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## Math 597F, Notes 5: A few examples of 2D boundary layers

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:

\displaystyle \left \{ \begin{aligned} u_t + u u_x + v u_z &= \mu u_{zz} - p_x, \qquad v = -\int_0^z u_x(x,\theta)\; d\theta \\ u_{\vert_{t=0}} &= u_0(x,z), \qquad u_{\vert_{z=0}} =0, \qquad \lim_{z\rightarrow \infty} u(t,x,z) = u^E(t,x), \end{aligned} \right. \ \ \ \ \ (1)

with the pressure gradient: ${p_x = -u_t^E- u^Eu^E_x(t,x)}$, where ${u^E}$ denotes the tangential component of Euler flow on the boundary ${y=0}$. Here, the tangential velocity component ${u}$ is an (only) unknown scalar function, and the normal velocity component ${v}$ is defined through the divergence-free condition. A simplest example: in the case ${u^E}$ is independent of ${x}$ and ${t}$, any solution to the following heat problem

$\displaystyle u_t = \mu u_{zz} , \qquad u_{\vert_{t=0}} = u_0(z), \qquad u_{\vert_{z=0}} =0, \qquad \lim_{z\rightarrow \infty} u(t,z) = u^E$

gives a boundary layer solution of (1) in the form ${[u,0]}$. 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 ${u^E}$ is a constant. One can introduce a shear flow ${[u_s,0]}$ solving the heat equation with the nonzero boundary condition at infinity in the Prandtl problem, and study the remainder ${\bar u = u - u_s}$ which solves

$\displaystyle \bar u_t + (u_s + \bar u) \bar u_x + \bar v (u_s' + \bar u_z) = \mu \bar u_{zz}, \qquad \bar v = -\int_0^z \bar u_x\; dz,$

together with zero boundary conditions for ${\bar u}$ at ${z=0}$ and ${z=\infty}$. The standard ${L^2}$ energy estimate immediately gives

$\displaystyle \frac 12 \frac{d}{dt} \| \bar u\|_{L^2}^2 + \mu \| \bar u_z\|_{L^2}^2 = - \iint u_s' \bar u \bar v.$

In the above energy balance, the right-hand side does not have a sign and even worse presents a loss of ${x}$-derivative: ${\bar v = -\int_0^z \bar u_x\; dz}$, 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 ${x}$-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-${\frac 74}$ class (an intermediate class between ${C^\infty}$ smooth and analytic). In fact, for ${C^\infty}$ 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:

$\displaystyle (t,x,z) \mapsto (\tau,\xi,\eta) = \Big (t,x,\frac{u(t,x,z)}{u^E(t,x)} \Big), \qquad w = \frac{u_z(t,x,z)}{u^E(t,x)}.$

A price to pay with this change of variables is the requirement of monotonicity of ${u}$ with respect to ${z}$ or equivalently the positivity of ${w}$, since the Jacobian determinant of the change of variables ${J = w}$. Positivity of ${w}$ allows one to change back to the original coordinates from the solution ${w(\tau, \xi, \eta)}$. We derive an equation for ${w = w(\tau, \xi, \eta)}$. Calculations yields

$\displaystyle \partial_t = \partial_\tau + \eta_t \partial_\eta , \qquad \partial_x = \partial_\xi + \eta_x \partial_\eta, \qquad \partial_z = \eta_z \partial_\eta$

in which ${\eta_t = u_t/u^E - u u_t^E/|u^E|^2}$, ${\eta_x = u_x/u^E - u u_x^E/|u^E|^2}$, and ${\eta_z = w}$. Differentiating the Prandtl equation with respect to ${z}$ yields

$\displaystyle \Big( \partial_t + u \partial_x + v\partial_z - \mu \partial_z^2 \Big) u_z = 0$

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

$\displaystyle \Big( \partial_t + u \partial_x + v\partial_z - \mu \partial_z^2 \Big) w = - \frac{u_z (\partial_t + u \partial_x) u^E}{|u^E|^2}.$

The left-hand side is simply

$\displaystyle \Big( \partial_t + u \partial_x + v\partial_z - \mu \partial_z^2 \Big) w = ( \partial_\tau + u \partial_\xi ) w + (\eta_t + u \eta_x + v \eta_z) w_\eta - \mu w w_\eta^2 - \mu w^2 w_{\eta \eta}$

Computing the middle term and using the Prandtl equation immediately give

$\displaystyle \eta_t + u \eta_x + v \eta_z = \mu w w_\eta - \frac{p_x}{u^E} - \frac{u (\partial_t + u \partial_x) u^E}{|u^E|^2}.$

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

\displaystyle \left \{ \begin{aligned} w_\tau + \eta u^E w_\xi + A w_\eta + B w- \mu w^2 w_{\eta \eta } & =0, \qquad (\tau, \xi, \eta)\in \mathbb{R}_+ \times \mathbb{R} \times [0,1] \\ w_{\vert{\tau=0}} = w_0(\xi, \eta), \qquad \mu w \partial_\eta w_{\vert{\eta =0}} &= \frac{p_x}{u^E},\qquad w_{\vert_{\eta =1}} =0 \end{aligned} \right. \ \ \ \ \ (2)

in which ${A,B}$ are defined by

$\displaystyle A(\tau, \xi, \eta): = - \frac{1}{u^E}\Big( (\eta - 1)u^E_t + (\eta^2-1) u^E u^E_x\Big)$

$B(\tau, \xi, \eta): = \frac{1}{u^E} (u^E_t + \eta u^E u^E_x).$

Again, in the simple situation when the boundary value of Euler flow ${u^E}$ is a constant, normalized to be one: ${u^E \equiv 1}$. The Prandtl problem is reduced to

\displaystyle \left \{ \begin{aligned} w_\tau + \eta w_\xi - \mu w^2 w_{\eta \eta } & = 0, \qquad (\tau, \xi, \eta)\in \mathbb{R}_+ \times \mathbb{R} \times [0,1] \\ w_{\vert{\tau=0}} = w_0(\eta), \qquad &\mu w \partial_\eta w_{\vert{\eta =0}} = 0,\qquad w_{\vert_{\eta =1}} =0 . \end{aligned} \right. \ \ \ \ \ (3)

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 ${[0,L]}$ or on the half-line in ${x}$-variable). Such an assumption persists for short time, thanks to the nature of the parabolic equations. More precisely, Oleinik considers the problem (2) on ${[0,T]\times [0,L] \times [0,1]}$ (that is, one also needs to prescribe boundary data for ${u}$ at ${x=0}$). Oleinik assumes that the boundary data ${u^E}$ is nonzero and sufficiently smooth and that there are positive constants ${c_0, C_0}$ so that initial data ${w_0}$ satisfies

$\displaystyle c_0 (1-\eta) \le w_0(\eta) \le C_0 (1-\eta),\qquad \eta \in [0,1],\ \ \ \ \ (4)$

and has bounded derivatives. Then, there exists a positive time ${T}$ such that the problem (3) has a unique smooth solution ${w(\tau, \xi, \eta)}$ in ${[0,T]\times [0,L] \times [0,1]}$. In addition, there hold

$\displaystyle \theta_1 (1-\eta) \;\le\; w (\tau, \xi,\eta)\;\le\; \theta_2(1-\eta), \qquad |w_\tau(\tau, \xi,\eta)|+ |w_\xi(\tau, \xi,\eta)|\;\le\; \theta_2(1-\eta),\ \ \ \ \ (5)$

uniformly for all ${(\tau, \xi,\eta) \in [0,T]\times [0,L]\times [0,1]}$, for positive constants ${\theta_1, \theta_2}$. 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 ${w(\tau, \xi, \eta) \approx 1-\eta}$ is equivalent to the estimate

$\displaystyle c_0 e^{-\theta_1 z} \le 1- \frac{u(t,x,z)}{u^E(t,x)} \le C_0 e^{-\theta_2 z}$

which asserts that the Prandtl solution ${u(t,x,z)}$ converges exponentially in ${z}$ to the boundary Euler flow ${u^E(t,x)}$. Note that in the Prandtl’s variables, there holds the relation

$\displaystyle z = \int_0^{u(t,x,z)/u^E(t,x)} \frac{1}{w(t,x,\eta')} d\eta'.$

Differentiating this identity with respect to ${t}$ and ${x}$, we immediately obtain

\displaystyle \begin{aligned} u_t &= u\frac{u^E_t}{u^E} + wu^E \int_0^{u/u^E}\frac{w_t}{w^2}(t,x,\eta')d\eta' \\ u_x &= u\frac{u^E_x}{u^E} + wu^E \int_0^{u/u^E}\frac{w_x}{w^2}(t,x,\eta')d\eta'. \end{aligned}\ \ \ \ \ (6)

The expression for ${v}$ can be obtained directly from the equation (noting ${u_z>0}$). We note that Oleinik was also able to construct global-in-${x}$ solution with a short time existence and global-in-time solution with a short distance in ${x}$ (that is, either ${T}$ or ${L}$ has to be sufficiently small). There is no global-in-time theory of smooth solutions, say in a fixed spatial domain ${[0,L]\times \mathbb{R}_+}$. 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 ${u^E \not \equiv0}$.

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: ${(x,z)\mapsto (x,\eta(x,z))}$ with

$\displaystyle \eta (x,z) =\int_{0}^{z} u(x,\theta)d\theta , \qquad \bar u(x,\eta ): = u (x,z) .$

It follows that ${\partial_x = \partial_x + \eta_x \partial_\eta = \partial_x - v \partial_\eta}$ and ${\partial_z = u \partial_\eta}$. This yields a parabolic equation for ${w}$:

$\displaystyle \bar u \bar u_x = \mu \bar u (\bar u \bar u_\eta)_\eta - p_x$

In fact, it appears more convenient to work with the function ${k = \bar u^2}$ which then solves

\displaystyle \left \{ \begin{aligned} k_x &= \mu \sqrt k k_{\eta \eta} - 2 p_x, \qquad x \in [0,L], \qquad \eta \ge 0 \\ k_{\vert_{x=0}} &= k_0(\eta), \qquad k_{\vert_{\eta =0}} = 0, \qquad \lim_{\eta \rightarrow \infty} k(x,\eta) = |u^E(x)|^2. \end{aligned} \right. \ \ \ \ \ (7)

The standard Maximum Principle can be applied to construct nonnegative local-in-${x}$ steady solutions to the Prandtl boundary layer equations. The main assumption in Oleinik’s work is that the tangential velocity component ${u}$ has boundary data satisfying ${u_z(0,0)>0}$. It is physically believed, but not rigorously proven, that there is a finite ${x_0}$ so that the local solution to the Prandtl problem satisfies ${u_z(x_0,0) =0}$ (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 ${u^E\equiv 1}$ (normalized). He searches for the stationary solution in term of self-similar stream function:

$\displaystyle u = \partial_z \psi, \qquad v = -\partial_x \psi , \qquad \psi = \sqrt {\mu x} f(\eta) , \qquad \eta = \frac{z}{\sqrt {\mu x}}.$

Here, ${\sqrt {\mu x}}$ accounts for the boundary layer thickness at the position ${x}$ on the boundary (or on the flat plate). Direct calculations yield ${\partial_x = \eta_x \partial_\eta = - \frac{\eta }{2x} \partial_\eta }$ and ${\partial_z = \frac{1}{\sqrt {\mu x}} \partial_\eta}$. This gives at once

$\displaystyle u = \psi_z = f'(\eta), \qquad v = -\psi_x = \frac{\eta\mu}{2\sqrt {\mu x}} f'(\eta) - \frac{\mu }{2\sqrt {\mu x}} f(\eta)$

and

$\displaystyle u_x = \psi_{xz} = -\frac{\eta }{2x} f''(\eta), \qquad u_z = \psi_{zz} = \frac{1}{\sqrt {\mu x}} f''(\eta), \qquad u_{zz} = \frac 1{\mu x} f'''(\eta).$

Putting all these into the steady Prandtl equation (with zero pressure gradient, since ${u^E}$ is a constant), we get an ODE problem for ${f = f(\eta)}$:

$\displaystyle f''' + \frac 12 f f'' = 0, \qquad f(0) = f'(0) = 0, \qquad \lim_{\eta\rightarrow \infty} f'(\eta) = 1.$

This ODE has been solved numerically, and provides a good approximation in describing the shape of boundary layers near the boundary (for instance, when ${\eta}$ is sufficiently small, one can solve the ODE problem using Taylor’s asymptotic expansions with respect to ${\eta}$). 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.