Local Well-posedness of Free Boundary Relativistic Barotropic Fluid Equation in Minkowski Space-time

Abstract

In this dissertation, we study the motion of a relativistic barotropic fluid with a free boundary. Relativistic barotropic fluids are fluids whose density and pressure are directly related. Such fluids are used as models in a wide range of scientific applications from meteorology to astrophysics. We prove that such a system of equations is locally well-posed, and the unique solution has the same regularity as the initial data. We use two methods to achieve the result. The first method is to construct approximate solutions to the equation by mollifying the nonlinear equation directly in the Lagrangian coordinate, and then passing to the limit so that the approximate solutions converge to the exact solution. The second method is to first consider the equation on a linear level, solve the linear equation using Galerkin approximation, and then solve the nonlinear equation by iterating on the solutions to these linear equations and passing to the limit.