Well-posedness for magnetoviscoelastic fluids in 3D

TL;DR

This paper proves local and global well-posedness of 3D magnetoviscoelastic fluid equations, showing solutions are smooth, unique, and converge to equilibrium states under certain conditions.

Contribution

It establishes the well-posedness and regularity of the magnetoviscoelastic fluid system in three dimensions using maximal Lp-regularity theory, a novel approach for this model.

Findings

Existence and uniqueness of local strong solutions.

Solutions become analytic in space and time.

Global convergence to equilibrium states.

Abstract

We show that the system of equations describing a magnetoviscoelastic fluid in three dimensions can be cast as a quasilinear parabolic system. Using the theory of maximal $L_p$-regularity, we establish existence and uniqueness of local strong solutions and we show that each solution is smooth (in fact analytic) in space and time. Moreover, we give a complete characterization of the set of equilibria and show that solutions that start out close to a constant equilibrium exist globally and converge to a (possibly different) constant equilibrium. Finally, we show that every solution that is eventually bounded in the topology of the state space exists globally and converges to the set of equilibria.