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### Open Access Journals

KRM

In this paper, we consider the Cauchy problem for the
non-cutoff Boltzmann equation in the soft potential case.
By using a singular change of velocity variables before and after collision, we prove the uniqueness of weak solutions to the Cauchy problem in the space of functions
with polynomial decay in the velocity variable.

KRM

We construct bounded classical solutions of the Boltzmann equation
in the whole space without specifying
any limit behaviors at the spatial infinity and without assuming the smallness condition on initial data.
More precisely, we show that if the initial data
is non-negative and belongs to a uniformly local Sobolev space in the space variable
and a standard Sobolev space with Maxwellian type decay property in the velocity variable,
then the Cauchy problem of the Boltzmann
equation possesses
a unique non-negative local solution in the same function
space,
both for the cutoff and non-cutoff collision cross section with
mild singularity.
The known solutions such as solutions on the torus (space periodic solutions) and in the vacuum (solutions
vanishing at the spatial infinity), and solutions in the whole space having a limit
equilibrium state at the spatial infinity
are included in our category.

KRM

The spatially homogeneous Boltzmann equation without angular cutoff
is discussed on the regularity of solutions for the modified hard
potential and Debye-Yukawa potential. When the angular singularity
of the cross section is moderate, any weak solution having the
finite mass, energy and entropy lies in the Sobolev space of
infinite order for any positive time, while for the general
potentials, it lies in the Schwartz space if it has moments of
arbitrary order.
The main ingredients of the proof are
the suitable choice of the mollifiers composed of
pseudo-differential operators and the sharp estimates of the
commutators of the Boltzmann collision operator and
pseudo-differential operators. The method developed here also
provides some new estimates on the collision operator.

keywords:
angular non-cutoff
,
commutator.
,
regularity
,
Boltzmann equation
,
pseudo-differential operators

DCDS

Most of the work on the Boltzmann equation is based on the Grad's
angular cutoff assumption. Even though the smoothing effect from the
singular cross-section without the angular cutoff corresponding to
the grazing collision is expected, there is no general mathematical
theory especially for the spatially inhomogeneous case. As a further
study on the problem in the spatially homogeneous situation, in this
paper, we will prove the Gevrey smoothing property of the solutions
to the Cauchy problem for Maxwellian molecules without angular
cutoff by using pseudo-differential calculus.
Furthermore, we apply
similar analytic techniques for the Sobolev space regularity to the
nonlinear equation, and prove the smoothing property of solutions
for the spatially homogeneous nonlinear Boltzmann equation with the
Debye-Yukawa potential.

KRM

The purpose of this paper is to extend the result concerning the existence and the uniqueness of
infinite energy solutions, given by Cannone-Karch, of the Cauchy problem for the spatially homogeneous Boltzmann equation of Maxwellian molecules
without Grad's angular cutoff assumption in the mild
singularity case, to the strong singularity case. This extension follows from a simple observation
of the symmetry on the unit sphere for the Bobylev formula which is the Fourier transform
of the Boltzmann collision term.

KRM

We prove that the linearized non-cutoff Boltzmann operator with Maxwellian molecules is exactly equal to a fractional power of the linearized Landau operator which is the sum of the harmonic oscillator and the spherical Laplacian. This result allows to display explicit sharp coercive estimates satisfied by the linearized non-cutoff Boltzmann operator for both Maxwellian and non-Maxwellian molecules.

keywords:
spectral analysis
,
anisotropy
,
microlocal analysis.
,
Boltzmann operator
,
Landau operator

KRM

Without Grad's angular cutoff assumption, the local existence of classical solutions to the Boltzmann equation is studied. There are two new improvements: the index of Sobolev spaces for the solution is related to the parameter of the
angular singularity; moreover, we do not assume that the initial data is close to a global equilibrium. Using the energy method, one important step in the analysis is the study of fractional derivatives of the collision operator and related commutators.

KRM

We consider the non-linear spatially homogeneous Landau equation with Maxwellian molecules in a close-to-equilibrium framework and show that the Cauchy problem for the fluctuation around the Maxwellian equilibrium distribution enjoys a Gelfand-Shilov regularizing effect in the class $S_{1/2}^{1/2}(\mathbb{R}^d)$, implying the ultra-analyticity and
the production of exponential moments of the fluctuation, for any positive time.

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