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

We now return to the Orr-Sommerfeld equations (the linearized Navier-Stokes equations around a boundary layer ; see this lecture):

For convenience, we denote and . The Orr-Sommerfeld equation simply reads

We shall construct solutions for each fixed pair and for sufficiently small .

As mentioned, there are unstable and stable profiles . We warm up ourselves by considering the unstable case, namely: we assume that there is a pair , with , so that the Rayleigh problem has a nonzero solution:

This latter assumption means precisely that the unstable eigenvalue of the linearized Euler operator is simple. This is equivalent to the condition that the Wronskian determinant, or the Evans function, satisfies . We introduce a perturbative analysis to construct a solution to the Orr-Sommerfeld problem, when is sufficiently small, so that as . This yields a growing mode to the linearized Navier-Stokes problem.

**7.1. A toy model**

Consider the problem

Let be the (bounded) inviscid solution: . Due to the outward characteristic to the boundary , bounded inviscid solutions need not to satisfy the zero boundary condition and the viscous equation. Boundary layers are needed, solving

for some source , which is due to the error introduced by the inviscid solution. Precisely,

leaving the error term: , which might not be well-defined (and so not contractive for the iteration to work!). We then correct this by introducing the iteration operator:

The Boundary Layer operator added is to regularize the possible singularity coming from the loss of derivatives in the error: . We can check that the new solution solves

Inductively, let be the error at the step and define

to be the next iterative solution. There holds

It suffices to show that is well-defined and contractive in some function space, and so and as , giving an exact solution to the problem . Let us work with the function space . The iteration solves , with , which gives

The boundary value can be taken to be the minus of that of the inviscid solution. For simplicity, we take . The iteration is contractive in , since

and

which is clearly bounded by . This proves that the Iter operator is contractive for sufficiently small , precisely: there holds for all bounded functions . It is worth noticing that the contraction is due to the localizedness of the boundary layers of size (and hence, norm is of order as seen above).

**7.2. Asymptotic behavior as **

In order to construct the independent solutions of Orr-Sommerfeld, let us study their possible behavior at infinity. One observes that as , solutions must behave like solutions of constant-coefficient limiting equation:

with . Solutions are thus of the form with or , where

Therefore, we can find two solutions with a “slow behavior” (one decaying and the other growing) and two solutions with a “fast behavior” where is of order (one decaying and the other growing). The first two slow-behavior solutions will be perturbations of eigenfunctions of the Rayleigh equation. The other two, , are specific to the Orr Sommerfeld equation and will be linked to the solutions of . The standard conjugation method for ODEs does not apply directly due to the -dependence in the equation, which makes the coefficients of the ODE decay exponentially only at a vanishing rate (in the fast variable) when tracking the fast modes. In fact, since the fast eigenvalues of the corresponding ODE system are well-separated from the slow ones, we will track these fast modes by diagonalization, keeping their variable-coefficients in our setting.

**7.3. Slow modes**

In this section, we iteratively construct “slow” modes of the Orr-Sommerfeld equations, starting from the Rayleigh solutions: constructed in the previous lecture. Our iteration follows similarly to that for the toy model problem. We start from a solution solving the Rayleigh equation: . Let denote the Orr-Sommerfeld operator:

Then, together with the definition of operator to solve for , there holds

in which the error term on the right is bounded by . Next, by induction, let us assume that we have constructed so that

with a sufficiently small error term in . We then improve the error term by constructing a new approximate solution so that it solves the Orr-Sommerfeld equations with a better error in . Precisely, we introduce

which clearly solves the Orr-Sommerfeld with a new error: . Again, by the definition of operator, we compute

which is bounded by . By Lemma 2 in the last lecture, we have

in which denotes the Evans function (or precisely, the Wronskian determinant) for the inviscid equations. Hence, whenever , converges to zero in and thus the series also converges to an exact solution of the Orr-Sommerfeld equations. We summarize the above into the following lemma, providing two independent slow modes of the Orr-Sommerfeld equations:

Lemma 1Let be the two independent Rayleigh solutions constructed in Lemma 2 in the previous lecture, and let be the Wronskian determinant of the two solutions (which is independent of ). For sufficiently small such that , there exist two independent solutions which solve the Orr-Sommerfeld equations

so that is close to the Rayleigh solutions in . Precisely, we have

for some positive constant independent of .

**7.4. Fast modes**

In this section, we shall construct two independent solutions, which asymptotically behave as , of the Orr-Sommerfeld equation: .

Lemma 2For sufficiently small , there exist two independent solutions which solve the Orr-Sommerfeld equations

so that satisfy

in which . In addition, the Wronskian determinant satisfies the estimate

As we are in the case of unstable profiles, we can assume that is away from the real axis and so is bounded below away from zero. Hence, . The Lemma 2 thus yields two independent solutions that behave as near the infinity.

*Proof:* We rewrite the Orr-Sommerfeld equation as

in which for convenience we have denoted and . We make a change of variables:

The equation for reads

It is convenient to write the above equation as the first order ode system. We introduce , with prime denoting the derivative with respect to . The system for reads

We note that is bounded away from zero, for . Hence, four eigenvalues of , two slow and two fast, satisfy

Let be the two fast eigenvalues with positive/negative real part, respectively. Notice that , and . We let

and for each case, we introduce

The function then solves

in which is defined as the lower block of the matrix , and the first row of the matrix is zero and by direct computation.

We further introduce

We write . Then, the above system yields

Consider the case. We observe that the eigenvalues of are the same as those of except for the eigenvalue . Thus, the eigenvalues of have real parts that are all positive and bounded away from zero. This proves that there is a positive constant so that

Taking in (5), we get

We denote by the right-hand side of the above integral form. We shall show that is contractive from into itself. Indeed, for any with , one calculates

By taking small enough, the above proves that is a contractive map, and thus there exists a (unique) solution solving (4) in the case. Furthermore, there holds

Next, consider the case in (5). In this case, the eigenvalues of have real parts that are all negative and bounded away from zero. This yields

for some . Taking in (5), we get

Again, if we denote by the right-hand side of the above integral form, we have for any with

This proves that is a contractive map for small , and thus there exists a (unique) solution solving (4) in the case. Furthermore, there holds

In summary, we have obtained two independent solutions to (3) such that

for arbitrary constants . By rescaling to the original variables, the lemma follows at once.

**7.5. Dispersion relation**

We are ready to construct a solution to the Orr-Sommerfeld problem, with . Indeed, we let and be two slow and fast modes that decay at infinity. A bounded solution to the Orr-Sommerfeld problem is constructed as a linear combination of these two decaying modes:

The zero boundary conditions on yield the linear dispersion relation:

We need to show that there are choices of so that the above relation holds for sufficiently small. From the construction of fast decaying modes, we have

Let us calculate the ratio for the slow modes. We recall that there holds the asymptotic expansion:

as long as . Here, we take the Rayleigh eigenfunction so that such that and on the boundary (we note that if vanishes on the boundary, then identically, since by the Rayleigh equation, all derivatives of vanish on the boundary). The dispersion relation then yields

Now, thanks to the assumption that the boundary layer is unstable to the inviscid problem, , with . The condition guarantees that there is a continuous dependence , with , so that the dispersion relation holds at for all sufficiently small. Note in addition that and so the smallness assumption: is valid. A growing mode of the linearized Navier-Stokes follows (for the unstable profiles ).

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