Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Sharp bounds for Boltzmann and Landau collision operators   The aim of the work is to provide a stable method to get sharp bounds for
Boltzmann and Landau operators in weighted Sobolev spaces and in anisotropic
spaces. All the sharp bounds are given for the original Boltzmann and Landau
operators. The sharpness means the lower and upper bounds for the operators are
consistent with the behavior of the linearized operators. Moreover, we make
clear the difference between the bounds for the original operators and those
for the linearized ones.
  According to the Bobylev's formula, we introduce two types of dyadic
decompositions performed in both phase and frequency spaces to make full use of
the interaction and the cancellation. It allows us to see clearly which part of
the operator behaves like a Laplace type operator and which part is dominated
by the anisotropic structure. It is the key point to get the sharp bounds in
weighted Sobolev spaces and in anisotropic spaces. Based on the geometric
structure of the elastic collision, we make a geometric decomposition to
capture the anisotropic structure of the collision operator. More precisely, we
make it explicit that the fractional Laplace-Beltrami operator really exists in
the structure of the collision operator. It enables us to derive the sharp
bounds in anisotropic spaces and then complete the entropy dissipation
estimates. The structures mentioned above are so stable that we can apply them
to the rescaled Boltzmann collision operator in the process of the grazing
collisions limit. Then we get the sharp bounds for the Landau collision
operator by passing to the limit. We remark that our analysis used here will
shed light on the investigation of the asymptotics from Boltzmann equation to
Landau equation.

18
The categories of this paper are: econ
0
The categories of this paper are: hep-ph
35
The categories of this paper are: mtrl-th
1
The categories of this paper are: cs
26
The categories of this paper are: bayes-an
13
The categories of this paper are: quant-ph
37
The categories of this paper are: supr-con
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
31
The categories of this paper are: ao-sci
36
The categories of this paper are: cmp-lg
16
The categories of this paper are: stat
11
The categories of this paper are: hep-lat
30
The categories of this paper are: q-alg
15
The categories of this paper are: nucl-ex
12
The categories of this paper are: nucl-th