Local existence of solutions to the Euler-Poisson system, including densities without compact support

TL;DR

This paper proves local existence and well-posedness of solutions to the Euler-Poisson system with densities that decay at infinity or have compact support, using weighted Sobolev spaces and a novel nonlinear estimate.

Contribution

It introduces a new analytical framework employing weighted Sobolev spaces and a nonlinear estimate to establish local solutions for the Euler-Poisson system with physically relevant densities.

Findings

Solutions have finite mass and energy.

Includes static spherical solutions for specific adiabatic constants.

The functional setting is suitable for physical models and Newtonian limit analysis.

Abstract

Local existence and well posedness for a class of solutions for the Euler Poisson system is shown. These solutions have a density $\rho$ which either falls off at infinity or has compact support. The solutions have finite mass, finite energy functional and include the static spherical solutions for the adiabatic constant $\gamma=\frac{6}{5}$. The result is achieved by using weighted Sobolev spaces of fractional order and a new non linear estimate which allows to estimate the physical density by the regularised non linear matter variable. Gamblin also has studied this setting but using very different functional spaces. However we believe that the functional setting we use is more appropriate to describe a physical isolated body and more suitable to study the Newtionan limit.