A local in time existence and uniqueness result of an inverse problem for the Kelvin-Voigt fluids

TL;DR

This paper establishes local in time existence and uniqueness results for an inverse problem involving Kelvin-Voigt fluids, focusing on reconstructing velocity and memory kernel from boundary measurements.

Contribution

It introduces a novel approach using contraction mapping to prove local and global existence and uniqueness for inverse problems in viscoelastic fluid models.

Findings

Local in time existence and uniqueness for the inverse problem

Global in time existence and uniqueness for Oseen type inverse problems

Application of contraction mapping in viscoelastic fluid inverse problems

Abstract

In this paper, we consider an inverse problem for three dimensional viscoelastic fluid flow equations, which arises from the motion of Kelvin-Voigt fluids in bounded domains (a hyperbolic type problem). This inverse problem aims to reconstruct the velocity and kernel of the memory term simultaneously, from the measurement described as the integral over determination condition. By using the contraction mapping principle in an appropriate space, a local in time existence and uniqueness result for the inverse problem of Kelvin-Voigt fluids are obtained. Furthermore, using similar arguments, a global in time existence and uniqueness results for an inverse problem of Oseen type equations are also achieved.