Well-posedness Theory of Kinetic Equations in a General Setting
DescriptionThere have been extensive studies on the mathematical theories of kinetic equations that include the classical equations and systems, like Boltzmann equation, Landau equation, Vlasov-Poisson(Maxwell)-Boltzmann(Landau) systems, etc. On the other hand, most if not all of the existence results on these equations and systems are obtained in the frameworks of perturbation of vacuum, or perturbation of a global equilibrium, or in a bounded domain, or space periodic solutions (in a torus), or the Cauchy problem with equilibrium states specified at infinity in space.One natural and interesting question that can be raised is that how about none of the above assumptions are satisfied. In this project, we will study the well-posedness theory, that is, existence, stability and uniqueness of solutions to some classical kinetic equations and systems in the whole space without specifying the behavior of the solutions at infinity in space.Take Boltzmann equation which is a fundamental equation in statistical physics and a key stone in kinetic theory as an example. The global existence of solutions has been established basically in the following settings: 1. in bounded domain or in a torus; 2. in the whole space or outside an obstacle as a perturbation of either vacuum or a global Maxwellian; 3. around some non-trivial profile governed by for example external force or wave patterns with global Maxwellian states given at infinity in space. One of the problems to be considered in this project is to study the Boltzmann equation in the whole space with initial data just as a bounded function in some function space. To study this problem, we need to apply the existing analytic techniques and to dicover some new ideas.To establish the well-posedness theory, we will start with the existence theory both local and global first. Some thoughts on the methods will be given later in the project. After obtaining the existence theory, we will then study the qualitative properties of the solutions, such as the stability, uniqueness, regularity and large time behavior.
|Effective start/end date
|1/12/11 → 16/03/15