Blowup of smooth solutions to the compressible Euler equations with radial symmetry on bounded domains

Jianwei DONG, Man Wai YUEN

Research output: Contribution to journalArticlespeer-review

Abstract

In this paper, we study the blowup of smooth solutions to the compressible Euler equations with radial symmetry on some fixed bounded domains (B= {x ∈ R: |x| ≤ R}, = 1,2,…) by introducing some new averaged quantities. We consider two types of flows: initially move inward and initially move outward on average. For the flow initially moving inward on average, we show that the smooth solutions will blowup in a finite time if the density vanishes at the origin only (ρ(t, 0) = 0, ρ(tr) > 0, 0 <rR) for N ≥ 1 or the density vanishes at the origin and the velocity field vanishes on the two endpoints (ρ(t, 0) = 0, v(t, R) = 0) for = 1. For the flow initially moving outward, we prove that the smooth solutions will break down in a finite time if the density vanishes on the two endpoints (ρ(t, R) = 0) for = 1. The blowup mechanisms of the compressible Euler equations with constant damping or time-depending damping are obtained as corollaries. Copyright © 2020 Springer Nature Switzerland AG. 
Original languageEnglish
Article number189
JournalZeitschrift fur Angewandte Mathematik und Physik
Volume71
Issue number6
Early online date24 Oct 2020
DOIs
Publication statusPublished - Dec 2020

Citation

Dong, J., & Yuen, M. (2020). Blowup of smooth solutions to the compressible Euler equations with radial symmetry on bounded domains. Zeitschrift fur Angewandte Mathematik und Physik, 71(6). Retrieved from https://doi.org/10.1007/s00033-020-01392-8

Keywords

  • Compressible Euler equations
  • Radial symmetry
  • Blowup

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