We consider the Kelvin-Helmholtz system describing the evolution of a
vortex-sheet near the circular stationary solution. Answering previous
numerical conjectures in the 90s physics literature, we prove an almost global
existence result for small-amplitude solutions. We first establish the
existence of a linear stability threshold for the Weber number, which
represents the ratio between the square of the background velocity jump and the
surface tension. Then, we prove that for almost all values of the Weber number
below this threshold any small solution lives for almost all times, remaining
close to the equilibrium. Our analysis reveals a remarkable stabilization
phenomenon: the presence of both non-zero background velocity jump and
capillarity effects enables to prevent nonlinear instability phenomena, despite
the inherently unstable nature of the classical Kelvin-Helmholtz problem. Questo
long-time existence would not be achievable in a setting where capillarity
alone provides linear stabilization, without the richer modulation induced by
the velocity jump. Our proof exploits the Hamiltonian nature of the equations.
Nello specifico, we employ Hamiltonian Birkhoff normal form techniques for
quasi-linear systems together with a general approach for paralinearization of
non-linear singular integral operators. This approach allows us to control
resonances and quasi-resonances at arbitrary order, ensuring the desired
long-time stability result.

Questo articolo esplora i giri e le loro implicazioni.

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