Well-posedness of evolution equations with time-dependent nonlinear mobility: a modified minimizing movement scheme

TL;DR

This paper establishes the existence of solutions for certain time-dependent nonlinear evolution equations using a modified gradient flow scheme, extending the classical approach to nonautonomous cases with explicit time dependence.

Contribution

It introduces a modified minimizing movement scheme tailored for evolution equations with time-dependent mobility functions, broadening the applicability of gradient flow methods.

Findings

Proves existence of nonnegative weak solutions for second and fourth order equations.

Extends gradient flow techniques to equations with explicit time-dependent mobility.

Provides a framework for potential extensions to bounded domains and more general mobility functions.

Abstract

We prove the existence of nonnegative weak solutions to a class of second and fourth order nonautonomous nonlinear evolution equations with an explicitly time-dependent mobility function posed on the whole space $\mathbb{R}^d$, for arbitrary $d\ge 1$. Exploiting a very formal gradient flow structure, the cornerstone of our proof is a modified version of the classical minimizing movement scheme for gradient flows. The mobility function is required to satisfy - at each time point separately - the conditions by which one can define a modified Wasserstein distance on the space of probability densities with finite second moment. The explicit dependency on the time variable is assumed to be at least of Lipschitz regularity. We also sketch possible extensions of our result to the case of bounded spatial domains and more general mobility functions.