{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T03:29:21Z","timestamp":1760239761024,"version":"build-2065373602"},"reference-count":32,"publisher":"MDPI AG","issue":"1","license":[{"start":{"date-parts":[[2020,12,31]],"date-time":"2020-12-31T00:00:00Z","timestamp":1609372800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100002261","name":"Russian Foundation for Basic Research","doi-asserted-by":"publisher","award":["18-29-10013"],"award-info":[{"award-number":["18-29-10013"]}],"id":[{"id":"10.13039\/501100002261","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100012708","name":"Foundation for the Advancement of Theoretical Physics and Mathematics","doi-asserted-by":"publisher","award":["19-7-1-13-3"],"award-info":[{"award-number":["19-7-1-13-3"]}],"id":[{"id":"10.13039\/501100012708","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Symmetry"],"abstract":"<jats:p>In this paper, we analyze various types of critical phenomena in one-dimensional gas flows described by Euler equations. We give a geometrical interpretation of thermodynamics with a special emphasis on phase transitions. We use ideas from the geometrical theory of partial differential equations (PDEs), in particular symmetries and differential constraints, to find solutions to the Euler system. Solutions obtained are multivalued and have singularities of projection to the plane of independent variables. We analyze the propagation of the shockwave front along with phase transitions.<\/jats:p>","DOI":"10.3390\/sym13010054","type":"journal-article","created":{"date-parts":[[2020,12,31]],"date-time":"2020-12-31T14:31:49Z","timestamp":1609425109000},"page":"54","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":1,"title":["Singularities in Euler Flows: Multivalued Solutions, Shockwaves, and Phase Transitions"],"prefix":"10.3390","volume":"13","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-6000-5063","authenticated-orcid":false,"given":"Valentin","family":"Lychagin","sequence":"first","affiliation":[{"name":"V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, 65 Profsoyuznaya Str., 117997 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-6573-9808","authenticated-orcid":false,"given":"Mikhail","family":"Roop","sequence":"additional","affiliation":[{"name":"V.A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, 65 Profsoyuznaya Str., 117997 Moscow, Russia"},{"name":"Faculty of Physics, Lomonosov Moscow State University, Leninskie Gory, 119991 Moscow, Russia"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2020,12,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Arnold, V. (1990). Singularities of Caustics and Wave Fronts, Springer.","DOI":"10.1007\/978-94-011-3330-2"},{"key":"ref_2","doi-asserted-by":"crossref","unstructured":"Arnold, V. (1984). 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