Too much regularity may force too much uniqueness

TL;DR

This paper examines time-dependent fractional-derivative problems, revealing that overestimating solution regularity imposes overly restrictive conditions, challenging common assumptions in the field.

Contribution

It demonstrates that assuming excessive regularity in fractional-derivative problems can lead to unnecessarily strict restrictions on solutions.

Findings

Overestimating regularity imposes severe restrictions

Common assumptions may be too restrictive

Regularity assumptions need careful reconsideration

Abstract

Time-dependent fractional-derivative problems $D_t^\delta u + Au = f$ are considered, where $D_t^\delta$ is a Caputo fractional derivative of order $\delta\in (0,1)\cup (1,2)$ and~$A$ is a classical elliptic operator, and appropriate boundary and initial conditions are applied. The regularity of solutions to this class of problems is discussed, and it is shown that assuming more regularity than is generally true---as many researchers do---places a surprisingly severe restriction on the problem.