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:

## Archive for April, 2015

## The onset of instability in first-order systems

Posted in Instabilities, New papers on April 19, 2015| Leave a Comment »

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

Posted in Math 597F: topics on boundary layers on April 19, 2015| Leave a Comment »

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:

with zero boundary conditions on and , in which denotes the Rayleigh operator and .

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

Posted in Math 597F: topics on boundary layers on April 19, 2015| Leave a Comment »

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 .