Isaac Neal, Calum Rickard, Steve Shkoller, Vlad Vicol
{"title":"A new type of stable shock formation in gas dynamics","authors":"Isaac Neal, Calum Rickard, Steve Shkoller, Vlad Vicol","doi":"10.3934/cpaa.2023118","DOIUrl":null,"url":null,"abstract":"From an open set of initial data, we construct a family of classical solutions to the 1D nonisentropic compressible Euler equations which form $ C^{0,\\nu} $ cusps as a first singularity, for any $ \\nu \\in [\\frac{1}{2}, 1) $. For this range of $ \\nu $, this is the first result demonstrating the stable formation of such $ C^{0,\\nu} $ cusp-type singularities, also known as pre-shocks. The proof uses a new formulation of the differentiated Euler equations along the fast acoustic characteristic, and relies on a novel set of $ L^p $ energy estimates for all $ 1<p<\\infty $, which may be of independent interest.","PeriodicalId":10643,"journal":{"name":"Communications on Pure and Applied Analysis","volume":null,"pages":null},"PeriodicalIF":1.0000,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications on Pure and Applied Analysis","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.3934/cpaa.2023118","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
From an open set of initial data, we construct a family of classical solutions to the 1D nonisentropic compressible Euler equations which form $ C^{0,\nu} $ cusps as a first singularity, for any $ \nu \in [\frac{1}{2}, 1) $. For this range of $ \nu $, this is the first result demonstrating the stable formation of such $ C^{0,\nu} $ cusp-type singularities, also known as pre-shocks. The proof uses a new formulation of the differentiated Euler equations along the fast acoustic characteristic, and relies on a novel set of $ L^p $ energy estimates for all $ 1
期刊介绍:
CPAA publishes original research papers of the highest quality in all the major areas of analysis and its applications, with a central theme on theoretical and numeric differential equations. Invited expository articles are also published from time to time. It is edited by a group of energetic leaders to guarantee the journal''s highest standard and closest link to the scientific communities.