Lpstability of solutions to Boltzmann equation with external forcing
帶外力的 Boltzmann 方程解的 Lp 穩定性
Student thesis: Master's Thesis
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Award date  2 Oct 2009 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(b0228f3ed2d94ac08d0d321de03c2c38).html 

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Abstract
The study of Cauchy problem of the Boltzmann equation is important in both theory
and applications. When there is no external force, existence of global solutions and
uniform stability of solutions in Lp space, where 1 ≤ p < ∞, were introduced in previous
works of Boltzmann equation. In this thesis, we will investigate the uniform stability of
solutions for the Cauchy problem of the Boltzmann equation when there is an external
force in the case of soft potentials for 1 ≤ p < ∞.
In the first part, we deal with the proof of uniform stability of solutions when the
external force may be large in L1 space for soft potentials. L1 stability in the case of
small external force was introduced in previous works of Boltzmann equation. Here, we
establish the uniform L1 stability by deriving a Gronwall type estimates for L1 distance
using the dispersion estimates of solutions to Boltzamnn equation with soft potentials
near vacuum.
In the second part, we discuss about the Lp stability of solutions for soft potentials
when p > 1 under the influence of two kinds of external forces whose size can be small or
large. When the external force is small, we first consider the time derivative of the pth
power of Lp norm of distance between two mild solutions f and g, then define nonlinear
functionals, and apply the Gronwall's inequality to obtain the Lp stability. For the case
of large external force, we can use the same method as in the case of L1. In other words,
we use the constructive assumptions on external force and Gronwall's inequality to derive
the Lp stability estimate. In addition, Minkowski's inequality is also used in this case.
 Cauchy problem, Mathematical models, Kinetic theory of gases, Transport theory