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### Open Access Journals

KRM

In this work, we show that integral estimates for a linear operator linked with Boltzmann
quadratic operator considered in [1] can also be obtained for the case of higher
singularities. Some estimates proven in this earlier work are improved, as in particular, we do not need any regularity with respect to the first function.

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

We review recent results about Boltzmann equation for singular or
non cutoff cross-sections. Both spatially homogeneous and
inhomogeneous Boltzmann equations are considered, and ideas related
to Landau equation are explained. Various technical tools are
presented, together with applications to existence and
regularization issues.

keywords:
hypoellipticity
,
singular kernels.
,
existence
,
cutoff hypothesis
,
harmonic analysis
,
Boltzmann equations

DCDS

We use Littlewood-Paley theory for the analysis of regularization properties of weak solutions of the homogeneous
Boltzmann equation. For non cutoff and non Maxwellian molecules, we show that such solutions are smoother than the
initial data. In particular, our method applies to any weak solution, though we assume that it belongs to a weighted $L^2$ space.

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

A Gaussian beam method is presented for the analysis of the energy of the high frequency solution to the mixed problem of the scalar wave equation in an open and convex subset $\Omega$ of $IR^n$, with initial conditions compactly supported in $\Omega$, and Dirichlet or Neumann type boundary condition. The transport of the microlocal energy density along the broken bicharacteristic flow at the high frequency limit is proved through the use of Wigner measures. Our approach consists first in computing explicitly the Wigner measures under an additional control of the initial data allowing to approach the solution by a superposition of first order Gaussian beams. The results are then generalized to standard initial conditions.

KRM

This paper is devoted to some a priori estimates for the homogeneous Landau equation with soft potentials. Using coercivity properties of the Landau operator for soft potentials, we prove that the global in time a priori estimates of weak solutions in $L^2$ space hold true for moderately soft potential cases $ \gamma \in[-2, 0) $ without any smallness assumption on the initial data. For very soft potential cases
$ \gamma \in[-3, -2) $, which cover in particular the Coulomb case $\gamma=-3$, we get local in time estimates of weak solutions in $L^{2}$.

In the proofs of these estimates, global ones for the special case $\gamma=-2$ and local ones for very soft potential cases $ \gamma \in[-3, -2) $, the control on time integral of some weighted Fisher information is required, which is an additional a priori estimate given by the entropy dissipation inequality.

In the proofs of these estimates, global ones for the special case $\gamma=-2$ and local ones for very soft potential cases $ \gamma \in[-3, -2) $, the control on time integral of some weighted Fisher information is required, which is an additional a priori estimate given by the entropy dissipation inequality.

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