On the quantum Boltzmann equation near Maxwellian and vacuum

Global well-posedness of the inhomogeneous kinetic wave equation near vacuum

“…Now, we present the two estimates which will be useful us in the proof of Proposition 3.4 and have been used in the context of Boltzmann-type equations, see e.g. [19,2,4,27]. For convenience of the reader, we provide the proofs below.…”

“…The main idea of the paper is that, for such dimensions, it is possible to connect the inhomogeneous kinetic wave equation to the cubic part of a quantum Boltzmann equation with moderately hard potential and no collisional averaging. Since the well-posedness of Boltzmann-type equations near vacuum has been widely studied [20,19,5,32,33,25,2,1,31,4,27] in the past, we employ techniques of the classical kinetic theory for the Boltzmann equation to address existence, uniqueness and stability of global in time mild solutions (see Section 2 for the precise definition of a mild solution) to the spatially inhomogeneous (KWE). Up to the author's knowledge, this is the first paper which addresses the global in time well-posedness of the inhomogeneous kinetic wave equation.…”

“…The Kaniel-Shinbrot iteration, introduced for the first time by Kaniel and Shinbrot in [20] for local solutions to the Boltzmann equation and used by Illner and Shinbrot [19] to provide global in time solutions to the Boltzmann equation, has been widely used for Boltzmann-type equations [5,32,33,25,2,1,31,4,27], but not in the context of wave turbulence so far.…”

Global existence and large time behavior of the quantum Boltzmann equation with small relative entropy

“…We intend to study the global well-posedness and decay of the quantum Boltzmann solutions F when it is close to the global quantum Maxwellian. Unlike the classical Boltzmann equation, the global Maxwellian for quantum Boltzmann equation (1.1) takes a different form (see [25]),…”

“…To our knowledge, there are very limited study on the well-posedness theory of the non-homogeneous quantum Boltzmann equation. Recently, using the nonlinear energy method and mild formulation, Ouyang-Wu [25] established the global well-posedness for the non-homogeneous quantum Boltzmann equation (1.1). We refer the interested readers to [1] for the global existence and large time behavior of the relativistic quantum Boltzmann equation.…”