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), we must analyze their behavior as z approaches positive infinity It becomes evident that, as z→ +∞, the solutions must resemble those of the constant-coefficient limiting equation: 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 limiting equation take the form Ce λz, with λ being either ±λ s or ±λ f Here, λ s is defined as ±α+O(α 2 √ ε) and λ f as ± 1.
In the analysis of the system, we identify two solutions, φ 1 and φ 2, exhibiting slow behavior with eigenvalues λ approximately equal to ±α, where one solution decays and the other grows Additionally, there are two solutions, φ 3 and φ 4, demonstrating fast behavior characterized by eigenvalues of order ±1/√ε, with one solution also decaying and the other growing The slow-behavior solutions φ 1 and φ 2 are perturbations of the eigenfunctions derived from 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 Ultimately, 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 solutions to the second primitive Airy equation exhibit exponential behavior, specifically exp(±|Z| 3/2), as |Z| approaches infinity In this context, Z is defined as η(z)/ǫ 1/3, representing the rapid variable near the critical layer, which has a magnitude of order ǫ 1/3 Additionally, η(z) corresponds to Langer’s variable, which asymptotically approaches z 2/3 as z tends to 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, while satisfying 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 focus will be narrowed to the analysis of 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 on the right-hand side of the equation Dif f(φ Ray ) = ǫ(∂ z 2 − α 2 ) 2 φ Ray is of order O(ǫ) in L ∞ Notably, the operator ∂ z 2 − α 2 and the term 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 ), resulting in φRay exhibiting the same singularity at z = zc Consequently, Dif f(φRay) contains singularities of orders log(z−z c ) and (z−z c ) − k for k = 1, 2, 3 To address these singularities, we apply 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.
We define φ₁ as φ Ray + Airy - 1 (Aₛ) + ∂ₓ - 2 A - a₁ (I₀), where Aₛ represents the singular part and is defined as χDiff(φRay), while I₀ denotes the regular part as (1 - χ)Diff(φRay) Additionally, ∂ₓ - 1 is expressed as -R ∞ z The function χ(z) is a smooth cut-off function that equals 1 on the interval [0, 1] and transitions to 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 properties within appropriate function spaces Our method circumvents the complexities of 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 Specifically, if Ray α (φ) equals f, this relationship is fundamental to our analysis.
When provided that U−c remains non-zero, particularly 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 derivative estimates diminish as the distance |z−z_c| decreases The crucial aspect is that the weight |z−z_c|^l effectively manages this singularity Furthermore, upon differentiating equation (2.11) l times, we find 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 |z−z_c| factor in the derivative estimates linking f and φ Additionally, since e^(±αz) is part of the kernel of ∂_z^2−α^2, if f decays like e^(−ηz), φ can only decay as e^(−αz) at infinity Therefore, if f is in X_(η,k), φ will be in Y_(α,k+2), achieving a gain of two derivatives and an extra |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 utilizing 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 Airy − 1 ◦ Dif f, which is formally of order 0 but exhibits singular behavior To manage this singularity, we employ two derivatives and incorporate a az−z c factor in the norms to ensure its smallness Through extensive calculations involving almost 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 bounded from X k η to X k η, as it involves simple multiplication by a bounded function By integrating these estimates, we can derive exact solutions to the Orr-Sommerfeld equations, utilizing solutions from either the Rayleigh or Airy equations This process results in four independent solutions, each represented as a convergent series that provides its expansion Ultimately, we combine the various terms from these solutions to formulate the dispersion relation of the Orr-Sommerfeld equations, and a thorough analysis of this relation leads to 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 of the Ray α operator 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.
In our analysis, we establish that φ 1,0 (z) = U(z) - c serves as an exact solution to Ray 0 (φ) = 0 Furthermore, the anticipated asymptotic expansion for φ 1,0 is validated for values of z close to z c, given that U(z c) equals c Subsequently, we develop a second particular solution, φ 2,0, by ensuring that the Wronskian determinant is equal 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 ensures that φ2,0 is holomorphic in C - {zc + R−} Specifically, when 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, meaning it is not holomorphic for z < zc Additionally, for a fixed z, φ2,0 is holomorphic in c as long as zc does not intersect.
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 \( (x, z) \) Additionally, it is important to note that the first derivative of this function exhibits a discontinuity at the point where \( x = z \) We will now present 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 Lemma 3.2, we identified two solutions, φ 1,0 and φ 2,0, for the equation Ray 0 (φ) = 0 As z approaches +∞, these solutions converge to a constant value, specifically φ 1,0 approaches U + −c We will now proceed to construct normal mode solutions for the Rayleigh equation under the condition that α is not equal to zero.
The spatially asymptotic limit of the Rayleigh equation reveals two normal mode solutions characterized by behaviors of 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 sources of singularities: one from the term 1/(U(x)−c) when x is close to zc, and the other from the (z−zc) log(z−zc) singularity of φ 2,0(z) This is evident when examining 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 function \( f(z) \) exhibits exponential decay at infinity, which is evident from the integral representation of \( Err R,α (f)(z) \) To analyze the behavior near the critical layer, we observe that for bounded values of \( z \), the order of singularity can be determined.
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 is derived directly by utilizing the additional factor of U−c in front of the integral (3.16) to constrain the log(z−z c) component Similar methods are applied to obtain the 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 as \( n \) approaches infinity, \( E_n \) converges to 0 in the space \( X_\eta \) provided that \( \alpha \log \text{Im} c \) remains sufficiently small Additionally, it is shown that the norm of \( \text{RaySolver}_\alpha (E_n) \) in \( Y_{2\alpha} \) is bounded by a constant multiplied by \( C_n \alpha^n (1 + |\log(\text{Im} c)|)^n \|f\|_\eta \) This establishes that the series \( \sum_{n \geq 0} \) converges.
(−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 in a distributional sense The estimates for values of \( z \) close to \( z_c \) are derived from analogous estimates on RaySolver \( \alpha (ã) \), as demonstrated in Lemma 3.5 Consequently, the proof of Proposition 3.1 is now 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 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 analyze the profileU's dependence more effectively, we employ a variable transformation known as Langer’s transformation, which is commonly utilized in physical literature This transformation maps the variables (z, φ) to (η, Φ), where η is defined by the equation η(z) = h3 By applying this classical method, 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).
In this section, we will provide estimates on G a to prepare for the upcoming discussions Specifically, we will examine the scenario where x is less than z Utilizing the estimates derived from the Airy functions as outlined in Lemma 4.1, we can draw significant conclusions.
In the case where fork is greater than or equal to zero and both x and z are significantly greater than zero, we can derive similar bounds for scenarios where x exceeds z Notably, the polynomial growth in x from the previous estimate can be substituted with the growth in z, while still maintaining 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 the space X η, indicating that Err a (f) is of order O(δ) within 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 \( X_\eta \) due to the small value of \( \delta \) According to Lemma 4.6, the inequality \( \| \phi_n \|_\eta \leq C \delta^{-1} \| E_n - 1 \|_\eta \leq C \delta^{-1} (C \delta)^{n-1} \| f \|_\eta \) holds true This indicates that \( \phi_n \) converges to zero in \( X_\eta \) as \( n \) increases, and additionally, the series converges as well.
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 maintaining the jump conditions 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:
The jump conditions for the function G(x, z) and its derivatives ∂zG(x, z) are derived from the conditions outlined in (4.24) and the behavior of G aux (X, Z) at the boundary X = Z Specifically, by defining 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 can easily establish the required jump conditions for ∂2zG and ∂z3G.
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 zc, 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 before Additionally, we must examine the scenario where z is less than or equal to the real part of zc.
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 obtain derivative bounds using the same methodology Ultimately, by applying the previously mentioned bounds on Ai(k, z) and Ci(k, z), we achieve the desired 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 term |Err A (x, z)| is bounded by Cδ, particularly when |z| ≤ 1 For the scenario 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, there exists an estimate for the error, expressed as kAiryErr(f)kη ≤Cδkfkη for all functions f in the space X η This indicates that the Airy error function, AiryErr(f), is of order O(δ) within the space 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 explores the convolution of the Green function associated with the Airy equation, focusing on the most singular term ∂z^4 f(z) Specifically, it examines the inverse of the Airy operator, which serves to smooth out 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 establish a consistent estimate for both AirySolver(ǫ∂ z 4 f) and AiryError(ǫ∂ z 4 f) The estimated results for the exact solver align with the methods outlined in Section 4.6 Therefore, 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 case where |z−z_c| ≥ δ, we observe that as z approaches infinity, Z = δ - 1/η(z) also tends to infinity, given that |η(z)| approximates (1 + |z|)^(2/3) for sufficiently large z We again divide the integral into two segments: |x−z_c| ≤ δ and |x−z_c| ≥ δ For the integral over the region where |x−z_c| ≤ δ, similar to previous analysis, we find that X and Z remain sufficiently distant from each other, leading to the conclusion that the result approaches 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 this analysis, we examine various scenarios based on the value of z When z is sufficiently distant from the critical point and boundary layer, specifically when z is greater than or equal to the absolute value of z_c plus a small positive value 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| + δ In this situation, since x is sufficiently distanced from the critical layer, we can effectively apply integration by parts three times to derive our results.
|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≤z≤ |z c | −δ and δ is much smaller than |z c |, the critical layer is positioned away from the boundary layer Under these conditions, the linear growth in Z becomes prominent, expressed as |Z|.(1 +|z c |/δ) Consequently, our analysis leads to significant findings in this context.
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) defines two integrals, I₁(z) and I₂(z), which represent the integration over the regions {|x−zₒ| ≤ δ} and {|x−zₒ| ≥ δ}, respectively It is important to note that (U(z)−c) ˙z² can be expressed as U′(zₒ)η(z), with Z defined as η(z)/δ For the second integral I₂(z), applying equations (5.2) and (5.3) alongside the bounds on the Green function for x values both near and away from z yields straightforward results.
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 estimate for I₁(z) within the region defined by |x - zc| ≤ δ is derived by applying integration by parts three times Initially, we set aside the boundary terms to focus on the integral term ǫ This approach allows us to clearly establish the bounds of the function, which is limited by Cδ(1 + z)e^(-ηz).
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 with respect to x Consequently, the integral can be easily bounded.
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, the proof of which will be provided later, demonstrating that these solutions serve as exact answers to the Orr-Sommerfeld equations, derived from the exact solution of the Rayleigh equation.
Proposition 6.1 states that for a given exact solution φ Ray in the space X α of the Rayleigh equation, denoted as Ray α (φ Ray ) = f, where f belongs to the space X η with η > 0, there exists an exact solution φ s (z) in X α that satisfies 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