Weak and mild solutions to the MHD equations and the viscoelastic Navier-Stokes equations with damping in Wiener amalgam spaces

TL;DR

This paper establishes the existence and properties of weak and mild solutions for 3D incompressible MHD and viscoelastic Navier-Stokes equations with damping in Wiener amalgam spaces, extending previous frameworks.

Contribution

It introduces new existence results for solutions in Wiener amalgam spaces and extends properties of local energy solutions for these complex fluid models.

Findings

Existence of mild solutions in Wiener amalgam spaces.

Construction of global-in-time local energy weak solutions.

Extension of regularity and uniqueness results for initial data in $L^2_{uloc}$.

Abstract

We study the three-dimensional incompressible magnetohydrodynamic (MHD) equations and the incompressible viscoelastic Navier-Stokes equations with damping. Building on techniques developed by Bradshaw, Lai, and Tsai (Math. Ann. 2024), we prove the existence of mild solutions in Wiener amalgam spaces that satisfy the corresponding spacetime integral bounds. In addition, we construct global-in-time local energy weak solutions in these amalgam spaces using the framework introduced by Bradshaw and Tsai (SIAM J. Math. Anal. 2021). As part of this construction, we also establish several properties of local energy solutions with $L^2_{\rm uloc}$ initial data, including initial and eventual regularity as well as small-large uniqueness, extending analogous results obtained for the Navier-Stokes equations by Bradshaw and Tsai (Comm. Partial Differential Equations 2020).