TY - JOUR
T1 - Invariant KAM Tori Around Annular Vortex Patches for 2D Euler Equations
AU - Hassainia, Zineb
AU - Hmidi, Taoufik
AU - Roulley, Emeric
N1 - Publisher Copyright:
© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2024.
PY - 2024/11
Y1 - 2024/11
N2 - We construct time quasi-periodic vortex patch solutions with one hole for the planar Euler equations. These structures are captured close to any annulus provided that its modulus belongs to a massive Borel set. The proof is based on Nash–Moser scheme and KAM theory applied with a Hamiltonian system governing the radial deformations of the patch. Compared to the scalar case discussed recently in Hassainia et al. (KAM theory for active scalar equations, arXiv:2110.08615), Hassainia and Roulley (Boundary effects on the existence of quasi-periodic solutions for Euler equations, arXiv:2202.10053), Hmidi and Roulley (Time quasi-periodic vortex patches for quasi-geostrophic shallow-water equations, arXiv:2110.13751) and Roulley (Dyn Partial Differ Equ 20(4):311–366, 2023), some technical issues emerge due to the interaction between the interfaces. One of them is related to a new small divisor problem in the second order Melnikov non-resonances condition coming from the transport equations advected with different velocities.
AB - We construct time quasi-periodic vortex patch solutions with one hole for the planar Euler equations. These structures are captured close to any annulus provided that its modulus belongs to a massive Borel set. The proof is based on Nash–Moser scheme and KAM theory applied with a Hamiltonian system governing the radial deformations of the patch. Compared to the scalar case discussed recently in Hassainia et al. (KAM theory for active scalar equations, arXiv:2110.08615), Hassainia and Roulley (Boundary effects on the existence of quasi-periodic solutions for Euler equations, arXiv:2202.10053), Hmidi and Roulley (Time quasi-periodic vortex patches for quasi-geostrophic shallow-water equations, arXiv:2110.13751) and Roulley (Dyn Partial Differ Equ 20(4):311–366, 2023), some technical issues emerge due to the interaction between the interfaces. One of them is related to a new small divisor problem in the second order Melnikov non-resonances condition coming from the transport equations advected with different velocities.
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U2 - 10.1007/s00220-024-05141-0
DO - 10.1007/s00220-024-05141-0
M3 - Article
AN - SCOPUS:85207849557
SN - 0010-3616
VL - 405
JO - Communications In Mathematical Physics
JF - Communications In Mathematical Physics
IS - 11
M1 - 270
ER -