Date: 14 April 2025 - 18 April 2025
The course is meant to be an introduction to the strategy of proof and technical tools used to prove that spectral stability implies nonlinear stability for traveling waves of partial differential equations of arbitrary amplitude.
Concepts and tools to be discussed include orbital stability, space-modulated stability, modulation systems, Floquet-Bloch transform, Laplace transform, Green functions, Evans functions, high-frequency damping estimates,…
We shall show those in practice by studying
- the stability theory for plane periodic waves of parabolic systems (as derived jointly with Mat Johnson, Pascal Noble and Kevin Zumbrun)
- the stabilty theory of singular waves of hyperbolic systems (still under construction, with some insights gained by the speaker jointly with Vincent Duchêne, Grégory Faye and Louis Garénaux)
- an example, for a family of viscous waves, of a stability result (obtained by the speaker jointly with Paul Blochas) uniform with respect to the singular vanishing viscosity limit despite shock layers.