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

We will justify two-fluid incompressible Navier-Stokes-Fourier-Maxwell system limits from  the cutoff and noncutoff 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 $\gamma\geq-1$. We believe that the Hilbert expansion method can deal with the other soft potential cases, and our proof here does not rely on Hilbert expansion.