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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.

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

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

In this work, we consider a spatially homogeneous Kac's equation
with a non cutoff cross section. We prove that the weak solution of
the Cauchy problem is in the Gevrey class for positive time. This
is a Gevrey regularizing effect for non smooth initial datum. The
proof relies on the Fourier analysis of Kac's operators and on an
exponential type mollifier.

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.

KRM

In this work, we study the Cauchy problem for the spatially homogeneous non-cutoff Boltzamnn equation with Maxwellian molecules.
We prove that this Cauchy problem enjoys Gelfand-Shilov's regularizing effect,
meaning that the smoothing properties
are the same as the Cauchy problem defined by the evolution equation associated to a fractional harmonic oscillator.
The power of the fractional exponent
is exactly the same as the singular index of the non-cutoff collisional
kernel of the Boltzmann equation.
Therefore, we get the sharp regularity of solutions in the Gevrey class
and also the sharp decay of solutions with an exponential weight.
We also give a method to construct the solution of the Boltzmann equation
by solving an infinite system of ordinary differential equations.
The key tool is the spectral decomposition of linear and non-linear Boltzmann operators.

KRM

By using the energy-type inequality, we obtain, in this paper, the
result on propagation of Gevrey regularity for the solution of the
spatially homogeneous Landau equation in the cases of Maxwellian
molecules and hard potential.

DCDS

In this work, we study the existence of $C^{\infty}$ local solutions
to $2$-Hessian equation in $\mathbb{R}^{3}$. We consider the case
that the right hand side function $f$ possibly vanishes, changes
the sign, is positively or negatively defined. We also give the convexities
of solutions which are related with the annulation or the sign of right-hand
side function $f$. The associated linearized operator are uniformly elliptic.

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