{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T07:31:00Z","timestamp":1777447860037,"version":"3.51.4"},"reference-count":9,"publisher":"SAGE Publications","issue":"1-2","license":[{"start":{"date-parts":[[2017,5,15]],"date-time":"2017-05-15T00:00:00Z","timestamp":1494806400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["Asymptotic Analysis"],"published-print":{"date-parts":[[2017,5,15]]},"abstract":"<jats:p>This short note concerns the formal limit presented in [ J. Comput Phys 230 (2011), 8057\u20138088] and [ Actes colloquia Rouen (2012)] between the compressible Euler equations with singular pressure (soft model) and the pressureless Euler system with unilateral constraint (hard model). These soft and hard models with maximal constraint on the density are used to reproduce congestion phenomena. In this paper, we are interested in the question how the different regions (congested and non-congested regions) fit together and how the transition occurs when congestion first develops in the initial system namely the compressible Euler equation with singular pressure. We shall develop a formal solution by matched asymptotics and show that a shock front may separate the congested region from the outside non-congested region. This, in some sense, shows that to justify the formal limit between the two models (soft and hard Euler systems), the shock fronts formation has to be considered and mathematically analyzed.<\/jats:p>","DOI":"10.3233\/asy-171421","type":"journal-article","created":{"date-parts":[[2017,5,16]],"date-time":"2017-05-16T12:54:53Z","timestamp":1494939293000},"page":"95-101","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":5,"title":["Development of congestion in compressible flow with singular pressure"],"prefix":"10.1177","volume":"103","author":[{"given":"Didier","family":"Bresch","sequence":"first","affiliation":[{"name":"LAMA UMR5127 CNRS, Bat. Le Chablais, Campus scientifique, Universit\u00e9 de Savoie Mont-Blanc, 73376 Le Bourget du Lac, France. E-mail:\u00a0"}]},{"given":"Michael","family":"Renardy","sequence":"additional","affiliation":[{"name":"Department of Mathematics, Virginia Tech, Blacksburg, VA 24061-0123, USA. E-mail:\u00a0"}]}],"member":"179","published-online":{"date-parts":[[2017,5,15]]},"reference":[{"key":"ref001","doi-asserted-by":"publisher","DOI":"10.1142\/S0218202502001635"},{"key":"ref002","doi-asserted-by":"publisher","DOI":"10.3934\/krm.2012.5.697"},{"key":"ref003","doi-asserted-by":"publisher","DOI":"10.1016\/j.crma.2014.06.009"},{"key":"ref004","unstructured":"G.Q.\u00a0Chen, Euler equations and related hyperbolic conservation laws, in: Handbook of Differential Equations, C.M.\u00a0Dafermos and E.\u00a0Feireisl, eds, Evolutionary Equations, Vol.\u00a02."},{"key":"ref005","doi-asserted-by":"publisher","DOI":"10.1137\/S0036141001399350"},{"key":"ref006","doi-asserted-by":"publisher","DOI":"10.1016\/j.jcp.2011.07.010"},{"key":"ref007","unstructured":"B.\u00a0Maury, Prise en compte de la congestion dans les mod\u00e8les de mouvements de foules, in: Actes des Colloques Caen 2012-Rouen 2011, 2012."},{"key":"ref008","unstructured":"C.\u00a0Perrin, Mod\u00e8les h\u00e9t\u00e9rog\u00e8nes en m\u00e9canique des fluides: ph\u00e9nom\u00e8nes de congestion, \u00e9coulements granulaires et mouvements collectifs, PhD thesis, Universit\u00e9 Grenoble-Alpes, 2016."},{"key":"ref009","doi-asserted-by":"publisher","DOI":"10.1080\/03605302.2015.1014560"}],"container-title":["Asymptotic Analysis"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-171421","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.3233\/ASY-171421","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.3233\/ASY-171421","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,28]],"date-time":"2026-04-28T12:36:02Z","timestamp":1777379762000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.3233\/ASY-171421"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2017,5,15]]},"references-count":9,"journal-issue":{"issue":"1-2","published-print":{"date-parts":[[2017,5,15]]}},"alternative-id":["10.3233\/ASY-171421"],"URL":"https:\/\/doi.org\/10.3233\/asy-171421","relation":{},"ISSN":["0921-7134","1875-8576"],"issn-type":[{"value":"0921-7134","type":"print"},{"value":"1875-8576","type":"electronic"}],"subject":[],"published":{"date-parts":[[2017,5,15]]}}}