发表机构
Tohoku University（东北大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究与双相泛函相关抛物方程中双相流朝奇异极小值演化，通过演化变分不等式证明流有限时间丧失光滑逼近性并定量估计时间，利用解析半群理论证明初始数据正则且泛函非退化时光滑逼近性短时持续，探究拉夫连季耶夫现象动力学。
AI 中文摘要
本文研究与双相泛函相关的抛物方程。已知双相泛函可能呈现拉夫连季耶夫现象，即存在奇异极小值。目的是研究相关双相流朝着奇异极小值演化的过程。为此，研究解是否能用光滑函数逼近。首先观察到有限时间内光滑逼近性丧失现象，证明流最终不再光滑可逼近，并建立丧失时间的定量估计。另一方面，研究光滑逼近性短时持续现象，证明当初始数据正则且泛函关于梯度变量非退化时，光滑逼近性在短时间内持续。有限时间丧失结果源于演化变分不等式，短时持续结果通过解析半群理论得到。本文新颖之处在于研究通常被视为稳态现象的拉夫连季耶夫现象的动力学方面。
英文摘要
This paper deals with parabolic equations associated with double phase functionals. It is known that double phase functionals may exhibit the Lavrentiev phenomenon, which indicates an existence of a singular minimizer. Our aim is to investigate the process of the associated double phase flow evolving toward the singular minimizer. For this purpose, we study whether solutions can be approximated by smooth functions. We first observe a phenomenon that we call finite-time loss of smooth approximability; more precisely, we prove that the flow eventually ceases to be smoothly approximable. We also establish quantitative estimates on the time of loss. On the other hand, we investigate a phenomenon that we call short-time persistence of smooth approximability; we prove that the smooth approximability persists for a short time provided that the initial datum is regular and that the functional is nondegenerate with respect to the gradient variable. The results concerning finite-time loss of smooth approximability are derived from the evolution variational inequality, whereas the short-time persistence result is obtained by applying analytic semigroup theory. The novelty of this paper lies in studying the dynamical aspect of the Lavrentiev phenomenon, which is usually regarded as a stationary phenomenon.
Comments19 pages, Remark 1.2 is corrected in the revised version
