Struwe-like solutions for an evolutionary model of magnetoviscoelastic fluids

TL;DR

This paper establishes the existence, uniqueness, and regularity of solutions for a complex system modeling magnetoviscoelastic fluids, combining advanced mathematical techniques to handle large initial data in a two-dimensional setting.

Contribution

It introduces a well-posedness theory for a coupled PDE system modeling magnetoviscoelastic fluids, proving smoothness of solutions except at discrete times.

Findings

Weak solutions are smooth except at discrete times

Uniqueness is proven using energy estimates in less regular frameworks

The analysis employs harmonic analysis and paradifferential calculus techniques

Abstract

In this work we investigate the existence and uniqueness of Struwe-like solutions for a system of partial differential equations modeling the dynamics of magnetoviscoelastic fluids. The considered system couples a Navier-Stokes type equation with a dissipative equation for the deformation tensor and a Landau-Lifshitz-Gilbert type equation for the magnetization field. The main purpose is to establish a well-posedness theory in a two-dimensional periodic domain under standard assumption of critical regularity for the (possibly large) initial data. We prove that the considered weak solutions are everywhere smooth, except for a discrete set of time values. The proof of the uniqueness is based on suitable energy estimates for the solutions within a functional framework which is less regular than the one of the Struwe energy level. These estimates rely on several techniques of harmonic analysis and paradifferential calculus.