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### Open Access Journals

DCDS

The Boltzmann equation
with a time-periodic inhomogeneous term
is solved on the existence of
a time-periodic solution that is close to an absolute Maxwellian and has
the same period as the inhomogeneous term, under some smallness assumption on
the inhomogeneous term and for the spatial dimension $n\ge 5$, and also
for the case $n=3$ and $ 4$ with an additional
assumption that the spatial integral of
the macroscopic component of the inhomogeneous term
vanishes.
This solution is a unique
time-periodic solution near the relevant Maxwellian
and asymptotically stable in time. Similar results are established also
with the space-periodic
boundary condition. As a special case,
our results cover the case
where the inhomogeneous term is time-independent,
proving the unique
existence and asymptotic stability of stationary solutions.
The proof is based on a combination of
the contraction mapping principle and time-decay estimates of
solutions to the linearized Boltzmann equation.

KRM

We consider the Euler equations governing
relativistic compressible fluids evolving in the Minkowski spacetime with several
spatial variables.
We propose a new symmetrization
which makes sense for solutions containing vacuum states
and, for instance, applies to the case of compactly supported solutions which are
important to model star dynamics.
Then, relying on these symmetrization and assuming that the velocity does not exceed some threshold
and remains bounded away from the light speed,
we deduce a local-in-time existence result for solutions containing vacuum states.
We also observe that the support of compactly supported solutions does not expand as time evolves.

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

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

Kinetic theory is probably one of the most efficient and important
theories allowing to bridge the microscopic and macroscopic
descriptions of a variety of dynamical phenomena in many fields of
science, technology, and more generally, in virtually all domains of
knowledge. Originally rooted in the theory of rarefied gases since
the seminal works of Boltzmann and Maxwell in the 19th century,
followed by landmarks established by Hilbert, Chapman and Enskog,
Carleman, Grad, and more recent mathematicians, kinetic theory has
expanded to many new areas of applications, ranging from physics to
economics and social sciences including especially modern fields
such as biology, epidemiology, and genetics.

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keywords:

DCDS

The exterior problem arising from the study of a flow past an
obstacle is one of the most classical and important subjects in gas
dynamics and fluid mechanics. The point of this problem is to assign
the bulk velocity at infinity, which is not a trivial driving force
on the flow so that some non-trivial solution profiles persist. In
this paper, we consider the exterior problem for the Boltzmann
equation when the Mach number of the far field equilibrium state is
small. The result here generalizes the previous one by Ukai-Asano on
the same problem to more general boundary conditions by crucially
using the velocity average argument.

CPAA

The existence of stationary solution to an exterior domain
of the Boltzmann equation was first studied by S. Ukai and K. Asano
in [25, 27] and was recently generalized by
S. Ukai, T. Yang, and H. J. Zhao in [29] to more
general boundary conditions. We note, however, that the results
obtained in [25, 29] require that the
temperature of the far field Maxwellian is the same as the one of
the Maxwellian preserved by the boundary conditions. The main
purpose of this paper is to discuss the case when these two
temperatures are different. The
analysis is based on some new estimates on the linearized collision
operator and the method introduced in [25, 27, 29].

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