KPP fronts in the 1-1-1 ABC flow speed up linearly in every direction
Year: 2026
Type: preprint
Abstract: Let $V(x,y,z)=(\sin z+\cos y,\ \sin x+\cos z,\ \sin y+\cos x)$ be the Arnold–Beltrami–Childress (ABC) flow with $A=B=C=1$, and let $c^*_e(K)$ be the minimal speed of pulsating fronts of $T_t+K\,V\cdot\nabla T=\Delta T+f(T)$ in the direction $e$, for a reaction $f$ of KPP type. We prove that $\lim_{K\to\infty}c^*_e(K)/K$ exists and is positive for every unit vector $e$. Zlatoš proved that this limit exists and equals a supremum over first integrals of the flow, and Xin, Yu and Zlatoš, who constructed periodic orbits of this flow, pointed out that a positive-measure tube of orbits moving roughly linearly would make the limit positive, but that a rigorous proof of such tubes was not known. We prove that such a tube exists. Two independent interval-arithmetic computations certify an elliptic periodic orbit with nonzero twist; Pöschel's KAM theorem with frequencies as parameters then gives invariant curves of the return map with a Lipschitz label, which yield a Lipschitz first integral supported on the tube. The deduction from the certified facts is checked in Lean, with Zlatoš's and Pöschel's theorems as cited inputs. This work has not yet been reviewed by experts. The proofs, the Lean formalization and the text were produced by AI models (from OpenAI and Anthropic), working under the author's direction. Written proof, with Lean and computer checks; not yet reviewed by experts. Produced by AI models under the author's direction.
Author: Graham, P.
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225633916
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