On the problem of stability of viscous shocks
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866911283836616704 |
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| author | Bolotin, Sergey Treschev, Dmitry |
| author_facet | Bolotin, Sergey Treschev, Dmitry |
| contents | We consider the problem of spectral stability of traveling wave solutions $u=γ(x-Wt)$ for a system of viscous conservation laws $\partial_t u + \partial_x F(u) = \partial^2_x u$. Such solutions correspond to heteroclinic trajectories $γ$ of a system of ODE. In general conditions of stability can be obtained only numerically. We propose a model class of piece-wise linear (discontinuous) vector fields $F$ for which the stability problem is reduced to a linear algebra problem. We show that the stability problem makes sense in such low regularity and construct several examples of stability loss. Every such example can be smoothed to provide a smooth example of the same phenomenon. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_19167 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the problem of stability of viscous shocks Bolotin, Sergey Treschev, Dmitry Analysis of PDEs Dynamical Systems We consider the problem of spectral stability of traveling wave solutions $u=γ(x-Wt)$ for a system of viscous conservation laws $\partial_t u + \partial_x F(u) = \partial^2_x u$. Such solutions correspond to heteroclinic trajectories $γ$ of a system of ODE. In general conditions of stability can be obtained only numerically. We propose a model class of piece-wise linear (discontinuous) vector fields $F$ for which the stability problem is reduced to a linear algebra problem. We show that the stability problem makes sense in such low regularity and construct several examples of stability loss. Every such example can be smoothed to provide a smooth example of the same phenomenon. |
| title | On the problem of stability of viscous shocks |
| topic | Analysis of PDEs Dynamical Systems |
| url | https://arxiv.org/abs/2511.19167 |