Singular Contour Dynamics and Paradifferential Reduction

We develop a direct paradifferential approach to nonlinear graph integral operators arising in contour dynamics. The main methodological result is an abstract structure-preserving paralinearization theorem, valid in arbitrary dimension, for a class of homogeneous kernels whose nonlinear dependence on the graph is expressed through normalized finite differences. The theorem covers both critical principal-value kernels and subcritical locally integrable kernels, gives explicit formulas for the principal symbols, and separates the finite-order symbolic contribution generated by the homogeneous singularity from order-zero far-field terms and arbitrarily smoothing remainders. A structural feature of the reduction is the cancellation of the intermediate symbolic order; at the critical endpoint the construction also preserves the exact skew-adjoint structure of the original singular integral. As a main application, we consider the three-dimensional two-phase free-boundary Euler equations with constant interfacial background vorticity. We first derive an autonomous contour-dynamics equation in canonical surface variables and then apply the abstract theorem directly to the singular integral operators generated by the Birkhoff–Rott formulation. This yields the complete paradifferential structure of the system without taking the Dirichlet–Neumann reduction as a starting point. After introducing an Alinhac good unknown and diagonalizing the resulting system, we obtain local well-posedness for small Sobolev perturbations in the stable Kelvin–Rayleigh–Taylor regime, uniformly on compact subsets of the stable parameter set.