The Compressible Euler and Acoustic Limits from Quantum Boltzmann Equation with Fermi–Dirac Statistics

This paper justifies the compressible Euler and acoustic limits from quantum Boltzmann equation with Fermi–Dirac statistics rigorously. By employing Hilbert expansion, in particular analyzing the nonlinear implicit transformation between the classical form of compressible Euler equations and the one obtained directly from BFD, and some new type of Grad–Caflisch type decay estimate of the linearized collision operator, we establish the compressible Euler limit from scaled BFD equation, which was formally derived by Zakrevskiy in (Kinetic models in the near-equilibrium regime. Thesis at Polytechnique, 2015) by moment method. Consequently, the acoustic limit is obtained in optimal scaling with respect to Knudsen number.