| Abstract: |
| We prove local well-posedness for a 2D vortex sheet separating two fluids of different densities in a 3D flow. The sheet evolves in a potential flow under the combined effects of gravity and surface tension. Our approach follows the method of Ambrose and Masmoudi, who treated the
density-matched case. The main challenge in the unequal-density setting is a more complicated equation governing the vortex sheet strength, which is determined by the jump in velocity potential across the interface. The velocity potential can be represented as a boundary integral of dipoles. In the density-matched case, the dipole strength $\mu$ satisfies an explicit time-evolution equation. In contrast, when the densities differ, $\mu_t$ appears implicitly in
a singular integral equation, which must first be inverted to find the evolution equation for $\mu$. The regularity of the vortex-sheet strength is established by combining new estimates with a bootstrap argument. This is joint work with Joseph Cavatchel (NJIT) and David Ambrose (Drexel). |
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