Trace operators for Riemann--Liouville fractional equations

TL;DR

This work develops a rigorous trace theory for Riemann–Liouville time-fractional equations on bounded domains by deriving a weak-solution framework and an explicit Mittag–Leffler series representation. It establishes existence and uniqueness for the RL fractional wave equation with $\alpha\in(1,2)$ (in particular $\alpha>\tfrac{3}{2}$) and analyzes regularity in interpolation spaces tied to the Laplacian, including trace regularity for the normal derivative on the boundary. A key contribution is the trace result obtained under weaker initial-data assumptions, together with precise conditions on the interpolation index $\mu_\alpha$ ensuring compatible gradient and RL-derivative regularity. The paper also proves energy decay estimates and develops a Lions-type duality between the RL problem and the Caputo problem, linking nonlocal time memory effects to conventional Caputo dynamics and providing a framework for boundary-control/observability analysis.

Abstract

We begin with a brief overview of the most commonly used fractional derivatives, namely the Caputo and Riemann-Liouville derivatives. We then focus on the study of the fractional time wave equation with the Riemann-Liouville derivative, addressing key questions such as well-posedness, regularity, and a trace result in appropriate interpolation spaces. Additionally, we explore the duality relationship with the Caputo fractional time derivative. The analysis is based on expanding the solution in terms of Mittag-Leffler functions.

Figures (1)

• Figure 1: Non-empty intersections

Theorems & Definitions (25)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Remark 2.4
• Lemma 2.5
• Definition 3.1
• Remark 3.2
• Theorem 3.3
• proof
• Theorem 4.1