On the global evolution of self-gravitating matter. Nonlinear interactions in Gowdy symmetry

TL;DR

This paper studies the evolution of self-gravitating fluids with Gowdy symmetry, analyzing the Euler equations, and establishing stability and regularity results for weak solutions in a gravitational setting.

Contribution

It introduces interaction functionals that control weak solutions and proves nonlinear stability of these solutions under Gowdy symmetry assumptions.

Findings

Uniform control of weak solutions via new interaction functionals

Establishment of local regularity and nonlinear stability

Proof of spurious matter field emergence under weak convergence

Abstract

We are interested in the evolution of a compressible fluid under its self-generated gravitational field. Assuming here Gowdy symmetry, we investigate the algebraic structure of the Euler equations satisfied by the mass density and velocity field. We exhibit several interaction functionals that provide us with a uniform control on weak solutions in suitable Sobolev norms or in bounded variation. These functionals allow us to study the local regularity and nonlinear stability properties of weakly regular fluid flows governed by the Euler-Gowdy system. In particular for the Gowdy equations, we prove that a spurious matter field arises under weak convergence, and we establish the nonlinear stability of weak solutions.