Weakly regular fluid flows with bounded variation on a Schwarzschild background

TL;DR

This paper develops a global existence theory for weak solutions with bounded variation for isothermal fluids around a Schwarzschild black hole, including shock waves and equilibrium states, using a novel random choice method.

Contribution

It introduces a new random choice method for weak BV solutions in curved spacetime and establishes stability and existence results for complex fluid behaviors near black holes.

Findings

Global existence of BV solutions with shocks

Exact preservation of equilibrium solutions

Stability of fluid equilibria under perturbations

Abstract

We study the global dynamics of isothermal fluids evolving in the domain of outer communication of a Schwarzschild black hole. We first formulate the initial value problem within a class of weak solutions with bounded variation (BV), possibly containing shock waves. We then introduce a version of the random choice method and establish a global-in-time existence theory for the initial value problem within the proposed class of weakly regular fluid flows. The initial data may have arbitrary large bounded variation and can possibly blow up near the horizon of the black hole. Furthermore, we study the class of possibly discontinuous, equilibrium solutions and design a version of the random choice method in which these fluid equilibria are exactly preserved. This leads us to a nonlinear stability property for fluid equilibria under small perturbations with bounded variation. Furthermore, we can also encompass several limiting regimes (stiff matter, non-relativistic flows, extremal black hole) by letting the physical parameters (mass of the black hole, light speed, sound speed) reach extremal values.