Finite-time growth of total variation in physical gas dynamics
Consider the $p$-system with polytropic pressure, $$v_t - u_x = 0, \qquad u_t + (v^{-\gamma})_x = 0 \qquad (\gamma > 1),$$ the equations of isentropic gas dynamics in Lagrangian coordinates, with specific volume $v$ and velocity $u$, and its mechanical entropy inequality $$\partial_t\left(\frac{u^2}{2} + \frac{v^{1-\gamma}}{\gamma - 1}\right) + \partial_x\left(u v^{-\gamma}\right) \le 0.$$ Write $\operatorname{TV}$ for total variation in $x \in \mathbb{R}$. Bressan, Chen and Zhang proved that if the total variation of an entropy weak solution whose density stays bounded away from zero blows up in finite time, then the solution contains infinitely many large shocks near some point of the $(t, x)$-plane; they also constructed piecewise smooth approximate solutions whose total variation blows up in finite time, in which the waves leaving each interaction have their exact strengths and rarefactions decay as in exact solutions (2018). Krupa gave, by a computer-assisted convex integration, a smooth pressure law and Riemann data with infinitely many bounded solutions of infinite total variation at every positive time; they contain no shocks, so satisfy the Lax and Liu admissibility conditions vacuously, and the construction is incompatible with a strictly convex entropy (2026).
Problem. Are there $\gamma > 1$, $T < \infty$ and a distributional solution $(v, u)$ on $[0, T) \times \mathbb{R}$, continuous in time with values in $L^1_{\mathrm{loc}}$ and satisfying the entropy inequality, with initial data of finite total variation and with $0 < a \le v \le b < \infty$ and $|u| \le M$ for some constants $a$, $b$, $M$, such that $$\operatorname*{ess\,sup}_{t \le T'} \bigl[\operatorname{TV} v(t) + \operatorname{TV} u(t)\bigr] < \infty$$ for every $T' < T$, while this quantity tends to infinity as $T' \uparrow T$?
Source: Bressan, Chen and Zhang (2018), §1, who name finite-time blowup of the total variation of entropy weak solutions with large BV data a major open problem for hyperbolic systems; stated in this form for the polytropic p-system by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- MP
- Keywords
- AIM Open Applied Problems, entropy solutions, hyperbolic conservation laws
References
@article{bressan2018,
author = {Bressan, Alberto and Chen, Geng and Zhang, Qingtian},
title = {On finite time {BV} blow-up for the p-system},
journal = {Comm. Partial Differential Equations},
volume = {43},
number = {8},
year = {2018},
pages = {1242--1280},
doi = {10.1080/03605302.2018.1499115},
eprint = {1710.03382},
archivePrefix = {arXiv},
}
@article{krupa2026,
author = {Krupa, Sam G.},
title = {Non-uniqueness and {BV} blowup for vacuously {L}iu-admissible solutions to p-system via computer-assisted proof},
journal = {J. Math. Fluid Mech.},
volume = {28},
number = {2},
year = {2026},
pages = {37},
doi = {10.1007/s00021-026-01018-5},
eprint = {2403.07784},
archivePrefix = {arXiv},
}
@misc{aim-p-system-finite-time-bv,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Finite-time growth of total variation in physical gas dynamics}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/24edaadc6bb272737d8da0745c662c992db07b22/problems/229-p-system-finite-time-bv.md},
note = {Entry 229 at commit 24edaad, 2026-10-15},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00631, "Finite-time growth of total variation in physical gas dynamics", clopen.solutions.