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This study proves that small-amplitude vortex sheet solutions to the Kelvin-Helmholtz system exist for almost all time. This is achieved by analyzing the linear stability threshold for the Weber number, demonstrating a stabilization phenomenon due to velocity jumps and surface tension.

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Area of Science:

  • Fluid Dynamics
  • Mathematical Physics
  • Nonlinear Analysis

Background:

  • The Kelvin-Helmholtz system models vortex sheet evolution near circular stationary solutions.
  • Previous numerical studies in the 1990s posed conjectures regarding solution existence.
  • The classical Kelvin-Helmholtz problem is inherently unstable.

Purpose of the Study:

  • To prove an almost global existence result for small-amplitude solutions to the Kelvin-Helmholtz system.
  • To establish the existence of a linear stability threshold for the Weber number.
  • To demonstrate a stabilization phenomenon preventing nonlinear instabilities.

Main Methods:

  • Hamiltonian Birkhoff normal form techniques for quasi-linear systems.
  • Paralinearization of nonlinear singular integral operators.
  • Analysis of resonances and quasi-resonances to arbitrary order.

Main Results:

  • Existence of a linear stability threshold for the Weber number (ratio of velocity jump squared to surface tension).
  • Small solutions exist for almost all times for Weber numbers below the threshold.
  • A stabilization phenomenon is revealed, preventing nonlinear instabilities.

Conclusions:

  • The combination of velocity jumps and capillarity effects leads to long-time existence of solutions.
  • This stabilization is more robust than capillarity alone, due to modulation effects from the velocity jump.
  • The findings answer previous numerical conjectures and highlight a novel stabilization mechanism.