The incompressible Navier-Stokes-Fourier-Maxwell system limits of the Vlasov-Maxwell-Boltzmann system for soft potentials: the noncutoff cases and cutoff cases

We obtain the global-in-time and uniform in Knudsen number $\epsilon$ energy estimate for the cutoff and non-cutoff scaled Vlasov-Maxwell-Boltzmann system for the soft potential. For the non-cutoff soft potential cases, our analysis relies heavily on additional dissipative mechanisms with respect to velocity, which are brought about by the strong angular singularity hypothesis, i.e. $\frac12\leq s<1$. In the case of cutoff cases, our proof relies on two new kinds of weight functions and complex construction of energy functions, and here we ask $\gamma\geq-1$. As a consequence, we justify the incompressible Navier-Stokes-Fourier-Maxwell equations with Ohm's law limit.