Let us give a few examples of boundary layer solutions to the Prandtl problem, derived in the last lecture. In 2D, we recall the Prandtl layer problem:

with the pressure gradient: , where denotes the tangential component of Euler flow on the boundary . Here, the tangential velocity component is an (only) unknown scalar function, and the normal velocity component is defined through the divergence-free condition. A simplest example: in the case is independent of and , any solution to the following heat problem

gives a boundary layer solution of (1) in the form . We call a solution of this form to be a shear flow.

Certainly, one would like to solve the boundary layer problem for more general data. Despite the fact that the scalar equation looks so simple, the well-posedness theory is quite delicate, even locally in time. The issue is the lack of the (usual) energy estimates. To see this, consider again the case is a constant. One can introduce a shear flow solving the heat equation with the nonzero boundary condition at infinity in the Prandtl problem, and study the remainder which solves

together with zero boundary conditions for at and . The standard energy estimate immediately gives

In the above energy balance, the right-hand side does not have a sign and even worse presents a loss of -derivative: , which is the main source of difficulty in building a local-in-time well-posedness theory of smooth solutions for the Prandtl problem. One could perform higher-order energy estimates for derivatives, but the loss of -derivatives remains. Sammartino and Caflisch (also, recently Kukavica and Vicol) were able to obtain local-in-time solutions in the analytic function spaces, with which the high energy estimates allow to have infinite loss of derivatives. Gérard-Varet and Masmoudi recently extended the local theory for solutions to be in a larger class: Gevrey- class (an intermediate class between smooth and analytic). In fact, for smooth solutions, the Prandtl problem is known to be ill-posed; for instance, Gérard-Varet and Dormy showed that the linearized problem around a non-monotonic shear profile has an exponentially growing solution in time. Fortunately, there are non-trivial examples of smooth solutions to the Prandtl problem, which shall be discussed below.

**1. Oleinik’s solutions**

In 1960s, Oleinik constructed Prandtl solutions using the Crocco’s transformation as follows. Introduce new variables:

A price to pay with this change of variables is the requirement of monotonicity of with respect to or equivalently the positivity of , since the Jacobian determinant of the change of variables . Positivity of allows one to change back to the original coordinates from the solution . We derive an equation for . Calculations yields

in which , , and . Differentiating the Prandtl equation with respect to yields

which is the vorticity equation for the boundary layer. This gives

The left-hand side is simply

Computing the middle term and using the Prandtl equation immediately give

Notice the term involving is cancelled out. This is the real advantage of this change of variables. Rearranging terms, we obtain a parabolic problem for :

Again, in the simple situation when the boundary value of Euler flow is a constant, normalized to be one: . The Prandtl problem is reduced to

Problems (2) and (3) are degenerate parabolic problems. Oleinik was able to construct local-in-time smooth solution to the above problems. Oleinik assumes the monotonicity on initial data and boundary data (she works on a bounded interval or on the half-line in -variable). Such an assumption persists for short time, thanks to the nature of the parabolic equations. More precisely, Oleinik considers the problem (2) on (that is, one also needs to prescribe boundary data for at ). Oleinik assumes that the boundary data is nonzero and sufficiently smooth and that there are positive constants so that initial data satisfies

and has bounded derivatives. Then, there exists a positive time such that the problem (3) has a unique smooth solution in . In addition, there hold

uniformly for all , for positive constants . There also hold point-wise bounds for the derivatives (In fact, one could construct Oleinik’s solutions via weighted energy estimates applying directly to the equation (2); see, for instance, this paper where we proved the stability of Oleinik’s solutions. I believe that similar estimates there should yield local-in-time solutions of (2)). Writing back to the Prandtl’s variables, it follows that the point-wise estimate is equivalent to the estimate

which asserts that the Prandtl solution converges exponentially in to the boundary Euler flow . Note that in the Prandtl’s variables, there holds the relation

Differentiating this identity with respect to and , we immediately obtain

The expression for can be obtained directly from the equation (noting ). We note that Oleinik was also able to construct global-in- solution with a short time existence and global-in-time solution with a short distance in (that is, either or has to be sufficiently small). There is no global-in-time theory of smooth solutions, say in a fixed spatial domain . In fact, there is a blow-up result in finite time of compactly supported smooth solutions of the Prandtl equations by E and Enquist (1998). It is noted that this blow-up solution does not satisfy nonzero boundary condition at infinity, when .

**2. Steady boundary layers**

Certainly, the above Crocco’s variables work for the steady (time-independent) boundary layers. To avoid the need of monotonicity along the boundary layer profiles, Oleinik uses the so-called von Mises transformation: with

It follows that and . This yields a parabolic equation for :

In fact, it appears more convenient to work with the function which then solves

The standard Maximum Principle can be applied to construct nonnegative local-in- steady solutions to the Prandtl boundary layer equations. The main assumption in Oleinik’s work is that the tangential velocity component has boundary data satisfying . It is physically believed, but not rigorously proven, that there is a finite so that the local solution to the Prandtl problem satisfies (vorticity vanishes). This is indeed the point on the plate at which boundary layers start to separate from the boundary: the boundary layer separation, which is believed to be inevitable. There also seems to be no mathematical theory to extend the boundary layer solution after the separation point.

**3. Blasius boundary layers**

Right after Prandtl derived his famous boundary layer equation, one of his students, Blasius, constructs a steady self-similar solution for steady flows and for constant boundary Euler flow (normalized). He searches for the stationary solution in term of self-similar stream function:

Here, accounts for the boundary layer thickness at the position on the boundary (or on the flat plate). Direct calculations yield and . This gives at once

and

Putting all these into the steady Prandtl equation (with zero pressure gradient, since is a constant), we get an ODE problem for :

This ODE has been solved numerically, and provides a good approximation in describing the shape of boundary layers near the boundary (for instance, when is sufficiently small, one can solve the ODE problem using Taylor’s asymptotic expansions with respect to ). Still, no mathematical analysis is available to prove the approximation of Blasius boundary layers from the exact Navier-Stokes solutions with an estimate on the approximation error. As a related result, in this paper with Y. Guo, we are able to establish the rigorous boundary layer approximation of steady Navier-Stokes over a short moving plate.

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