The Well-posedness And Hydrodynamic Limits Of Quantum Boltzmann Equation With Fermi-Dirac Statistics

This thesis is a study on the theory of quantum Boltzmann equation with FermiDirac statistics.Two main results are established in this paper.The first result.We consider the scaled Boltzmann-Fermi-Dirac equation(briefly,BFD)in incompressible time scaling.By employing new estimates on trilinear terms in collision integral,we prove the global existence of the classical solution to BFD equation near equilibrium.Furthermore,the limit from BFD equation to incompressible NavierStokes-Fourier equations is justified rigorously,which was formally derived in the thesis of Zakrevskiy [58].The second result.Our second result is devoted to the compressible Euler limit from BFD equation.This limit was formally derived in Zakrevskiy’s thesis [58] by moment method.We employ the Hilbert approach to obtain the same fluid limit and then to justify the limiting process rigorously.The forms of the classical compressible Euler system and the one derived from BFD are different.Our proof is based on the analysis of the nonlinear implicit transformation between the solution to Euler equation from BFD equation and the parameters of local Fermi-Dirac distribution.Furthermore,we give the novel nonlinear estimates on the microscopic part of coefficients of Hilbert expansion in details,which is different from the conclusion given by Caflisch [14] in the case of classical Boltzmann equation.We also derive the acoustic limit form BFD equation since the acoustic equations are the linearization of compressible Euler equations about the constant state(f,u,T)=(1,0,1).