Bistable traveling fronts in strong shear flows: speed and profile asymptotics
Abstract
We study the strong-shear limit of bistable traveling fronts in infinite cylinders with periodic transverse boundary conditions. We prove that, for every sufficiently regular periodic shear profile, the front speed normalized by the flow amplitude converges as the amplitude tends to infinity. After suitable longitudinal rescaling and translations, the corresponding profiles converge along subsequences to full fronts of the limiting degenerate equation. Furthermore, under a Hörmander-type non-degeneracy condition on the shear, the limiting front is proved to be regular and unique up to translation, and the whole normalized family converges uniformly. A main difficulty is that the sign-changing bistable reaction allows the limiting transition to split through intermediate transverse equilibria. We rule out this possibility by exploiting the instability of such equilibria together with suitable regularization and comparison arguments. Finally, in contrast with the combustion case, where every nonconstant mean-zero shear yields a positive limiting speed, we construct an example showing that, for a fixed bistable reaction, smooth mean-zero shears can produce negative, zero, or positive limiting speeds. In particular, the example shows that a shear which accelerates propagation at small amplitudes may reverse the propagation direction when its amplitude becomes large.
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BibTeX
@misc{ding2026bistable,
title = {{Bistable traveling fronts in strong shear flows: speed and profile asymptotics}},
author = {Weiwei Ding and Mingmin Zhang and Zhaoyun Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25855},
eprint = {2609.25855},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25855},
}
RIS
TY - UNPB
TI - Bistable traveling fronts in strong shear flows: speed and profile asymptotics
AU - Weiwei Ding
AU - Mingmin Zhang
AU - Zhaoyun Zhang
PY - 2026
DA - 2026/09/22
PB - arXiv
DO - 10.48550/arxiv.2609.25855
UR - https://arxiv.org/abs/2609.25855
AB - We study the strong-shear limit of bistable traveling fronts in infinite cylinders with periodic transverse boundary conditions. We prove that, for every sufficiently regular periodic shear profile, the front speed normalized by the flow amplitude converges as the amplitude tends to infinity. After suitable longitudinal rescaling and translations, the corresponding profiles converge along subsequences to full fronts of the limiting degenerate equation. Furthermore, under a Hörmander-type non-degeneracy condition on the shear, the limiting front is proved to be regular and unique up to translation, and the whole normalized family converges uniformly. A main difficulty is that the sign-changing bistable reaction allows the limiting transition to split through intermediate transverse equilibria. We rule out this possibility by exploiting the instability of such equilibria together with suitable regularization and comparison arguments. Finally, in contrast with the combustion case, where every nonconstant mean-zero shear yields a positive limiting speed, we construct an example showing that, for a fixed bistable reaction, smooth mean-zero shears can produce negative, zero, or positive limiting speeds. In particular, the example shows that a shear which accelerates propagation at small amplitudes may reverse the propagation direction when its amplitude becomes large.
ER -