Double phase flow under Lavrentiev phenomenon
arXiv:2607.25296
The paper investigates how solutions to parabolic equations derived from double phase functionals evolve, showing that smooth approximability can be lost in finite time due to the Lavrentiev phenomenon, while also proving short‑time persistence of smoothness under certain regularity conditions.
Abstract
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.
19 pages