Low regularity of non-l²(ℝⁿ) local solutions to GMHD-α systems

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-α) system, which differs from the original MHD system by including an additional non-linear terms (indexed by α), and replacing the Laplace operators by more general Fourier multipliers with symbols of the form −|ξ|^γ /g(|ξ|). In [8], the problem was considered with initial data in the Sobolev space H^s,2(ℝⁿ) with n ≥ 3. Here we consider the problem with initial data in H^s,p(ℝⁿ) with n ≥ 3 and p > 2. Our goal is to minimizing the regularity required for obtaining uniqueness of a solution.