Periodic orbits of the ABC flow with A= B= C= 1

Jack Xin Yifeng Yu Andrej Zlatos

Analysis of PDEs mathscidoc:1912.43888

SIAM Journal on Mathematical Analysis, 48, (6), 4087-4093, 2016.12
In this paper, we prove that the celebrated Arnold--Beltrami--Childress (ABC) flow with parameters A=B=C=1, given by $ \dot x =\sin z+\cos y,\ \dot y = \sin x+\cos z,\ \dot z =\sin y + \cos x $, has periodic orbits on (2\pi\mathbb T)^3 with rotation vectors parallel to (1,0,0), (0,1,0), and (0,0,1). Despite ABC flows being studied since the 1960s, this seems to be the first time that existence of nonperturbative periodic orbits has been established for them. The main difficulty here is the lack of a variational structure for these flows, and our proof instead relies on their symmetry properties. As an application of our result, we show that the well-known G-equation model of turbulent combustion with this ABC flow on (2\pi\mathbb T)^3 has a linear (i.e., maximal possible) flame speed enhancement rate as the flow amplitude grows to infinity. To the best of our knowledge, this is the first time an asymptotic flame speed growth law has been established for a
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  title={Periodic orbits of the ABC flow with A= B= C= 1},
  author={Jack Xin, Yifeng Yu, and Andrej Zlatos},
  booktitle={SIAM Journal on Mathematical Analysis},
Jack Xin, Yifeng Yu, and Andrej Zlatos. Periodic orbits of the ABC flow with A= B= C= 1. 2016. Vol. 48. In SIAM Journal on Mathematical Analysis. pp.4087-4093.
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