Formation of singularities and existence of global continuous solutions for the compressible euler equations
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 form...
| Päätekijät: | , , |
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| Aineistotyyppi: | Journal article |
| Kieli: | English |
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Society for Industrial and Applied Mathematics
2021
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| _version_ | 1826267647429312512 |
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| author | Chen, G Chen, G-QG Zhu, S |
| author_facet | Chen, G Chen, G-QG Zhu, S |
| author_sort | Chen, G |
| collection | OXFORD |
| description | 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. |
| first_indexed | 2024-03-06T20:57:23Z |
| format | Journal article |
| id | oxford-uuid:39b75106-a2b4-4726-a2ba-ad039a5b2e47 |
| institution | University of Oxford |
| language | English |
| last_indexed | 2024-03-06T20:57:23Z |
| publishDate | 2021 |
| publisher | Society for Industrial and Applied Mathematics |
| record_format | dspace |
| spelling | oxford-uuid:39b75106-a2b4-4726-a2ba-ad039a5b2e472022-03-26T13:57:14ZFormation of singularities and existence of global continuous solutions for the compressible euler equationsJournal articlehttp://purl.org/coar/resource_type/c_dcae04bcuuid:39b75106-a2b4-4726-a2ba-ad039a5b2e47EnglishSymplectic ElementsSociety for Industrial and Applied Mathematics2021Chen, GChen, G-QGZhu, SWe 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. |
| spellingShingle | Chen, G Chen, G-QG Zhu, S Formation of singularities and existence of global continuous solutions for the compressible euler equations |
| title | Formation of singularities and existence of global continuous solutions for the compressible euler equations |
| title_full | Formation of singularities and existence of global continuous solutions for the compressible euler equations |
| title_fullStr | Formation of singularities and existence of global continuous solutions for the compressible euler equations |
| title_full_unstemmed | Formation of singularities and existence of global continuous solutions for the compressible euler equations |
| title_short | Formation of singularities and existence of global continuous solutions for the compressible euler equations |
| title_sort | formation of singularities and existence of global continuous solutions for the compressible euler equations |
| work_keys_str_mv | AT cheng formationofsingularitiesandexistenceofglobalcontinuoussolutionsforthecompressibleeulerequations AT chengqg formationofsingularitiesandexistenceofglobalcontinuoussolutionsforthecompressibleeulerequations AT zhus formationofsingularitiesandexistenceofglobalcontinuoussolutionsforthecompressibleeulerequations |