Small-data shock formation in solutions to 3D quasilinear wave equations: An overview
In his 2007 monograph, Christodoulou proved a remarkable result giving a detailed description of shock formation, for small H[superscript s]-initial conditions (with ss sufficiently large), in solutions to the relativistic Euler equations in three space dimensions. His work provided a significant ad...
| Main Authors: | , , , |
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| Other Authors: | |
| Format: | Article |
| Language: | en_US |
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World Scientific
2016
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| Online Access: | http://hdl.handle.net/1721.1/105230 https://orcid.org/0000-0001-5020-3568 |
| _version_ | 1826198978444656640 |
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| author | Holzegel, Gustav Klainerman, Sergiu Wong, Willie Wai-Yeung Speck, Jared R. |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Holzegel, Gustav Klainerman, Sergiu Wong, Willie Wai-Yeung Speck, Jared R. |
| author_sort | Holzegel, Gustav |
| collection | MIT |
| description | In his 2007 monograph, Christodoulou proved a remarkable result giving a detailed description of shock formation, for small H[superscript s]-initial conditions (with ss sufficiently large), in solutions to the relativistic Euler equations in three space dimensions. His work provided a significant advancement over a large body of prior work concerning the long-time behavior of solutions to higher-dimensional quasilinear wave equations, initiated by John in the mid 1970’s and continued by Klainerman, Sideris, Hörmander, Lindblad, Alinhac, and others. Our goal in this paper is to give an overview of his result, outline its main new ideas, and place it in the context of the above mentioned earlier work. We also introduce the recent work of Speck, which extends Christodoulou’s result to show that for two important classes of quasilinear wave equations in three space dimensions, small-data shock formation occurs precisely when the quadratic nonlinear terms fail to satisfy the classic null condition. |
| first_indexed | 2024-09-23T11:12:50Z |
| format | Article |
| id | mit-1721.1/105230 |
| institution | Massachusetts Institute of Technology |
| language | en_US |
| last_indexed | 2024-09-23T11:12:50Z |
| publishDate | 2016 |
| publisher | World Scientific |
| record_format | dspace |
| spelling | mit-1721.1/1052302022-10-01T02:06:21Z Small-data shock formation in solutions to 3D quasilinear wave equations: An overview Holzegel, Gustav Klainerman, Sergiu Wong, Willie Wai-Yeung Speck, Jared R. Massachusetts Institute of Technology. Department of Mathematics Speck, Jared R. In his 2007 monograph, Christodoulou proved a remarkable result giving a detailed description of shock formation, for small H[superscript s]-initial conditions (with ss sufficiently large), in solutions to the relativistic Euler equations in three space dimensions. His work provided a significant advancement over a large body of prior work concerning the long-time behavior of solutions to higher-dimensional quasilinear wave equations, initiated by John in the mid 1970’s and continued by Klainerman, Sideris, Hörmander, Lindblad, Alinhac, and others. Our goal in this paper is to give an overview of his result, outline its main new ideas, and place it in the context of the above mentioned earlier work. We also introduce the recent work of Speck, which extends Christodoulou’s result to show that for two important classes of quasilinear wave equations in three space dimensions, small-data shock formation occurs precisely when the quadratic nonlinear terms fail to satisfy the classic null condition. National Science Foundation (U.S.) (Grant DMS-1162211) Solomon Buchsbaum AT&T Research Fund 2016-11-07T19:16:35Z 2016-11-07T19:16:35Z 2016-03 2014-07 Article http://purl.org/eprint/type/JournalArticle 0219-8916 1793-6993 http://hdl.handle.net/1721.1/105230 Holzegel, Gustav et al. “Small-Data Shock Formation in Solutions to 3D Quasilinear Wave Equations: An Overview.” Journal of Hyperbolic Differential Equations 13.1 (2016): 1–105. https://orcid.org/0000-0001-5020-3568 en_US http://dx.doi.org/10.1142/s0219891616500016 Journal of Hyperbolic Differential Equations Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf World Scientific arXiv |
| spellingShingle | Holzegel, Gustav Klainerman, Sergiu Wong, Willie Wai-Yeung Speck, Jared R. Small-data shock formation in solutions to 3D quasilinear wave equations: An overview |
| title | Small-data shock formation in solutions to 3D quasilinear wave equations: An overview |
| title_full | Small-data shock formation in solutions to 3D quasilinear wave equations: An overview |
| title_fullStr | Small-data shock formation in solutions to 3D quasilinear wave equations: An overview |
| title_full_unstemmed | Small-data shock formation in solutions to 3D quasilinear wave equations: An overview |
| title_short | Small-data shock formation in solutions to 3D quasilinear wave equations: An overview |
| title_sort | small data shock formation in solutions to 3d quasilinear wave equations an overview |
| url | http://hdl.handle.net/1721.1/105230 https://orcid.org/0000-0001-5020-3568 |
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