Formation of Point Shocks for 3D Compressible Euler
We consider the 3D isentropic compressible Euler equations with the ideal gas law. We provide a constructive proof of the formation of the first point shock from smooth initial datum of finite energy, with no vacuum regions, with nontrivial vorticity present at the shock, and under no symmetry assum...
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| Published in | Communications on pure and applied mathematics Vol. 76; no. 9; pp. 2073 - 2191 |
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| Main Authors | , , |
| Format | Journal Article |
| Language | English |
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Melbourne
John Wiley & Sons Australia, Ltd
01.09.2023
John Wiley and Sons, Limited |
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| Abstract | We consider the 3D isentropic compressible Euler equations with the ideal gas law. We provide a constructive proof of the formation of the first point shock from smooth initial datum of finite energy, with no vacuum regions, with nontrivial vorticity present at the shock, and under no symmetry assumptions. We prove that for an open set of Sobolev‐class initial data that are a small L∞ perturbation of a constant state, there exist smooth solutions to the Euler equations which form a generic stable shock in finite time. The blowup time and location can be explicitly computed, and solutions at the blowup time are smooth except for a single point, where they are of cusp‐type with Hölder C1/3 regularity. Our proof is based on the use of modulated self‐similar variables that are used to enforce a number of constraints on the blowup profile, necessary to establish global existence and asymptotic stability in self‐similar variables. © 2022 Wiley Periodicals LLC. |
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| AbstractList | We consider the 3D isentropic compressible Euler equations with the ideal gas law. We provide a constructive proof of the formation of the first point shock from smooth initial datum of finite energy, with no vacuum regions, with nontrivial vorticity present at the shock, and under no symmetry assumptions. We prove that for an open set of Sobolev‐class initial data that are a small L∞ perturbation of a constant state, there exist smooth solutions to the Euler equations which form a generic stable shock in finite time. The blowup time and location can be explicitly computed, and solutions at the blowup time are smooth except for a single point, where they are of cusp‐type with Hölder C1/3 regularity. Our proof is based on the use of modulated self‐similar variables that are used to enforce a number of constraints on the blowup profile, necessary to establish global existence and asymptotic stability in self‐similar variables. © 2022 Wiley Periodicals LLC. We consider the 3D isentropic compressible Euler equations with the ideal gas law. We provide a constructive proof of the formation of the first point shock from smooth initial datum of finite energy, with no vacuum regions, with nontrivial vorticity present at the shock, and under no symmetry assumptions . We prove that for an open set of Sobolev‐class initial data that are a small L ∞ perturbation of a constant state, there exist smooth solutions to the Euler equations which form a generic stable shock in finite time. The blowup time and location can be explicitly computed, and solutions at the blowup time are smooth except for a single point , where they are of cusp‐type with Hölder C 1/3 regularity. Our proof is based on the use of modulated self‐similar variables that are used to enforce a number of constraints on the blowup profile, necessary to establish global existence and asymptotic stability in self‐similar variables. © 2022 Wiley Periodicals LLC. |
| Author | Shkoller, Steve Vicol, Vlad Buckmaster, Tristan |
| Author_xml | – sequence: 1 givenname: Tristan surname: Buckmaster fullname: Buckmaster, Tristan email: buckmaster@math.princeton.edu organization: Department of Mathematics, Princeton University – sequence: 2 givenname: Steve surname: Shkoller fullname: Shkoller, Steve email: shkoller@math.ucdavis.edu organization: Department of Mathematics, UC Davis – sequence: 3 givenname: Vlad surname: Vicol fullname: Vicol, Vlad email: vicol@cims.nyu.edu organization: Courant Institute, New York University |
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| Cites_doi | 10.1007/BF02392822 10.1016/s0294-1449(16)30114-7 10.4171/192 10.1002/cpa.3160460402 10.1016/0022-0396(79)90082-2 10.2307/121020 10.1007/BF01210741 10.1063/1.1704154 10.1007/s00222-018-0799-8 10.1090/S0002-9947-02-02982-3 10.1093/imrn/rny012 10.1215/S0012-7094-97-08605-1 10.1002/cpa.21956 10.1017/S0022112096002406 10.1088/0951-7715/22/1/001 10.1007/978-1-4612-1116-7_1 10.4007/annals.2005.161.157 10.1002/cpa.3160270307 10.1007/s10240-019-00110-z 10.1007/978-3-642-04048-1 |
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| Title | Formation of Point Shocks for 3D Compressible Euler |
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