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| Main Authors: | , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.15619 |
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| _version_ | 1866909359604236288 |
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| author | Buckmaster, Tristan Chen, Jiajie |
| author_facet | Buckmaster, Tristan Chen, Jiajie |
| contents | In this paper, we prove blowup for the defocusing septic complex-valued nonlinear wave equation in $\mathbb{R}^{4+1}$. This work builds on the earlier results of Shao, Wei, and Zhang [SWZ2024a,SWZ2024b], reducing the order of the nonlinearity from $29$ to $7$ in $\mathbb{R}^{4+1}$. As in [SWZ2024a,SWZ2024b], the proof hinges on a connection between solutions to the nonlinear wave equation and the relativistic Euler equations via a front compression blowup mechanism. More specifically, the problem is reduced to constructing smooth, radially symmetric, self-similar imploding profiles for the relativistic Euler equations.
As with implosion for the compressible Euler equations, the relativistic analogue admits a countable family of smooth imploding profiles. The result in [SWZ2024a] represents the construction of the first profile in this family. In this paper, we construct a sequence of solutions corresponding to the higher-order profiles in the family. This allows us to saturate the inequalities necessary to show blowup for the defocusing complex-valued nonlinear wave equation with an integer order of nonlinearity and radial symmetry via this mechanism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_15619 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Blowup for the defocusing septic complex-valued nonlinear wave equation in $\mathbb{R}^{4+1}$ Buckmaster, Tristan Chen, Jiajie Analysis of PDEs In this paper, we prove blowup for the defocusing septic complex-valued nonlinear wave equation in $\mathbb{R}^{4+1}$. This work builds on the earlier results of Shao, Wei, and Zhang [SWZ2024a,SWZ2024b], reducing the order of the nonlinearity from $29$ to $7$ in $\mathbb{R}^{4+1}$. As in [SWZ2024a,SWZ2024b], the proof hinges on a connection between solutions to the nonlinear wave equation and the relativistic Euler equations via a front compression blowup mechanism. More specifically, the problem is reduced to constructing smooth, radially symmetric, self-similar imploding profiles for the relativistic Euler equations. As with implosion for the compressible Euler equations, the relativistic analogue admits a countable family of smooth imploding profiles. The result in [SWZ2024a] represents the construction of the first profile in this family. In this paper, we construct a sequence of solutions corresponding to the higher-order profiles in the family. This allows us to saturate the inequalities necessary to show blowup for the defocusing complex-valued nonlinear wave equation with an integer order of nonlinearity and radial symmetry via this mechanism. |
| title | Blowup for the defocusing septic complex-valued nonlinear wave equation in $\mathbb{R}^{4+1}$ |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2410.15619 |