The global smooth symmetric solution to 2-D full compressible Euler system of Chaplygin gases

TL;DR

This paper proves the global existence of smooth symmetric solutions to the 2-D full compressible Euler system of Chaplygin gases, highlighting differences from 3-D cases and employing weighted energy estimates.

Contribution

It establishes the first global symmetric solution result for 2-D Chaplygin gases using novel weighted energy estimates and continuous induction.

Findings

Global smooth symmetric solutions are proven to exist for 2-D Chaplygin gases.

Weighted energy estimates are crucial for controlling solutions.

The approach differs from 3-D cases due to dimensional effects.

Abstract

For one dimensional or multidimensional compressible Euler system of polytropic gases, it is well known that the smooth solution will generally develop singularities in finite time. However, for three dimensional Chaplygin gases, due to the crucial role of "null condition" in the potential equation which is derived by the irrotational and isentropic flow, P.Godin in [9] has proved the global existence of a smooth 3-D spherically symmetric flow with variable entropy when the initial data are of small smooth perturbations with compact supports to a constant state. It is noted that there are some essential differences on the global solution or blowup problems between 2-D and 3-D hyperbolic systems. In this paper, we will focus on the global symmetric solution problem of 2-D full compressible Euler system of Chaplygin gases. Through carrying out involved analysis and finding an appropriate weight we can derive some uniform weighted energy estimates on the small symmetric solution to 2-D compressible Euler system of Chaplygin gases and further establish the global existence of smooth solution by continuous induction method.