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## The onset of instability in first-order systems

Nicolas Lerner, Ben Texier and I have just submitted to arxiv our long paper on “The onset of instability in first-order systems”, in which we prove the Hadamard’s instability for first-order quasilinear systems that lose its hyperbolicity in positive times. Precisely, we consider the Cauchy problem for the following first-order systems of partial differential equations:

## Math 597F, Notes 8: unstable Orr-Sommerfeld solutions for stable profiles

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:

$\displaystyle Ray_\alpha(\phi) = \epsilon \Delta_\alpha ^2 \phi,$

with zero boundary conditions on ${\phi}$ and ${\phi'}$, in which ${Ray_\alpha(\cdot)}$ denotes the Rayleigh operator and ${\Delta_\alpha = \partial_z^2 - \alpha^2}$.

## Math 597F, Notes 7: unstable Orr-Sommerfeld solutions for unstable profiles

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

\displaystyle \left \{ \begin{aligned} (U-c) (\partial_z^2 - \alpha^2) \phi - U'' \phi &= \epsilon (\partial_z^2 - \alpha^2)^2 \phi \\ \phi_{\vert_{z=0}} = \phi'_{\vert_{z=0}} &= 0, \qquad \lim_{z\rightarrow \infty} \phi(z) =0. \end{aligned} \right. \ \ \ \ \ (1)

For convenience, we denote ${Ray_\alpha(\phi) = (U-c) (\partial_z^2 - \alpha^2) \phi - U'' \phi}$ and ${\Delta_\alpha = \partial_z^2 - \alpha^2}$. The Orr-Sommerfeld equation simply reads

$\displaystyle Ray_\alpha(\phi) = \epsilon \Delta_\alpha ^2 \phi.$

We shall construct solutions for each fixed pair ${(\alpha,c)}$ and for sufficiently small ${\epsilon}$.