Singular Contour Dynamics and Paradifferential Reduction
Abstract
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.
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BibTeX
@misc{li2026singular,
title = {{Singular Contour Dynamics and Paradifferential Reduction}},
author = {Xingyu Li and Emeric Roulley and Stefano Scrobogna},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.03091},
eprint = {2610.03091},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.03091},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Singular Contour Dynamics and Paradifferential Reduction
AU - Xingyu Li
AU - Emeric Roulley
AU - Stefano Scrobogna
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.03091
UR - https://arxiv.org/abs/2610.03091
AB - 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.
ER -