Jun

22

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 KamalinejadPh.D. Advisor:Almut BurchardThesis 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.

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