# Generalized model solutions for physical systems modeled by PDEs and their linear stability

#### Abtract

In this lecture consisting of 6 sequences of 1hr30 minutes, we propose to study various physical system such as conservative systems of equations supplemented with a reaction diffusion equation (Zeldovich equations), isentropic Navier Stokes equations coupled with magnetohydrodynamics, over dense plasma physics equations, Low mach number system of equations for fluid mechanics, the propagation of nerve influx (the Hodgkin-Huxley equations). We first reduce the study of a particular solution of these systems into the solution of a system of ODEs depending only on $$x$$ or $$x-ct$$, $$x$$ being a chosen spatial coordinate and $$t$$ being the time. The theory of system of ODEs on $$[X0,+\infty[$$ or on $$[X0,X1[$$ is then developed, where $$X1$$ is a regular singular point of the system. This amounts for identifying an analytic solution in a half plane of the form $$Rz \geq \delta 0$$ or in a neighborhood of $$X1$$. Using this particular solution, one derives the perturbation methods which identify the stability and instability regimes in the high frequency setup (wavelength of the transverse to x perturbation small with respect to the characteristic dimensions of the problem.

#### Date

18.05.2015: 10:15-11:45
19.05.2015: 10:15-11:45
20.05.2015: 08:30-10:00
20.05.2015: 10:15-11:45

MI 02.12.020