Determination of time dependent source terms for Stokes systems in unbounded domains

TL;DR

This paper investigates the unique determination of time-dependent, vector-valued source terms in Stokes systems within unbounded domains, using boundary measurements and advanced mathematical techniques, including controllability and harmonic functions.

Contribution

It introduces new conditions for uniquely identifying general time-dependent sources in Stokes systems, incorporating unknown divergence, with a reconstruction algorithm and numerical examples.

Findings

Established unique determination conditions for source terms.

Developed a reconstruction algorithm for inverse problems.

Provided numerical examples demonstrating the method's effectiveness.

Abstract

This article is devoted to the analysis of inverse source problems for Stokes systems in unbounded domains where the corresponding velocity flow is observed on a surface. Our main objective is to study the unique determination of general class of time-dependent and vector-valued source terms with potentially unknown divergence. Taking into account the challenges inherent in this class of inverse source problems, we aim to identify the most precise conditions that ensure their resolution. Motivated by various fluid motion problems, we explore several class of boundary measurements. Our proofs are based on different arguments, including unique continuation properties for Stokes systems, approximate controllability, complex analysis, and the application of explicit harmonic functions and explicit solutions for Stokes systems. This analysis is complemented by a reconstruction algorithm and examples of numerical computations.