On the problem of stability of viscous shocks

Sergey Bolotin; Dmitry Treschev

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arXiv · 2511.19167

On the problem of stability of viscous shocks

Sergey Bolotin·Dmitry Treschev

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Abstract

We consider the problem of spectral stability of traveling wave solutions $u=\gamma(x-Wt)$ for a system of viscous conservation laws $\partial_t u + \partial_x F(u) = \partial^2_x u$. Such solutions correspond to heteroclinic trajectories $\gamma$ 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.

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Sergey Bolotin, Dmitry Treschev. 2025-11-24. On the problem of stability of viscous shocks. https://arxiv.org/abs/2511.19167

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