Incompressible magnetohydrodynamic limit of the Vlasov-Maxwell-Boltzmann equations

The hydrodynamic limit of the Vlasov-Maxwell-Boltzmann equations is considered for weak solutions. Using relative entropy estimate about an absolute Maxwellian, an incompressible Electron-Magnetohydrodynamics-Fourier limit for solutions of the Vlasov-Maxwell-Blotzmann equations over any periodic spatial domain in $\R^3$ is studied. It is shown that any properly scaled sequence of renormalized solutions of the Vlasov-Maxwell-Boltzmann equations has fluctuations that (in the weak $L^1$ topology) converge to an infinitesimal Maxwellian with fluid variables that satisfy the incompressibility and Boussinesq relations. It is also shown that the limits of the velocity, the electric field, and the magnetic field are governed by a weak solution of an incompressible electron-magnetohydrodynamics system for all time.