Math 285J Spring 2009

Topics in Nonlinear PDE

Instructor: Andrea Bertozzi

This course will cover some advanced topics in nonlinear partial differential equations, with applications to image processing and aggregation phenomena. The syllabus will be broken up by the type of phenomena of interest. Students will attend lectures, read current and recent literature, and do projects on current research topics. This course is specially designed for students to work on a miniproject connected with the course that could result in a refereed journal publication. Students getting credit for the course will be assigned a research problem near the beginning of the course and will do additional reading related to that problem. They will be expected to do an in class presentation on that problem at some point during the course, as well as to write a paper about the problem. Students who take this course should have a minimum of the 266 or 269 sequence, preferably both. The course is designed for those who have passed the ADE qual and the Numerical qual, although it is possible to take the course with only one qual.


Part I: Global Existence and Finite time singularities in nonlinear PDE.

This part of the course will cover finite time singularities in nonlinear PDE. Suggested reading:

Michael Taylor Partial Differential Equations, Volume III, Springer-Verlag, Applied Mathematical Sciences Series, Volume 117, 1991.
John Ball, 1977, Remarks on the Blow-Up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math. Oxford (2) (1977), 473-486.
A. J. Bernoff, Andrea L Bertozzi, Singularities in a Modified Kuramoto-Sivashinsky Equation Describing Interface Motion for Phase Transition, Physica D 85 (1995) 375-404.
Andrew J. Majda and Andrea L. Bertozzi, Vorticity and Incompressible Flow, Cambridge, Univ. Press, 2002.
Andrea L. Bertozzi and Thomas Laurent, Finite time blowup in an aggregation equation on R^n preprint, 2006. Andrea L. Bertozzi and Jeremy Brandman, Finite-time blow-up of L^\infty-weak solutions of an aggregation equation accepted in Communications in the Mathematical Sciences, 2008, special issue in honor of Andrew Majda's 60th birthday.
Andrea L. Bertozzi Jose Carrillo, and Thomas Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels Nonlinearity, 22, 683-710, 2009.
Self-similar blowup solutions to an aggregation equation, Y. Huang and A. L. Bertozzi, preprint.

Part II: Pattern formation - aggregation phenomena, diffuse interfaces, and coarsening.

Front migration in the nonlinear Cahn-Hilliard equation, R.L. Pego, Proc. Roy. Soc. London, Ser. A, Vol. 422, No. 1863, pp. 261-278.
R.V. Kohn and F. Otto, "Upper bounds on coarsening rates," Comm. Math. Phys. 229 (2002) 375-395.
J. Dobrosotskaya and A. L. Bertozzi A wavelet-Laplace variational technique for image deconvolution and inpainting, IEEE Trans. Im. Proc. vol 17, no. 5., May 2008.
M. B. Short, M. R. D'Orsogna, V. B. Pasour, G. E. Tita, P. J. Brantingham, A. L. Bertozzi, and L. Chayes, A statistical model of criminal behavior, M3AS: Mathematical Models and Methods in Applied Sciences, special issue on Traffic, Crowds, and Swarms, volume 18, Supp., pages 1249-1267, 2008.
G. O. Mohler, M. B. Short, P. J. Brantingham, F. P. Schoenberg, and G. E. Tita, Self-exciting point process modeling of crime preprint 2008.

Course schedule: The course will meet MWF from 1-2pm.

Final Presentations