257 research outputs found
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
Regularity theory for the spatially homogeneous Boltzmann equation with cut-off
We develop the regularity theory of the spatially homogeneous Boltzmann
equation with cut-off and hard potentials (for instance, hard spheres), by (i)
revisiting the Lp-theory to obtain constructive bounds, (ii) establishing
propagation of smoothness and singularities, (iii) obtaining estimates about
the decay of the sin- gularities of the initial datum. Our proofs are based on
a detailed study of the "regularity of the gain operator". An application to
the long-time behavior is presented.Comment: 47 page
Ghost effect by curvature in planar Couette flow
We study a rarefied gas, described by the Boltzmann equation, between two
coaxial rotating cylinders in the small Knudsen number regime. When the radius
of the inner cylinder is suitably sent to infinity, the limiting evolution is
expected to converge to a modified Couette flow which keeps memory of the
vanishing curvature of the cylinders (ghost effect). In the 1-d stationary case
we prove the existence of a positive isolated L_2-solution to the Boltzmann
equation and its convergence. This is obtained by means of a truncated
bulk-boundary layer expansion which requires the study of a new Milne problem,
and an estimate of the remainder based on a generalized spectral inequality.Comment: Revised version of the paper in Kinetic and related models, vol. 4
(2011) 109-13
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
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