Low Regularity of Self-Similar Solutions of Two-Dimensional Riemann problems with Shocks for the Isentropic Euler system
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arXiv
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| Natura: | Preprint |
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2025
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| author | Chen, Gui-Qiang G. Feldman, Mikhail Xiang, Wei |
| author_facet | Chen, Gui-Qiang G. Feldman, Mikhail Xiang, Wei |
| contents | We are concerned with the low regularity of self-similar solutions of two-dimensional Riemann problems for the isentropic Euler system. We establish a general framework for the analysis of the local regularity of such solutions for a class of two-dimensional Riemann problems for the isentropic Euler system, which includes the regular shock reflection problem, the Prandtl reflection problem, the Lighthill diffraction problem, and the four-shock Riemann problem. We prove that the velocity is not in $H^1$ in the subsonic domain for the self-similar solutions of these problems in general. This indicates that the self-similar solutions of the Riemann problems with shocks for the isentropic Euler system are of much more complicated structure than those for the Euler system for potential flow; in particular, the velocity is not necessarily continuous in the subsonic domain. The proof is based on a regularization of the isentropic Euler system to derive the transport equation for the vorticity, a renormalization argument extended to the case of domains with boundary, and DiPerna-Lions-type commutator estimates. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2504_07034 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Low Regularity of Self-Similar Solutions of Two-Dimensional Riemann problems with Shocks for the Isentropic Euler system Chen, Gui-Qiang G. Feldman, Mikhail Xiang, Wei Analysis of PDEs Mathematical Physics Pattern Formation and Solitons Fluid Dynamics Primary: 35M10, 35M12, 35B65, 35L65, 35J70, 76H05, 35L67, 35B45, 35B40, 35B36, 76N10, Secondary: 35R35, 35L03, 35L04, 35J67, 76L05, 76J20, 76N20, 76G25 We are concerned with the low regularity of self-similar solutions of two-dimensional Riemann problems for the isentropic Euler system. We establish a general framework for the analysis of the local regularity of such solutions for a class of two-dimensional Riemann problems for the isentropic Euler system, which includes the regular shock reflection problem, the Prandtl reflection problem, the Lighthill diffraction problem, and the four-shock Riemann problem. We prove that the velocity is not in $H^1$ in the subsonic domain for the self-similar solutions of these problems in general. This indicates that the self-similar solutions of the Riemann problems with shocks for the isentropic Euler system are of much more complicated structure than those for the Euler system for potential flow; in particular, the velocity is not necessarily continuous in the subsonic domain. The proof is based on a regularization of the isentropic Euler system to derive the transport equation for the vorticity, a renormalization argument extended to the case of domains with boundary, and DiPerna-Lions-type commutator estimates. |
| title | Low Regularity of Self-Similar Solutions of Two-Dimensional Riemann problems with Shocks for the Isentropic Euler system |
| topic | Analysis of PDEs Mathematical Physics Pattern Formation and Solitons Fluid Dynamics Primary: 35M10, 35M12, 35B65, 35L65, 35J70, 76H05, 35L67, 35B45, 35B40, 35B36, 76N10, Secondary: 35R35, 35L03, 35L04, 35J67, 76L05, 76J20, 76N20, 76G25 |
| url | https://arxiv.org/abs/2504.07034 |