Research on the Measure Valued Solution to the Spatially Homogeneous Boltzmann Equation
關於空間齊次的玻爾茲曼方程的測度解的研究
Student thesis: Doctoral Thesis
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Award date  23 Jun 2016 
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Permanent Link  https://scholars.cityu.edu.hk/en/theses/theses(7635668a22c54520916e1084d5e29411).html 

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Abstract
This thesis focuses on the mathematical study of the Boltzmann equation, which models the motion of the dilute gas. The problem we considered is the Cauchy problem to the homogeneous Boltzmann equation without angular cutoff, where the initial datum is assumed to be a probability measure.
The research mainly consists of two parts. In the first part, we considered the homogeneous Boltzmann equation for Maxwellian molecules. Assume the initial datum is a probability measure with only some finite moments, (which may be of infinite entropy and infinite energy), then the existence and uniqueness of the solution are proved in the framework of M^{α}, M^{α}, M^{α}_{k}, which are the spaces of characteristic functions. Moreover, if the initial datum isn’t a single Dirac measure, in fact, the solution belongs to H^{∞} as long as the time t > 0.
In the second part, we developed a uniform approach to the homogeneous Boltzmann equation for both hard potential with finite energy, and soft potential with finite or infinite energy, by using Toscani metric. Under the angular noncutoff assumption on the crosssection, the solutions obtained are shown to be in the Schwartz space in the velocity variable as long as the initial data is not a single Dirac mass without any extra moment condition for hard potential, and with the boundedness on moments of any order for soft potential. Besides, the problem related to DebyeYukawa potential is also considered in this thesis, which behaves very like the hard potential case.
The research mainly consists of two parts. In the first part, we considered the homogeneous Boltzmann equation for Maxwellian molecules. Assume the initial datum is a probability measure with only some finite moments, (which may be of infinite entropy and infinite energy), then the existence and uniqueness of the solution are proved in the framework of M^{α}, M^{α}, M^{α}_{k}, which are the spaces of characteristic functions. Moreover, if the initial datum isn’t a single Dirac measure, in fact, the solution belongs to H^{∞} as long as the time t > 0.
In the second part, we developed a uniform approach to the homogeneous Boltzmann equation for both hard potential with finite energy, and soft potential with finite or infinite energy, by using Toscani metric. Under the angular noncutoff assumption on the crosssection, the solutions obtained are shown to be in the Schwartz space in the velocity variable as long as the initial data is not a single Dirac mass without any extra moment condition for hard potential, and with the boundedness on moments of any order for soft potential. Besides, the problem related to DebyeYukawa potential is also considered in this thesis, which behaves very like the hard potential case.