Low regularity of non-$L^2(R^n)$ local solutions to gMHD-alpha systems

TL;DR

This paper investigates the minimal regularity needed for uniqueness of solutions to the generalized MHD-alpha system with initial data in certain Sobolev spaces, extending previous work to broader function spaces.

Contribution

It extends the analysis of the gMHD-alpha system to initial data in $H^{s,p}$ spaces with $p > 2$, aiming to identify the lowest regularity for solution uniqueness.

Findings

Identifies lower regularity thresholds for uniqueness.

Extends previous Sobolev space results to $H^{s,p}$ spaces.

Provides insights into the regularity requirements for generalized MHD systems.

Abstract

The Magneto-Hydrodynamic (MHD) system of equations governs viscous fluids subject to a magnetic field and is derived via a coupling of the Navier-Stokes equations and Maxwell's equations. Recently it has become common to study generalizations of fluids-based differential equations. Here we consider the generalized Magneto-Hydrodynamic alpha (gMHD-$\alpha$) system, which differs from the original MHD system by including an additional non-linear terms (indexed by $\alpha$), and replacing the Laplace operators by more general Fourier multipliers with symbols of the form $-|\xi|^\gamma / g(|\xi|)$. In a paper by Pennington, the problem was considered with initial data in the Sobolev space $H^{s,2}(\mathbb{R}^n)$ with $n \geq 3$. Here we consider the problem with initial data in $H^{s,p}(\mathbb{R}^n)$ with $n \geq 3$ and $p > 2$. Our goal is to minimize the regularity required for obtaining uniqueness of a solution.