发表机构
Università degli Studi di Trieste; Università Statale di Milano（的里雅斯特大学; 米兰大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出直接仿微分方法处理轮廓动力学中的非线性图积分算子，建立结构保持的仿线性化定理，并应用于三维两相欧拉方程，在稳定Kelvin-Rayleigh-Taylor区域获得局部适定性。
AI 中文摘要
我们发展了一种直接仿微分方法，用于处理轮廓动力学中出现的非线性图积分算子。主要方法论结果是一个抽象的结构保持的仿线性化定理，适用于任意维度，针对一类齐次核，其关于图的非线性依赖通过归一化有限差分表达。该定理同时涵盖临界主值核和次临界局部可积核，给出主符号的显式公式，并将由齐次奇异性产生的有限阶符号贡献与零阶远场项及任意光滑化余项分离。约化的一个结构特征是中间符号阶的抵消；在临界端点，构造还保留了原始奇异积分的精确斜伴随结构。作为主要应用，我们考虑具有常数界面背景涡量的三维两相自由边界欧拉方程。我们首先在规范曲面变量中推导自治轮廓动力学方程，然后直接将抽象定理应用于Birkhoff-Rott公式生成的奇异积分算子。这在不以Dirichlet-Neumann约化为起点的情况下给出了系统的完整仿微分结构。在引入Alinhac好未知量并对所得系统进行对角化后，我们获得了稳定Kelvin-Rayleigh-Taylor区域中小Sobolev扰动的局部适定性，且在稳定参数集的紧子集上一致成立。
英文摘要
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.
Comments201 pages
