Graduate Blog

Everyone welcome. Refreshments will be served in the Math Lounge
before the exam.

Thursday, June 28, 2012, 2:10 p.m., in BA 6183, 40 St. George Street

Ph.D. Candidate:  Ehsan Kamalinejad

Ph.D. Advisor:  Almut Burchard

Thesis Title:  Optimal Transport Approach to Non-linear Evolution
               Equations
	       (http://www.math.toronto.edu/~ehsan/ut-thesis.pdf)

Thesis Abstract:

Gradient flows of energy functionals on the space of probability
measures with Wasserstein metric has proved to be a strong tool
in studying mass conserved evolution equations. The solutions of
such gradient flows provide an alternate formulation for the solutions
of the corresponding evolution equations. The main condition which
guarantees existence, uniqueness, and continuous dependence on
initial data is known to be displacement convexity of the
corresponding energy on the Wasserstein space. We provide a relaxed
notion of displacement convexity and we use it to establish short
time existence and uniqueness of Wasserstein gradient flows for
higher order energy functionals which are not displacement convex
in the standard sense. This extends the applicability of the gradient
flow approach to larger family of energies. As an application,
local and global well-posedness of different higher order non-linear
evolution equations are derived. Examples include the thin-film
equation and the quantum drift diffusion equation in one
spatial variable.