(this post was also posted here on my new blog address:http://blog.toannguyen.org/ )

We now turn to the delicate case: Orr-Sommerfeld solutions for stable profiles to Rayleigh. The results reported here are in a joint work with E. Grenier and Y. Guo, directing some tedious details of the proof to our paper. We consider the Orr-Sommerfeld problem:

with zero boundary conditions on and , in which denotes the Rayleigh operator and .

We assume that the boundary layer is stable to the Rayleigh operator, that is, there is no unstable eigenvalue with to the Rayleigh problem (equivalently, the spectrum of Euler lies precisely on the imaginary axis). In term of the Wronskian determinant, this asserts that for all with . The dispersion relation studied in the last section shows that there is no eigenvalue to the Orr-Sommerfeld problem, with being bounded away from zero in the limit . Hence, in search for a possible unstable point spectrum of Navier-Stokes, we will construct solutions to Orr-Sommerfeld equations in the regime where all parameters and are sufficiently small. In particular, , in order to take the inverse of the operator as studied in the last section. Throughout the notes, denotes a complex number so that , for each fixed complex number .

Clearly, is not a good iteration operator, since by a view of Rayleigh solutions, typically has a singularity of the form and therefore has a singularity of order . To deal with the singularity, we need to examine the leading operator in Orr-Sommerfeld equations near the singular point . Indeed, introducing the blow-up variable: and search for the ansatz solution , which is in the literature often referred as the critical layer. It follows that approximately solves

with the critical layer size . That is, within the critical layer, solves the classical Airy equation. In the light of the toy model problem in the previous section, we introduce the following iteration operator:

It suffices to show that the Iter operator is contractive in a suitable function space. This would take care o the singularity. (in the proof, we in fact need a slightly more complicated iteration operator to take care of a possible linear growth in of the Rayleigh solutions at infinity). Here, the Airy operator is defined by

Several issues to overcome. We need to appropriately define and , and most of all, to study the smoothing effect of Airy operator near the critical layer. The inverse is studied at length in the previous lecture. Similarly, the inverse will be constructed through its Green kernel.

**8.1. Green kernel**

We recall the basic construction of the Green kernel of a linear, variable-coefficient, second-order ODE operator:

with never vanishing. For simplicity, we assume that there are two independent solutions to the homogenous equation , with being bounded and unbounded at infinity . We search for a particular (bounded) solution to the non-homogenous equation in the integral form:

in which is called the Green kernel of , which by the equation satisfies , where denotes the delta function centered at . Comparing the singularity in the equation for , one asserts that must be continuous and has a jump across ; precisely,

We now construct a Green kernel (which is non unique, unless we impose a boundary condition). As , solves the homogenous equation and so can be constructed as a linear combination of the other two independent solutions . This yields

The jump conditions across yield the linear equations:

which gives and . Hence, the Green kernel can be defined as

in which denotes the Wronskian determinant, which can be computed through the Abel’s identity: . In particular, when , the Wronskian determinant is constant. Note that the Green kernel is not unique, as we did not take care of the boundary conditions. Certainly, this construction yields a particular solution in the integral form (2).

**8.2. Airy operator**

We now study the Airy operator, defined as in (1). Note that does not exactly solve the classical Airy equation: . We shall make a change of variables and unknowns in order to go back to the classical Airy equation. This change is very classical in physical literature, and called the Langer’s transformation: , with defined by

in which and is the inverse of the map . By a view of the definition (4), we note that , with . The following lemma links the Airy operator (1) with the classical Airy equation.

Lemma 1Let be the Langer’s transformation defined as above. The function solves the classical Airy equation:

*Proof:* The lemma follows from direct calculations.

*8.2.1. The classical Airy.* Thanks to the Langer’s transformation, we first solve the classical Airy equation (6) for . Let us denote

to be the critical layer size, and introduce the notation . Then solves the truly classical Airy: . We use the following classical lemma:

Lemma 2The classical Airy equation has two independent solutions and so that the Wronskian determinant of and equals to one. In addition, and converge to as ( being real), respectively. Furthermore, there hold asymptotic bounds:

and

in which , for , and is the primitives of for and is defined by the inductive path integrals

so that the integration path is contained in the sector with . The Airy functions for are defined similarly.

Thus, we can now introduce the Green kernel of the classical Airy equation (6):

in which . By definition, we have

Lemma 3There hold pointwise estimates:

for all .

*8.2.2. An approximate Green kernel for Airy operator.* Let us take and where is the Langer’s transformation and denote and . By a view of (5), we define the function so that

in which the factor was added simply to normalize the jump of . It then follows from Lemma 1 together with that

That is, is indeed an approximate Green function of the Airy operator, defined as in (1), up to a small error term of order . It remains to solve (9) for , retaining the jump conditions on across . In view of primitive Airy functions, let us denote

and

Thus, together with our convention that the Green function should vanish as goes to for each fixed , we are led to introduce

in which are chosen so that the jump conditions (see below) hold. Clearly, by definition, solves (9), and hence (10). Here the jump conditions on the Green function read:

We note that from (9) and the jump conditions on across , the above jump conditions of and follow easily. In order for the jump conditions on and , we take

This yields an approximate Green kernel for the Airy operator. Direct, but much tedious, calculations give sufficient point-wise estimates on the Green kernel, yielding the following convolution estimates; see this paper for details of the proof.

Lemma 4Let be the approximate Green kernel constructed as above. There holds

for arbitrary , and for , denoting the function space that consists of decaying functions at an exponential rate of .

*8.2.3. Inverse for Airy operator.* We study the inhomogeneous Airy equation: , for some source . Thanks to the above construction of an approximate Green kernel, we introduce an approximate inverse of operator:

where the error operator is introduced due to the approximation of the Green kernel and thus is defined as

From the convolution estimates, Lemma 4, there hold

for all and . Recalling that , and so the error is indeed small of order . It is worth noting that unlike , the maps from into itself. The reason is that the slight loss of an exponential decay in is due precisely to the linear growth in in the approximate Green function , whereas this linear growth vanishes in the calculation for . We recover the same rate of decay for as that of .

We now construct an inverse for Airy operator by iteration. Let us start with a fixed . Define

for all , with . It follows by induction that

for all . By induction, and hence in as , for sufficiently small . In addition, , which shows that converges to zero in for arbitrary fixed as . Furthermore the series

in as , for some . We then denote , for each . In addition, we have that is, is the exact inverse for the Airy operator. Summarizing, we have proved the following theorem.

Lemma 5Let be positive numbers. Assume that is sufficiently small. There exists an exact solver , which is a bounded operator from to , so that

**8.3. Singularities and contraction of Iter operator**

In this section, we study the smoothing effect of the modified Airy function. Precisely, let us consider the Airy equation with a singular source:

in which , that is and its derivatives decay exponentially at infinity and behaves as near the critical layer . The singular source arises as an error of the inviscid solution when solving the full viscous problem. The key for the contraction of the iteration operator lies in the following lemma:

Lemma 6Assume that . Let be the exact Airy solver of the operator constructed as in Proposition 5 and let . There holds the estimate:

*Proof:* The rough idea is that the convolution can be computed as

in which is bounded and is localized near the critical layer of the size of order . This indicates the bound by as stated in the estimate (18). The factor is precisely due to the linear growth in in the Green kernel . We refer to the paper, Section 5, for details of the proof.

Having provided the estimates on and , and the convolution estimates, we can obtain the contraction of the operator, defined by

Precisely, we can prove the following lemma:

Lemma 7For , the operator is a well-defined map from to . Furthermore, there holds

**8.4. Slow modes**

In this paragraph we explicitly compute the boundary contribution of the first terms in the expansion of the slow Orr-Sommerfeld modes, which are obtained from the Rayleigh solutions:

in which the second term is obtained by the interation via the Iter operator, plus higher order terms. We recall that the Rayleigh solution, again obtained via a perturbative analysis, is of the form:

It is crucial to note that the possible singularity in the Rayleigh solution arises only at the order of . That is, we apply the Airy smoothing operator, Lemma 6, precisely to the term, yielding

This yields at once the following lemma:

Lemma 8Let be the slow mode constructed above. For small , such that and , there hold

**8.5. Fast modes**

Similarly, the fast modes are constructed as a perturbation from the second primitive Airy solutions:

Here, is to normalize the possible blow-up value of on the boundary , since could be arbitrarily large. By construction, there holds the following expansion of the fast mode on the boundary :

By definition, we have and

In the study of the linear dispersion relation, we are interested in the ratio . From the above estimates on and , it follows at once that

As will be calculated below, . Hence, the above ratio is estimated by

Here, we recall that , and . Therefore, we are interested in the ratio for complex , for being in a small neighborhood of . Without loss of generality, in what follows, we consider . Directly from the asymptotic behavior of the Airy functions, we obtain the following lemma:

Lemma 9Let be defined as above. Then, is uniformly bounded on the ray for . In addition, there holds

for all large . At , we have

This yields at once the following estimate on the ratio (22):

Lemma 10As long as is sufficiently large, there holds

In particular, the imaginary part of becomes negative when is large (the ratio has a positive imaginary part when is small).

**8.6. Linear dispersion relation**

Any linear combination of the slow and fast modes is an exact solution to the Orr-Sommerfeld equation. The zero boundary conditions yield the dispersion relation:

We shall show that for some ranges of , the dispersion relation yields the existence of unstable eigenvalues . By Lemmas 8 and 10, the linear dispersion relation (24) simply becomes

We are interested in the region where is large. The dispersion relation yields

Hence as , the eigenvalue converges to and so , from the equation . The existence of a unique near so that the linear dispersion (25) holds, when are sufficiently small, follows easily from the Implicit Function Theorem.

8.6.1 Lower stability branch:

Let us consider the case , for some constant . We recall that . That is, for fixed constant . In addition, since , we have

Thus, we are in the case that the critical layer goes up to the boundary with staying bounded in the limit . We obtain the following lemma.

Lemma 11Let . For sufficiently large, there exists a critical constant so that the eigenvalue has its imaginary part changing from negative (stability) to positive (instability) as increases past . In particular,

*Proof:* The imaginary part of the dispersion relation (25) yields

which gives and so . Next, also from Lemma 10, the right-hand side is positive when is small, and becomes negative when . Consequently, together with (27), there must be a critical number so that for all , the right-hand side is positive, yielding the lemma as claimed.

8.6.2. Intermediate zone:

Let us now turn to the intermediate case when

with . In this case and hence . That is, the critical layer is away from the boundary: . We have the following lemma.

Lemma 12Let with . For arbitrary fixed positive , the eigenvalue always has positive imaginary part (instability) with

*Proof:* As mentioned above, is unbounded in this case. Since , we indeed have

as since . By Lemma 10, the right hand side of the dispersion relation reads

which is positive, since . It is crucial to note that in this case which can be neglected in the dispersion relation (28) as compared to the size of the imaginary part of . This yields the lemma.

**8.6.3 Upper stability branch: **

The upper branch of marginal stability is more delicate to handle. This is the case when the term of order on the left hand side of the dispersion relation has the same order as the right-hand side, and hence it is not clear whether remains positive. We expect that the Rayleigh solutions will dominate when , and the imaginary part of will change from positive (instability) to negative (stability).

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