On global smooth solutions of 3-D compressible Euler equations with vanishing density in infinitely expanding balls

Gang Xu; Huicheng Yin

doi:10.3934/dcds.2020112

Article Contents

2020, Volume 40, Issue 4: 2213-2265. Doi: 10.3934/dcds.2020112

This issue Previous Article Semilinear elliptic system with boundary singularity Next Article Spectral decomposition for rescaling expansive flows with rescaled shadowing

On global smooth solutions of 3-D compressible Euler equations with vanishing density in infinitely expanding balls

Gang Xu 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: April 30, 2019

Revised: September 30, 2019

Published: December 2019

The first author is supported by NSFC grant No.11571141, No.11971237, and the second author is supported by NSFC grant No.11571177, No.11731007

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Abstract

In this paper, we are concerned with the global smooth solution problem for 3-D compressible isentropic Euler equations with vanishing density in an infinitely expanding ball. It is well-known that the classical solution of compressible Euler equations generally forms the shock as well as blows up in finite time due to the compression of gases. However, for the rarefactive gases, it is expected that the compressible Euler equations will admit global smooth solutions. We now focus on the movement of compressible gases in an infinitely expanding ball. Because of the conservation of mass, the fluid in the expanding ball becomes rarefied meanwhile there are no appearances of vacuum domains in any part of the expansive ball, which is easily observed in finite time. We will confirm this interesting phenomenon from the mathematical point of view. Through constructing some anisotropy weighted Sobolev spaces, and by carrying out the new observations and involved analysis on the radial speed and angular speeds together with the divergence and rotations of velocity, the uniform weighted estimates on sound speed and velocity are established. From this, the pointwise time-decay estimate of sound speed is obtained, and the smooth gas fluids without vacuum are shown to exist globally.

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Mathematics Subject Classification: Primary: 35L70, 35L65, 35L67; Secondary: 76N15.

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Figure 1. Polytropic gases lie in a 3D expanding ball

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Figure 2. Continuous transonic flow in an infinite long de Laval nozzle

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Figure 3. 2-D supersonic flow without vacuum in a divergent nozzle

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Figure 4. 3-D supersonic flow without vacuum in a divergent nozzle

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