Vacuum and singularity formation for compressible Euler equations with time-dependent damping

Ying Sui; Weiqiang Wang; Huimin Yu

doi:10.3934/dcds.2022184

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2023, Volume 43, Issue 5: 1905-1925. Doi: 10.3934/dcds.2022184

This issue Previous Article Fredholm theory of families of discrete dynamical systems and its applications to bifurcation theory Next Article Square-tiled tori

Vacuum and singularity formation for compressible Euler equations with time-dependent damping

1.
School of Mathematics and Statistics, Shandong Normal University, Jinan 250014, China
2.
Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

^*Corresponding author: Huimin Yu
^*Corresponding author: Huimin Yu

Received: March 2022

Revised: November 2022

Early access: December 29, 2022

Published online: December 29, 2022

The last author is supported by [NSFC Grant No. 12271310 and Natural Science Foundation of Shandong Province ZR2022MA088.].

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Abstract

In this paper, vacuum and singularity formation are considered for compressible Euler equations with time-dependent damping. For $ 1<\gamma{\leq} 3 $, by constructing some new control functions ingeniously, we obtain the lower bounds estimates on density for arbitrary classical solutions. Basing on these lower estimates, we succeed in proving the singular formation theorem for all $ \lambda $, which was open in [19] for some cases. Moreover, the singularity formation of the compressible Euler equations when $ \gamma = 3 $ is investigated, too.

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Mathematics Subject Classification: Primary: 35Q31, 76N10; Secondary: 76L05.

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