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The Magneto-Hydrodynamic (MHD) system of equations governs viscous fluidssubject to a magnetic field and is derived via a coupling of the Navier-Stokesequations and Maxwell's equations. Recently it has become common to studygeneralizations of fluids-based differential equations. Here we consider thegeneralized Magneto-Hydrodynamic alpha (gMHD-$\alpha$) system, which differsfrom the original MHD system by including an additional non-linear terms(indexed by $\alpha$), and replacing the Laplace operators by more generalFourier multipliers with symbols of the form $-|\xi|^\gamma / g(|\xi|)$. In apaper by Pennington, the problem was considered with initial data in theSobolev space $H^{s,2}(\mathbb{R}^n)$ with $n \geq 3$. Here we consider theproblem 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 ofa solution.

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