A kinetic equation for spin polarized Fermi systems

This paper a kinetic Boltzmann equation having a general type of collision kernel and modelling spin-dependent Fermi gases at low temperatures modelled by a kinetic equation of Boltzmann type. The distribution functions have values in the space of positive hermitean 2x2 complex matrices. Global existence of bounded weak solutions is proved in L1 to the initial value problem in a periodic box.Comment: Replacement with extended results, to appear in Kinetic and Related Model

Well-posedness of the Cauchy problem for a space-dependent anyon Boltzmann equation

A fully non-linear kinetic Boltzmann equation for anyons and large initial data is studied in a periodic 1d setting. Strong L1 solutions are obtained for the Cauchy problem. The main results concern global existence, uniqueness, and stability.Comment: 22 pages. In this version an earlier error has been corrected, and with it a study of the time asymptotics moved to a future paper. arXiv admin note: text overlap with arXiv:1207.059

On a Boltzmann equation for Haldane statistics

The study of quantum quasi-particles at low temperatures including their statistics, is a frontier area in modern physics. In a seminal paper F.D. Haldane proposed a definition based on a generalization of the Pauli exclusion principle for fractional quantum statistics. The present paper is a study of quantum quasi-particles obeying Haldane statistics in a fully non-linear kinetic Boltzmann equation model with large initial data on a torus. Strong L1 solutions are obtained for the Cauchy problem. The main results concern existence, uniqueness and stability. Depending on the space dimension and the collision kernel, the results obtained are local or global in time.Comment: 24 pages. arXiv admin note: text overlap with arXiv:1406.0265 This is the published version of the paper. The condition (2.3) on the collision kernel is strengthened, as required for the result to hol

On the Cauchy problem with large data for the space-dependent Boltzmann Nordheim equation III

This paper studies the quantum Boltzmann Nordheim equation from a Boltzmann equation for Haldane statistics. Strong solutions are obtained for the Cauchy problem with initial data in L1 and uniformly bounded on a one (resp. two or three)-dimensional torus for three-dimensional velocities and pseudo-Maxwellian (resp. very soft) forces. The main results are existence, uniqueness and stability of solutions conserving mass, momentum, and energy, with the uniform bound exploding if the solutions are only local in time.Comment: 20 pages. arXiv admin note: text overlap with arXiv:1711.10357, arXiv:1601.06927, arXiv:1611.0747

On stationary solutions to normal, coplanar, discrete Boltzmann equation models

The paper proves existence of renormalized solutions for a class of velocity-discrete coplanar stationary Boltzmann equations with given indata. The proof is based on the construction of a sequence of approximations with L1 compactness for an integrated collision frequency and gain term. The compactness is obtained using the Kolmogorov Riesz theorem.Comment: to appear in Communications in Mathematical Science

The Enskog Process

The existence of a weak solution to a McKean-Vlasov type stochastic differential system corresponding to the Enskog equation of the kinetic theory of gases is established under natural conditions. The distribution of any solution to the system at each fixed time is shown to be unique. The existence of a probability density for the time-marginals of the velocity is verified in the case where the initial condition is Gaussian, and is shown to be the density of an invariant measure.Comment: 38 page