On Local Singularities in Ideal Potential Flows with Free Surface
Despite important advances in the mathematical analysis of the Euler equations for water waves, especially over the last two decades, it is not yet known whether local singularities can develop from smooth data in well-posed initial value problems. For ideal free-surface flow with zero surface tensi...
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| Veröffentlicht in: | Chinese annals of mathematics. Serie B 2019-11, Vol.40 (6), p.925-948 |
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| container_title | Chinese annals of mathematics. Serie B |
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| creator | Liu, Jian-Guo Pego, Robert L. |
| description | Despite important advances in the mathematical analysis of the Euler equations for water waves, especially over the last two decades, it is not yet known whether local singularities can develop from smooth data in well-posed initial value problems. For ideal free-surface flow with zero surface tension and gravity, the authors review existing works that describe “splash singularities”, singular hyperbolic solutions related to jet formation and “flip-through”, and a recent construction of a singular free surface by Zubarev and Karabut that however involves unbounded negative pressure. The authors illustrate some of these phenomena with numerical computations of 2D flow based upon a conformal mapping formulation. Numerical tests with a different kind of initial data suggest the possibility that corner singularities may form in an unstable way from specially prepared initial data. |
| doi_str_mv | 10.1007/s11401-019-0167-z |
| format | Article |
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For ideal free-surface flow with zero surface tension and gravity, the authors review existing works that describe “splash singularities”, singular hyperbolic solutions related to jet formation and “flip-through”, and a recent construction of a singular free surface by Zubarev and Karabut that however involves unbounded negative pressure. The authors illustrate some of these phenomena with numerical computations of 2D flow based upon a conformal mapping formulation. Numerical tests with a different kind of initial data suggest the possibility that corner singularities may form in an unstable way from specially prepared initial data.</description><identifier>ISSN: 0252-9599</identifier><identifier>EISSN: 1860-6261</identifier><identifier>DOI: 10.1007/s11401-019-0167-z</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Applications of Mathematics ; Boundary value problems ; Conformal mapping ; Euler-Lagrange equation ; Free surfaces ; Mathematics ; Mathematics and Statistics ; Potential flow ; Singularity (mathematics) ; Surface tension ; Two dimensional flow ; Water waves ; Well posed problems</subject><ispartof>Chinese annals of mathematics. Serie B, 2019-11, Vol.40 (6), p.925-948</ispartof><rights>The Editorial Office of CAM and Springer-Verlag Berlin Heidelberg 2019</rights><rights>Copyright Springer Nature B.V. 2019</rights><rights>Copyright © Wanfang Data Co. Ltd. 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Numerical tests with a different kind of initial data suggest the possibility that corner singularities may form in an unstable way from specially prepared initial data.</description><subject>Applications of Mathematics</subject><subject>Boundary value problems</subject><subject>Conformal mapping</subject><subject>Euler-Lagrange equation</subject><subject>Free surfaces</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Potential flow</subject><subject>Singularity (mathematics)</subject><subject>Surface tension</subject><subject>Two dimensional flow</subject><subject>Water waves</subject><subject>Well posed problems</subject><issn>0252-9599</issn><issn>1860-6261</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2019</creationdate><recordtype>article</recordtype><recordid>eNp1kEFLAzEQhYMoWKs_wNuCB0-rM9lNsnssxWqhUKF6DjGdrVtrtia7VPvrTVmhJw_DDMP33jCPsWuEOwRQ9wExB0wBy1hSpfsTNsBCQiq5xFM2AC54WoqyPGcXIawBMFcCBmw0d8mssWaTLGq36jbG121NIaldMl1SXD83Lbm2jtNk0-xCsqvb92TiiZJF5ytj6ZKdVWYT6OqvD9nr5OFl_JTO5o_T8WiW2iwv2rTglgohRCWNUIUtObeHyaiMqzcjjTJWLivJCQkVYUkFWMWBIwhRUL7Mhuy2990ZVxm30uum8y5e1OHbfWji8XeQAHkkb3py65uvjkJ7RHmGQuQCoIwU9pT1TQieKr319afxPxpBHzLVfaY6-upDpnofNbzXhMi6Ffmj8_-iX20Hd6U</recordid><startdate>20191101</startdate><enddate>20191101</enddate><creator>Liu, Jian-Guo</creator><creator>Pego, Robert L.</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><general>Department of Physics and Department of Mathematics, Duke University, Durham, NC 27708, USA%Department of Mathematical Sciences and Center for Nonlinear Analysis, Carnegie Mellon University,Pittsburgh, Pennsylvania, PA 12513, USA</general><scope>AAYXX</scope><scope>CITATION</scope><scope>2B.</scope><scope>4A8</scope><scope>92I</scope><scope>93N</scope><scope>PSX</scope><scope>TCJ</scope></search><sort><creationdate>20191101</creationdate><title>On Local Singularities in Ideal Potential Flows with Free Surface</title><author>Liu, Jian-Guo ; Pego, Robert L.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c348t-82ce8555f6a578c922c6a57a7327ba6a7ac6df62e1e17e19e80c720210558e4d3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2019</creationdate><topic>Applications of Mathematics</topic><topic>Boundary value problems</topic><topic>Conformal mapping</topic><topic>Euler-Lagrange equation</topic><topic>Free surfaces</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Potential flow</topic><topic>Singularity (mathematics)</topic><topic>Surface tension</topic><topic>Two dimensional flow</topic><topic>Water waves</topic><topic>Well posed problems</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Liu, Jian-Guo</creatorcontrib><creatorcontrib>Pego, Robert L.</creatorcontrib><collection>CrossRef</collection><collection>Wanfang Data Journals - Hong Kong</collection><collection>WANFANG Data Centre</collection><collection>Wanfang Data Journals</collection><collection>万方数据期刊 - 香港版</collection><collection>China Online Journals (COJ)</collection><collection>China Online Journals (COJ)</collection><jtitle>Chinese annals of mathematics. 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| ispartof | Chinese annals of mathematics. Serie B, 2019-11, Vol.40 (6), p.925-948 |
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| source | Springer Nature - Complete Springer Journals; Alma/SFX Local Collection |
| subjects | Applications of Mathematics Boundary value problems Conformal mapping Euler-Lagrange equation Free surfaces Mathematics Mathematics and Statistics Potential flow Singularity (mathematics) Surface tension Two dimensional flow Water waves Well posed problems |
| title | On Local Singularities in Ideal Potential Flows with Free Surface |
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