Local solvability and loss of smoothness of the Navier-Stokes-Maxwell equations with large initial data

TL;DR

This paper investigates the local existence and smoothness loss of solutions to the Navier-Stokes-Maxwell equations with large initial data, using a Fourier-based approach to handle nonlinearities and Maxwell's equations.

Contribution

It introduces a novel application of Fujita-Kato's method in Fourier coefficient spaces to analyze the MHD system's local solvability and smoothness loss.

Findings

Proves local-in-time existence of solutions for periodic initial data.

Shows loss of smoothness due to the damped-wave operator's properties.

Utilizes $ ext{ell}^1$ Fourier coefficient spaces for effective nonlinear estimates.

Abstract

Existence of local-in-time unique solution and loss of smoothness of full Magnet-Hydro-Dynamics system (MHD) is considered for periodic initial data. The result is proven using Fujita-Kato's method in $\ell^1$ based (for the Fourier coefficients) functional spaces enabling us to easily estimate nonlinear terms in the system as well as solutions to Maxwells's equations. A loss of smoothness result is shown for the velocity and magnetic field. It comes from the damped-wave operator which does not have any smoothing effect.