Existence and stability of waves in hydrodynamic models with dissipation

We investigate existence and stability of waves in hydrodynamic models with dissipation. In chapter 1 we study a two-fluid description for high and low temperature components of the electron velocity distribution in an idealized tokamak plasma evolving on a cylindrical domain, and taking into account nonlinear drift effects only. We refine previous results on the laminar steady state stability and include viscosity. Taking the temperature difference as the primary parameter we show that linear instabilities and bifurcations occur within a finite interval and for small enough viscosity only, while the steady state is globally stable for parameters sufficiently far outside the interval. We find that primary instabilities always stem from the lowest spatial harmonics for aspect ratios of poloidal vs. radial extent below some value larger than 2. Moreover, we show that any codimension-one bifurcation of the laminar state is supercritical, yielding spatio-temporal oscillations in the form of traveling waves, hence locally stable for such bifurcations destabilizing the laminar state. In the degenerate case, where the instability region in the temperature difference is a point, these solutions form an arc connecting the bifurcation points. We also provide numerical simulations to illustrate and corroborate the analysis, and find additional bifurcations of the traveling waves. In chapter 2 we study two hydrodynamic models. The first is a 1-D compressible Euler system with diffusion-dispersion terms. We prove existence of traveling waves or dispersive shocks for this system. Moreover we analyze the spectral properties of the linearized system around steady states and profiles. We performed numerics about the Evans function and provide numerical evidence for the stability of a wave. The second system is a modification of the first with a nonlinear dissipation term. We prove existence of profiles for it.