Effects of the Background Viscosity in Hyperbolic Systems of Conservation Laws

Abstract

Hyperbolic conservation laws naturally arise when second-order viscous terms are neglected in partial differential equations. As discontinuities can occur in weak solutions of conservation laws, it is necessary to have their counterparts in the viscous setting. This thesis is divided into two parts. The first part focuses on systems from magnetohydrodynamics (MHD). We analyze small-amplitude shock profiles and show that their existence sensitively depends on the background viscosity. This is linked to heteroclinic bifurcations in the associated dynamical systems, and we derive explicit conditions under which such bifurcations occur. Subsequently, we study two classes of planar, quadratic systems derived from MHD. The first, the MHD-Burgers model, serves as a prototype of a non-strictly hyperbolic system and illustrates how the structure of Riemann solutions depends on background viscosity. The second, denoted the Brio-Rosenau model, essentially captures the left- and right-propagating waves in the Riemann problem for barotropic MHD. We construct explicit solutions to the Riemann problem and analyze their viscosity dependence using wave-pattern maps. We distinguish between classical and nonclassical shock structures depending on the background viscosity. The second part of the thesis investigates a model for relativistic fluid dynamics proposed by Bemfica, Disconzi, and Noronha. We analyze the dynamical system governing shock profiles in this model, focusing on bifurcation phenomena and the existence of heteroclinic orbits. The system is shown to exhibit a singular perturbation structure. Additionally, we identify oscillatory heteroclinic orbits, which represent a qualitative difference from the non-relativistic case. We then explore these profile equations from the perspective of bifurcation theory, identifying a generic structure involving a saddle-node bifurcation combined with a singular perturbation.