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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2507.21470 |
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| _version_ | 1866908470781935616 |
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| author | Mei, Yu Yuan, Peng |
| author_facet | Mei, Yu Yuan, Peng |
| contents | This paper investigates the large-time behavior of the viscous shock profile for the one-dimensional system of viscoelasticity, subject to initial perturbations that approach space-periodic functions at far fields. We specifically address the case with non-convex constitutive stress relations and non-degenerate Lax's shock. Under the assumptions of suitably small initial perturbations satisfying a zero-mass type condition, we prove that the solution of the system converges to a viscous shock profile with a shift, which is partially determined by the space-periodic perturbation. Notably, our result imposes no amplitude restrictions on the viscous shock waves. This work extends the result of Kawashima-Matsumura (\textit{Commun. Pure Appl. Math.} \textbf{47}, 1994) by simultaneously handling both large-amplitude shocks and space-periodic perturbations, while also generalizing the result of Huang-Yuan (\textit{Commun. Math. Phys.} \textbf{387}, 2021) by allowing for a non-convex constitutive relation. The key ingredient of proof is decomposing the large-amplitude shock wave into small-amplitude shocks and, for each, introducing suitable transform and weight functions to counteract the adverse effects of non-convex constitutive relations encountered during weighted energy estimates on the system in effective velocity and deformation gradient variables. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_21470 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Stability of Large-Amplitude Viscous Shock Under Periodic Perturbation for 1-d Viscoelasticity with Non-Convex Constitutive Relations Mei, Yu Yuan, Peng Analysis of PDEs 35B40, 35L65, 35L67, 76A10 This paper investigates the large-time behavior of the viscous shock profile for the one-dimensional system of viscoelasticity, subject to initial perturbations that approach space-periodic functions at far fields. We specifically address the case with non-convex constitutive stress relations and non-degenerate Lax's shock. Under the assumptions of suitably small initial perturbations satisfying a zero-mass type condition, we prove that the solution of the system converges to a viscous shock profile with a shift, which is partially determined by the space-periodic perturbation. Notably, our result imposes no amplitude restrictions on the viscous shock waves. This work extends the result of Kawashima-Matsumura (\textit{Commun. Pure Appl. Math.} \textbf{47}, 1994) by simultaneously handling both large-amplitude shocks and space-periodic perturbations, while also generalizing the result of Huang-Yuan (\textit{Commun. Math. Phys.} \textbf{387}, 2021) by allowing for a non-convex constitutive relation. The key ingredient of proof is decomposing the large-amplitude shock wave into small-amplitude shocks and, for each, introducing suitable transform and weight functions to counteract the adverse effects of non-convex constitutive relations encountered during weighted energy estimates on the system in effective velocity and deformation gradient variables. |
| title | Stability of Large-Amplitude Viscous Shock Under Periodic Perturbation for 1-d Viscoelasticity with Non-Convex Constitutive Relations |
| topic | Analysis of PDEs 35B40, 35L65, 35L67, 76A10 |
| url | https://arxiv.org/abs/2507.21470 |