On global axisymmetric solutions to 2D compressible full Euler equations of Chaplygin gases

Fei Hou; Huicheng Yin

doi:10.3934/dcds.2020083

Article Contents

2020, Volume 40, Issue 3: 1435-1492. Doi: 10.3934/dcds.2020083

This issue Previous Article Long-time solvability for the 2D dispersive SQG equation with improved regularity Next Article Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay

On global axisymmetric solutions to 2D compressible full Euler equations of Chaplygin gases

Fei Hou and
Huicheng Yin^,

School of Mathematical Sciences and Mathematical Institute, Nanjing Normal University, Nanjing 210023, China

^* Corresponding author: Huicheng Yin
^* Corresponding author: Huicheng Yin

Received: January 31, 2019

Revised: June 30, 2019

Published: December 2019

The authors are supported by NSFC (No. 11571177, No. 11731007)

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Abstract

For 2D compressible full Euler equations of Chaplygin gases, when the initial axisymmetric perturbation of a rest state is small, we prove that the smooth solution exists globally. Compared with the previous references, there are two different key points in this paper: both the vorticity and the variable entropy are simultaneously considered, moreover, the usual assumption on the compact support of initial perturbation is removed. Due to the appearances of the variable entropy and vorticity, the related perturbation of solution will have no decay in time, which leads to an essential difficulty in establishing the global energy estimate. Thanks to introducing a nonlinear ODE which arises from the vorticity and entropy, and considering the difference between the solutions of the resulting ODE and the full Euler equations, we can distinguish the fast decay part and non-decay part of solution to Euler equations. Based on this, by introducing some suitable weighted energies together with a class of weighted $ L^\infty $-$ L^\infty $ estimates for the solutions of 2D wave equations, we can eventually obtain the global energy estimates and further complete the proof on the global existence of smooth solution to 2D full Euler equations.

Keywords:

Compressible full Euler equations,
Chaplygin gases,
global solution,
vorticity,
null condition,
weighted $ L^\infty $-$ L^\infty $ estimates.

Mathematics Subject Classification: Primary: 35L45, 76N15; Secondary: 35L67.

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