Asymptotic analysis for a very fast diffusion equation arising from the 1D quantization problem

TL;DR

This paper analyzes the long-term behavior of a fast diffusion PDE in 1D, demonstrating exponential convergence to equilibrium and establishing conditions for stability, motivated by measure quantization problems.

Contribution

It provides the first asymptotic analysis of a very fast diffusion equation related to measure quantization, including stability conditions and convergence rates.

Findings

Proves exponential convergence to equilibrium.

Establishes conditions for $W_2$-stability.

Links PDE behavior to measure quantization.

Abstract

In this paper we study the asymptotic behavior of a very fast diffusion PDE in 1D with periodic boundary conditions. This equation is motivated by the gradient flow approach to the problem of quantization of measures introduced in \cite{CGI1}. We prove exponential convergence to equilibrium under minimal assumptions on the data, and we also provide sufficient conditions for $W_2$-stability of solutions.