# Regularity theory for time-fractional advection-diffusion-reaction equations

## TL;DR

This paper studies the regularity and derivative estimates of solutions to linear time-fractional advection-diffusion-reaction equations with variable coefficients, addressing challenges posed by nonlocal fractional derivatives for numerical analysis.

## Contribution

It introduces new energy-based analytical techniques and fractional inequalities to establish estimates for solutions with low regularity, aiding numerical error analysis.

## Key findings

- Derived estimates for time derivatives of solutions
- Addressed low regularity initial data
- Developed novel energy methods for fractional PDEs

## Abstract

We investigate the behavior of the time derivatives of the solution to a linear time-fractional, advection-diffusion-reaction equation, allowing space- and time-dependent coefficients as well as initial data that may have low regularity. Our focus is on proving estimates that are needed for the error analysis of numerical methods. The nonlocal nature of the fractional derivative creates substantial difficulties compared with the case of a classical parabolic PDE. In our analysis, we rely on novel energy methods in combination with a fractional Gronwall inequality and certain properties of fractional integrals.