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1. 偏微分方程-ODI原理与柱状平均曲率流

The PDE-ODI principle and cylindrical mean curvature flows

作者: Richard H. Bamler, Yi Lai

类别: math.DG, math.AP

发布日期: 2025-12-31

arXiv ID: 2512.25050v1

链接: http://arxiv.org/abs/2512.25050v1

摘要: We introduce a new approach for analyzing ancient solutions and singularities of mean curvature flow that are locally modeled on a cylinder. Its key ingredient is a general mechanism, called the \emph{PDE–ODI principle}, which converts a broad class of parabolic differential equations into systems of ordinary differential inequalities. This principle bypasses many delicate analytic estimates used in previous work, and yields asymptotic expansions to arbitrarily high order. As an application, we establish the uniqueness of the bowl soliton times a Euclidean factor among ancient, cylindrical flows with dominant linear mode. This extends previous results on this problem to the most general setting and is made possible by the stronger asymptotic control provided by our analysis. In the other case, when the quadratic mode dominates, we obtain a complete asymptotic expansion to arbitrary polynomial order, which will form the basis for a subsequent paper. Our framework also recovers and unifies several classical results. In particular, we give new proofs of the uniqueness of tangent flows (due to Colding-Minicozzi) and the rigidity of cylinders among shrinkers (due to Colding-Ilmanen-Minicozzi) by reducing both problems to a single ordinary differential inequality, without using the Łojasiewicz-Simon inequality. Our approach is independent of prior work and the paper is largely self-contained.

2. 关于紧扰动无穷维系统的精确可观测性研究

On exact Observability for Compactly perturbed infinite dimension system

作者: Nisrine Charaf, Faouzi Triki

类别: math.AP

发布日期: 2025-12-31

arXiv ID: 2512.25041v1

链接: http://arxiv.org/abs/2512.25041v1

摘要: In this paper, we study the observability of compactly perturbed infinite dimensional systems. Assuming that a given infinite-dimensional system with self-adjoint generator is exactly observable we derive sufficient conditions on a compact self adjoint perturbation to guarantee that the perturbed system stays exactly observable. The analysis is based on a careful asymptotic estimation of the spectral elements of the perturbed unbounded operator in terms of the compact perturbation. These intermediate results are of importance themselves.