On I. Symmetry breaking under perturbations, and, II. Relativistic fluid dynamics.

Abstract

I have studied two different problems, from bifurcation theory and fluid dynamics respectively. I first consider the symmetry breaking problem for radial solutions of certain semilinear elliptic equations. My main result here is that symmetry breaking occurs along an infinite family of solutions to these equations. In order to do this I first show that symmetry must break infinitesimally, i.e., the linearized operator has a non-radial element in it's kernel. Then utilizing the equivariant Conley index and noting that certain spaces differ as representations of the orthogonal group, I show that actual symmetry breaking occurs. The second question I study is from relativistic fluid dynamics. I consider the system of hyperbolic conservation laws describing the conservation of baryon number and momentum in one dimensional relativistic fluid dynamics. When the equation of state is that for an ideal isentropic gas, I demonstrate the existence of solutions with shocks for the Cauchy problem. I construct, using the Glimm scheme, bounded weak solutions of the initial value problem in a two dimensional Minkowski spacetime. The analysis is based upon showing that a certain nonlocal functional is decreasing on the approximate solutions generated via Glimm's method. The main technical points in the analysis involve showing that the shock curves based at different points are congruent in the plane of Riemann invariants and that the system satisfies the shock interaction condition. In the case of an isentropic fluid with a polytropic equation of state, I demonstrate the existence of a (unique) weak solution to the Riemann problem. An invariant region for the solution of the Riemann problem is established and it is proved that all speeds remain bounded by the speed of light throughout the solution.