| Date: | November 5, 2018 |
| Time: | 4:00pm |
| Location: | Blocker 117 |
| Speaker: | Peter Markowich, KAUST |
| Title: | On Wigner and Bohmian Measures in Semiclassical Quantum Mechanics |
| Abstract: | We present different approaches to the semiclassical limit of the Schrödinger equation,including the (position space based) WKB, the (phase space based) Wigner and the fluid-like Bohmian approach. We point out the interrelations of these approaches, which give a ’transition’ from quantum to classical mechanics, illustrate their strengths and weaknesses and connections to classical fluid/gas dynamics. |
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| Date: | November 6, 2018 |
| Time: | 4:00pm |
| Location: | Blocker 117 |
| Speaker: | Peter Markowich, KAUST |
| Title: | Continuum Modeling of Transportation Networks with Differential Equations |
| Abstract: | We present a discrete ODE and continuum PDE modeling framework for biological transportation networks, which can be used to describe leaf venation, neuronal networks, blood vessel networks etc. The differential equations, which are based on first principles, are capable of modeling network formation, evolution and adaptation. Many mathematical difficulties arise, like instabilities, formation of (thin patterns, multiple steady states, singular scaling limits etc. |
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| Date: | November 7, 2018 |
| Time: | 4:00pm |
| Location: | Blocker 220 |
| Speaker: | Peter Markowich, KAUST |
| Title: | Propagation of Monokinetic Phase Space Measures with Rough Momentum Profiles |
| Abstract: | Consider a Radon measure which is a phase space probability density carried by the graph of a Lipschitz continuous (or even less regular) vector field. We study the structure of the push-forward of such a measure by a Hamiltonian flow. In particular, we provide an estimate on the number of folds in the support of the transported measure. We also study in detail the type of singularities in the projection of the transported measure in configuration space (averaging out the momentum variable). We analyze the conditions under which this projected measure can have atoms, and give an example in which the projected measure is singular with respect to the Lebesgue measure and diffuse. We discuss applications of our results to the classical limit of the Schrödinger equation. Finally we present various examples and counterexamples showing that our results are sharp. |
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