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## Math 597F, Notes 6: Linear inviscid stability theory

We study the linearization of 2D Navier-Stokes around a boundary layer ${\vec v_\mathrm{app} =[\bar u, \sqrt \nu \bar v](t', x', y'/\sqrt \nu)}$, in which ${(t',x',y')}$ denotes the original coordinates for Navier-Stokes. Materials in this lecture are drawn from the joint paper(s) with Grenier and Guo (here and here).

## 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.