Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves
© 2016, Springer International Publishing AG. In an influential 1964 article, P. Lax studied 2 × 2 genuinely nonlinear strictly hyperbolic PDE systems (in one spatial dimension). Using the method of Riemann invariants, he showed that a large set of smooth initial data lead to bounded solutions whose...
| Main Authors: | , , , |
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| Format: | Article |
| Language: | English |
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Springer Science and Business Media LLC
2021
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| Online Access: | https://hdl.handle.net/1721.1/133378 |
| _version_ | 1826205620932444160 |
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| author | Speck, Jared Holzegel, Gustav Luk, Jonathan Wong, Willie |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Speck, Jared Holzegel, Gustav Luk, Jonathan Wong, Willie |
| author_sort | Speck, Jared |
| collection | MIT |
| description | © 2016, Springer International Publishing AG. In an influential 1964 article, P. Lax studied 2 × 2 genuinely nonlinear strictly hyperbolic PDE systems (in one spatial dimension). Using the method of Riemann invariants, he showed that a large set of smooth initial data lead to bounded solutions whose first spatial derivatives blow up in finite time, a phenomenon known as wave breaking. In the present article, we study the Cauchy problem for two classes of quasilinear wave equations in two spatial dimensions that are closely related to the systems studied by Lax. When the data have one-dimensional symmetry, Lax’s methods can be applied to the wave equations to show that a large set of smooth initial data lead to wave breaking. Here we study solutions with initial data that are close, as measured by an appropriate Sobolev norm, to data belonging to a distinguished subset of Lax’s data: the data corresponding to simple plane waves. Our main result is that under suitable relative smallness assumptions, the Lax-type wave breaking for simple plane waves is stable. The key point is that we allow the data perturbations to break the symmetry. Moreover, we give a detailed, constructive description of the asymptotic behavior of the solution all the way up to the first singularity, which is a shock driven by the intersection of null (characteristic) hyperplanes. We also outline how to extend our results to the compressible irrotational Euler equations. To derive our results, we use Christodoulou’s framework for studying shock formation to treat a new solution regime in which wave dispersion is not present. |
| first_indexed | 2024-09-23T13:15:59Z |
| format | Article |
| id | mit-1721.1/133378 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T13:15:59Z |
| publishDate | 2021 |
| publisher | Springer Science and Business Media LLC |
| record_format | dspace |
| spelling | mit-1721.1/1333782024-01-03T18:30:33Z Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves Speck, Jared Holzegel, Gustav Luk, Jonathan Wong, Willie Massachusetts Institute of Technology. Department of Mathematics © 2016, Springer International Publishing AG. In an influential 1964 article, P. Lax studied 2 × 2 genuinely nonlinear strictly hyperbolic PDE systems (in one spatial dimension). Using the method of Riemann invariants, he showed that a large set of smooth initial data lead to bounded solutions whose first spatial derivatives blow up in finite time, a phenomenon known as wave breaking. In the present article, we study the Cauchy problem for two classes of quasilinear wave equations in two spatial dimensions that are closely related to the systems studied by Lax. When the data have one-dimensional symmetry, Lax’s methods can be applied to the wave equations to show that a large set of smooth initial data lead to wave breaking. Here we study solutions with initial data that are close, as measured by an appropriate Sobolev norm, to data belonging to a distinguished subset of Lax’s data: the data corresponding to simple plane waves. Our main result is that under suitable relative smallness assumptions, the Lax-type wave breaking for simple plane waves is stable. The key point is that we allow the data perturbations to break the symmetry. Moreover, we give a detailed, constructive description of the asymptotic behavior of the solution all the way up to the first singularity, which is a shock driven by the intersection of null (characteristic) hyperplanes. We also outline how to extend our results to the compressible irrotational Euler equations. To derive our results, we use Christodoulou’s framework for studying shock formation to treat a new solution regime in which wave dispersion is not present. 2021-10-27T19:52:26Z 2021-10-27T19:52:26Z 2016 2021-04-28T16:52:19Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/133378 en 10.1007/S40818-016-0014-4 Annals of PDE [21992576] Creative Commons Attribution-Noncommercial-Share Alike http://creativecommons.org/licenses/by-nc-sa/4.0/ application/pdf Springer Science and Business Media LLC arXiv |
| spellingShingle | Speck, Jared Holzegel, Gustav Luk, Jonathan Wong, Willie Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves |
| title | Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves |
| title_full | Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves |
| title_fullStr | Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves |
| title_full_unstemmed | Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves |
| title_short | Stable Shock Formation for Nearly Simple Outgoing Plane Symmetric Waves |
| title_sort | stable shock formation for nearly simple outgoing plane symmetric waves |
| url | https://hdl.handle.net/1721.1/133378 |
| work_keys_str_mv | AT speckjared stableshockformationfornearlysimpleoutgoingplanesymmetricwaves AT holzegelgustav stableshockformationfornearlysimpleoutgoingplanesymmetricwaves AT lukjonathan stableshockformationfornearlysimpleoutgoingplanesymmetricwaves AT wongwillie stableshockformationfornearlysimpleoutgoingplanesymmetricwaves |