Existence of strong solutions for the Oldroyd model with multivalued right-hand side

TL;DR

This paper proves the existence of strong solutions for a viscoelastic fluid model with multivalued forces, using advanced fixed-point theorems, extending previous results to more complex, set-valued systems.

Contribution

It establishes local and global existence of strong solutions for the Oldroyd model with multivalued right-hand side, generalizing classical results to differential inclusions.

Findings

Strong solutions exist locally for the coupled system.

Global solutions exist for small initial data.

The approach uses a generalized Kakutani fixed-point theorem.

Abstract

The initial value problem for a coupled system is studied. The system consists of a differential inclusion and a differential equation and models the fluid flow of a viscoelastic fluid of Oldroyd type. The set-valued right-hand side of the differential inclusion satisfies certain measurability, continuity and growth conditions. The local existence (and global existence for small data) of a strong solution to the coupled system is shown using a generalisation of Kakutani's fixed-point theorem and applying results from the single-valued case.