Abstract

We are concerned with the formation of singularities and the existence of global continuous solutions of the Cauchy problem for the one-dimensional nonisentropic Euler equations for compressible fluids. For the isentropic Euler equations, we pinpoint a necessary and sufficient condition for the formation of singularities of solutions with large initial data that allow a far-field vacuum---there exists a compression in the initial data. For the nonisentropic Euler equations, we identify a sufficient condition for the formation of singularities of solutions with large initial data that allow a far-field vacuum---there exists a strong compression in the initial data. Furthermore, we identify two new phenomena---decompression and de-rarefaction---for the nonisentropic Euler flows, different from the isentropic flows, via constructing two respective solutions. For the decompression phenomenon, we construct a first global continuous nonisentropic solution, even though the initial data contain a weak compression, by solving a backward Goursat problem, so that the solution is smooth, except on several characteristic curves across which the solution has a weak discontinuity (i.e., only Lipschitz continuity). For the de-rarefaction phenomenon, we construct a continuous nonisentropic solution whose initial data contain isentropic rarefactions (i.e., without compression) and a locally stationary varying entropy profile, for which the solution still forms a shock wave in a finite time.

Keywords

  1. Euler equations
  2. nonisentropic
  3. compressible flow
  4. continuous solution
  5. singularity formation
  6. far-field vacuum

MSC codes

  1. 76N15
  2. 35L65
  3. 35L67
  4. 35Q31
  5. 35A01
  6. 35B44

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Information & Authors

Information

Published In

cover image SIAM Journal on Mathematical Analysis
SIAM Journal on Mathematical Analysis
Pages: 6280 - 6325
ISSN (online): 1095-7154

History

Submitted: 3 February 2020
Accepted: 9 July 2021
Published online: 4 November 2021

Keywords

  1. Euler equations
  2. nonisentropic
  3. compressible flow
  4. continuous solution
  5. singularity formation
  6. far-field vacuum

MSC codes

  1. 76N15
  2. 35L65
  3. 35L67
  4. 35Q31
  5. 35A01
  6. 35B44

Authors

Affiliations

Funding Information

Newton International Fellowships Alumni : AL/201021
Australian Research Council https://doi.org/10.13039/501100000923 : DP170100630
National Science Foundation https://doi.org/10.13039/100000001 : DMS-1715012
Royal Society https://doi.org/10.13039/501100000288 : WM090014, NF170015
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/E035027/1, EP/L015811/1

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