MATH6042 - Topics in Differential Equations II - 2021/22
General Information
Lecturer
-
Dr. Kunlun QI
- Office: LSB 232A
- Tel: 3943 7978
- Email:
Time and Venue
- Lecture: Th 2:30PM - 5:15PM, LSB 222
Course Description
In this course, the development of the spatially homogeneous theory to the Boltzmann equation will be introduced, especially for the well-posedness result of the Cauchy problem in the space of probability measure space. On the other side, the numerical simulation about the homogeneous Boltzmann equation, mainly the deterministic Spectral Method will also be presented; furthermore, some corresponding stability/error analysis frameworks will be discussed in a suitable manner.
Textbooks
- Cedric, Villani. A review of mathematical topics in collisional kinetic theory. in Handbook of Mathematical Fluid Mechanics, volume~I. North-Holland, 2002.
- Carlo Cercignani, Reinhard Illner, and Mario Pulvirenti. The mathematical theory of dilute gases. Vol. 106. Springer Science & Business Media, 2013.
- Robert Glassey. The Cauchy problem in kinetic theory. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996.
- Pierre Degond, Lorenzo Pareschi and Giovanni Russo. Modeling and computational methods for kinetic equations. Birkhäuser Boston, Inc., Boston, MA, 2004
References
- A. V. Bobylev. A class of invariant solutions of the Boltzmann equation. Dokl. Akad. Nauk SSSR, 231(3):571--574, 1976.
- A. Pulvirenti and G. Toscani. The theory of the nonlinear Boltzmann equation for Maxwell molecules in Fourier representation. Ann. Mat. Pura Appl. (4), 171:181-204, 1996.
- J. A. Carrillo and G. Toscani. Contractive probability metrics and asymptotic behavior of dissipative kinetic equations. Riv. Mat. Univ. Parma, 7(6):75–198, 2007.
- M. Cannone and G. Karch. Infinite energy solutions to the homogeneous Boltzmann equation. Comm. Pure Appl. Math., 63(6):747--778, 2010.
- Y. Morimoto, S. Wang, and T. Yang. Measure valued solutions to the spatially homogeneous Boltzmann equation without angular cutoff. J. Stat. Phys., 165(5):866--906, 2016.
- K. Qi. On the Measure Valued Solution to the Inelastic Boltzmann Equation with Soft Potentials, J. Stat. Phys., 183(27), 2021.
- L. Pareschi and G. Russo. Numerical solution of the Boltzmann equation I: spectrally accurate approximation of the collision operator. SIAM J. Numer. Anal.,37,1217-1245, 2000.
- J. Hu and K. Qi A fast Fourier spectral method for the homogeneous Boltzmann equation with non-cutoff collision kernels. J. Comput. Phys., 423, p.109806, 2020.
Pre-class Notes
Lecture Notes
- LectureNote_Part_I_(Typed_Version)
- LectureNote_Part_II_(Typed_Version)
- LectureNote_Part_III_(Handwritten)
- LectureNote_Part_IV_(Handwritten)
- LectureNote_Part_V_(Handwritten)
- LectureNote_Part_VI_(Handwritten)
- LectureNote_Part_VII_(Handwritten)
- LectureNote_Part_VIII_(Handwritten)
- LectureNote_Part_IX_(Handwritten)
Assessment Scheme
| Each student will be asked to report on a topic assigned by the teacher (the topic can be a paper related to the course or report on the content learned in the class). | 100% |
Honesty in Academic Work
The Chinese University of Hong Kong places very high importance on honesty in academic work submitted by students, and adopts a policy of zero tolerance on cheating and plagiarism. Any related offence will lead to disciplinary action including termination of studies at the University. Although cases of cheating or plagiarism are rare at the University, everyone should make himself / herself familiar with the content of the following website:
http://www.cuhk.edu.hk/policy/academichonesty/and thereby help avoid any practice that would not be acceptable.
Assessment Policy Last updated: April 13, 2022 22:42:27