Can every bounded entire Burgers profile recur at late times?
For $\alpha < \beta$ and a Borel probability measure $\mu$ supported in $[\alpha, \beta]$, define $$\phi_\mu(x) = \frac{\int z\, e^{-zx/2}\, d\mu(z)}{\int e^{-zx/2}\, d\mu(z)}.$$ For bounded initial data, the viscous Burgers equation $$u_t + u u_x = u_{xx}, \qquad u(0, x) = u_0(x),$$ has a bounded solution given by the Cole–Hopf transformation. As Gallay and Scheel recall, every entire solution with values in $[\alpha, \beta]$ is $$u(t, x) = \frac{\int z\, e^{-zx/2 + z^2 t/4}\, d\mu(z)}{\int e^{-zx/2 + z^2 t/4}\, d\mu(z)}$$ for a unique such $\mu$, so $\phi_\mu$ is its profile at time zero. They constructed bounded initial data whose $\omega$-limit set contains entire solutions describing the merging of two shocks.
Problem. For every $\alpha < \beta$ and every Borel probability measure $\mu$ supported in $[\alpha, \beta]$, does there exist $u_0 \in L^\infty(\mathbb{R})$ with $\alpha \le u_0 \le \beta$ almost everywhere such that the solution of the equation above admits sequences $t_k \to \infty$ and $x_k \in \mathbb{R}$ for which $u(t_k, x + x_k) \to \phi_\mu(x)$ uniformly on every compact interval? The measure may be continuous or have infinitely many atoms.
Source: Gallay and Scheel (2024), the conjecture following Proposition 6.2; stated in this form by AIM — Open Applied Problems.
Imported from github.com/MColbrook/AIM.
Classification
- Primary
- AP — Analysis of PDEs
- Secondary
- DS
- Keywords
- AIM Open Applied Problems, Cole–Hopf transformation, omega-limit sets, viscous Burgers equation
References
@article{gallay2024,
author = {Gallay, Thierry and Scheel, Arnd},
title = {Viscous shocks and long-time behavior of scalar conservation laws},
journal = {Commun. Pure Appl. Anal.},
volume = {23},
number = {10},
year = {2024},
pages = {1448--1482},
doi = {10.3934/cpaa.2023119},
eprint = {2306.13341},
archivePrefix = {arXiv},
}
@article{hopf1950,
author = {Hopf, Eberhard},
title = {The partial differential equation $u_t + uu_x = \mu u_{xx}$},
journal = {Comm. Pure Appl. Math.},
volume = {3},
number = {3},
year = {1950},
pages = {201--230},
doi = {10.1002/cpa.3160030302},
}
@misc{aim-burgers-entire-limits,
author = {Brunton, Steve and Colbrook, Matthew J. and de Hoop, Maarten V. and Stepaniants, George and Townsend, Alex and Ward, Rachel},
title = {{Can every bounded entire Burgers profile recur at late times?}},
howpublished = {AIM --- Open Applied Problems, GitHub repository},
year = {2026},
url = {https://github.com/MColbrook/AIM/blob/1b7c7fd251c1643bb41f0e3cef38704bcef410fd/problems/351-burgers-entire-limits.md},
note = {Entry 351 at commit 1b7c7fd, 2026-10-13},
}
Resolution claims
We construct bounded initial data for the viscous Burgers equation whose translated late-time limits contain any prescribed profile represented by a compactly supported probability measure. A uniform fixed-time locality estimate for the Cole--Hopf formula allows backward histories to be placed in disjoint spatial windows. The construction also produces a single initial datum realizing every probability-measure profile in a fixed value …
Status history
| When | Change | By | Reason |
|---|---|---|---|
| 2026-10-16 | Needs verification → Solution claimed | — | Recorded here: AIM @ a153f46, 351-burgers-profile-realization/aim351_burgers_realization.pdf. |
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-00668, "Can every bounded entire Burgers profile recur at late times?", clopen.solutions.