On the Onsager's energy conservation for the convergence dynamics of the 3D-Leray-$\alpha$ gaseous star model

TL;DR

This paper investigates energy conservation in the convergence dynamics of a 3D-Leray-alpha model for gaseous stars, testing Onsager's hypothesis by analyzing the model's behavior as it approaches the Euler equations.

Contribution

It introduces a Leray-alpha gaseous star model as an inviscid regularization of Euler equations and examines its energy conservation properties in the limit as regularization vanishes.

Findings

The model converges to Euler equations as regularization parameter approaches zero.

Energy conservation is analyzed in the context of Onsager's hypothesis.

The model provides insights into the regularity conditions for energy conservation.

Abstract

This research presents a model that accurately represents the motions of gaseous stars We employ the Navier-Stokes-Poisson system to transform compressible Euler equations into non-compressible ones by combining quasineutral and inviscid conditions. We intend to put Onsager's hypothesis to the test using the Leray-alpha gaseous star model. This conjecture connects energy conservation with the regularity of weak solutions in the Euler equations. The model used in this work functions as an inviscid regularization of the Euler equations. It technically converges to the Euler equations as the regularization length scale $\alpha$ approaches $0^{+}$.