Description
- Abstract:
- In this thesis, we study some boundary problems of the Boltzmann equation and the Boltzmann equation with the large external potential.If the gas is contained in a bounded domain or flows past a solid bodies, the Boltzmann equation must be accompanied by boundary conditions, which describe the interaction of the gas with the solid walls. The basic boundary conditions are in-flow injection, diffuse reflection, bounce-back, and specular reflection boundary conditions.First, we demonstrate that discontinuity of the Boltzmann solution is created at the non-convex part of the grazing boundary, then propagate only along the forward characteristics inside the domain before it hits on the boundary again with the in-flow injection, diffuse reflection and bounce-back boundary conditions. The main ingredient is the new regularizing effect of the gain term.Second, we prove the global well-posedness of the Boltzmann equation with the specular reflection boundary condition when the domain is the cylindrical shape. The main ingredients are introducing collision-pair map to show the transversality and developing the graph comparison method to characterize grazing velocities for the non-convex domains.If the gas particles are charged and there is background electonic potential, the Boltzmann equation is accompanied by the field term ∇χΦ⋅∇ᵥF in the equation. The global well-posedness in the presence of the large external potential is an important open problem. We use Asano's work, showing the transversality in the presence of potentials, in L<sup>p</sup> - L<sup>∞</sup> framework to resolve this problem.
- Notes:
- Thesis (Ph.D. -- Brown University (2011)
Access Conditions
- Rights
- In Copyright
- Restrictions on Use
- Collection is open for research.
Citation
Kim, Chanwoo,
"Initial Boundary Value Problem of the Boltzmann Equation"
(2011).
Mathematics Theses and Dissertations.
Brown Digital Repository. Brown University Library.
https://doi.org/10.7301/Z0XD0ZX8
