Global Energy Conservation Solution for the N − abc Family of Camassa-Holm Type Equation

Global Energy Conservation Solution for the N − abc Family of Camassa-Holm Type Equation | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Global Energy Conservation Solution for the N − abc Family of Camassa-Holm Type Equation ZAIYUN ZHANG, ZHENHAI LIU, YOUJUN DENG This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-1913302/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract In this paper, we investigate the N − abc family of Camassa-Holm type equation with (N + 1)-order nonlinearities. Using both the lower order and the higher order energy conservation laws, as well as the characteristic method, we establish the global existence and uniqueness of the Hölder continuous energy weak solution to the N − abc family of Camassa-Holm type equation in the energy space H 1 (R) × W 1,2N (R). Moreover, we show that a very natural and interesting problem is to study how the regularity of solution changes with respect to N. More precisely, we establish Hölder continuous energy weak solutions with the exponent 1 − 1 2N. This result precisely shows how the regularity of solution changes with respect to the the power of nonlinear wave speed N, and it reveals an intrinsic relation between Camassa-Holm equation (linear wave speed u 1), Novikov equation (quadratic wave speed u 2) and the N − abc family of Camassa-Holm type equation (wave speed u N). This result improves earlier ones in the literatures, such as [1], [8], [45] and [50]. N - abc family of Camassa-Holm type equation Global conservative weak solution Uniqueness Characteristic Cusp singularity Full Text Additional Declarations No competing interests reported. Cite Share Download PDF Status: Posted Version 1 posted You are reading this latest preprint version Research Square lets you share your work early, gain feedback from the community, and start making changes to your manuscript prior to peer review in a journal. As a division of Research Square Company, we’re committed to making research communication faster, fairer, and more useful. We do this by developing innovative software and high quality services for the global research community. Our growing team is made up of researchers and industry professionals working together to solve the most critical problems facing scientific publishing. Also discoverable on Platform About Our Team In Review Editorial Policies Advisory Board Help Center Resources Author Services Accessibility API Access RSS feed Manage Cookie Preferences © Research Square 2026 | ISSN 2693-5015 (online) Privacy Policy Terms of Service Do Not Sell My Personal Information {"props":{"pageProps":{"initialData":{"identity":"rs-1913302","acceptedTermsAndConditions":true,"allowDirectSubmit":true,"archivedVersions":[],"articleType":"Research Article","associatedPublications":[],"authors":[{"id":129264341,"identity":"7391658c-f0ab-45d6-95dc-91eddcce7a04","order_by":0,"name":"ZAIYUN ZHANG","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAABDElEQVRIiWNgGAWjYDACCQjFAyIOfKiwkWNjbz5AtBbGgzPOpBnz8RxLIEoLCDAf5m07lDhPIkcBrw752c3PHn75YydjcCP3wAEetgPpbQw5DAw/Krbh1MI455i5sQxPMo/BjbyEAxI8d3LbGM4eYOw5cxunFmaJBDNpCQlmoJYcgwMGEs9y2xj7EpgZ23BrYZNI/yYtYVAP0ZJgcDidDagdrxYeiRwzyQ8JhyFaDiQcTmBjI6BFQiKnTJrhwHEeyTNvDA42HEgzbONhSziIzy/yM9K3Sf74U23PdzzH+PPffzby8vMfH3zwowK3FnAQgOJR4QCSyAHsChGA8QfIugZCykbBKBgFo2DEAgDKO1oKPe47MgAAAABJRU5ErkJggg==","orcid":"","institution":"Hunan Institute of Science and Technology","correspondingAuthor":true,"submittingAuthor":false,"prefix":"","firstName":"ZAIYUN","middleName":"","lastName":"ZHANG","suffix":""},{"id":129264342,"identity":"96de0d06-86d2-4127-b4cb-f879239dbd02","order_by":1,"name":"ZHENHAI LIU","email":"","orcid":"","institution":"Guangxi University for Nationalities","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"ZHENHAI","middleName":"","lastName":"LIU","suffix":""},{"id":129264343,"identity":"3363af7b-dcb2-44ed-87a9-7992806aad0b","order_by":2,"name":"YOUJUN DENG","email":"","orcid":"","institution":"Central South University","correspondingAuthor":false,"submittingAuthor":false,"prefix":"","firstName":"YOUJUN","middleName":"","lastName":"DENG","suffix":""}],"badges":[],"createdAt":"2022-07-31 01:59:07","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-1913302/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-1913302/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":25283343,"identity":"fbd4e47d-8dcc-4ec7-b708-9eb52b6c0388","added_by":"auto","created_at":"2022-08-16 19:09:54","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":518562,"visible":true,"origin":"","legend":"","description":"","filename":"GlobalconservationsolutionforCHtypeequation.pdf","url":"https://assets-eu.researchsquare.com/files/rs-1913302/v1_covered.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"\u003cp\u003eGlobal Energy Conservation Solution for the \u003cem\u003eN − abc\u003c/em\u003e Family of Camassa-Holm Type Equation\u003c/p\u003e","fulltext":[{"header":"Full Text","content":"This preprint is available for \u003ca href='/article/rs-1913302/latest.pdf' target='_blank'\u003edownload as a PDF\u003c/a\u003e."}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":false,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":true,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":true,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"N - abc family of Camassa-Holm type equation, Global conservative weak solution, Uniqueness, Characteristic, Cusp singularity","lastPublishedDoi":"10.21203/rs.3.rs-1913302/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-1913302/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"In this paper, we investigate the N − abc family of Camassa-Holm type equation with (N + 1)-order nonlinearities. Using both the lower order and the higher order energy conservation laws, as well as the characteristic method, we establish the global existence and uniqueness of the Hölder continuous energy weak solution to the N − abc family of Camassa-Holm type equation in the energy space H 1 (R) × W 1,2N (R). Moreover, we show that a very natural and interesting problem is to study how the regularity of solution changes with respect to N. More precisely, we establish Hölder continuous energy weak solutions with the exponent 1 − 1 2N. This result precisely shows how the regularity of solution changes with respect to the the power of nonlinear wave speed N, and it reveals an intrinsic relation between Camassa-Holm equation (linear wave speed u 1), Novikov equation (quadratic wave speed u 2) and the N − abc family of Camassa-Holm type equation (wave speed u N). 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