LOW REGULARITY OF NON-L²(Rⁿ) LOCAL SOLUTIONS TO GMHD-ALPHA 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. It has recently 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 the presence of additional nonlinear terms (indexed by the choice of α) and replacing the Laplace operators in the equations by more general Fourier multipliers with symbols of the form − |ξ|^γ /g(|ξ|). In [Pennington, 2017], one of the authors considered the problem with initial data in Sobolev spaces of the form Hs,²(Rⁿ) with n ≥ 3. Here we consider the problem with initial data in H^s,p(Rⁿ) with n ≥ 3 and p > 2, with the goal of minimizing the regularity required to obtain unique existence results.

MSC Codes 35B65, 35A02, 76W05