufficiently large R These α j (R) values delineate the lower and upper branches of the marginal (in)stability curve for the boundary layer U Specifically, we will demonstrate the existence of a critical constant Ac1, such that with 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