Long-Time Dynamics for the Kelvin-Helmholtz Equations Close to Circular Vortex Sheets

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.