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CHARLEMAGNE Distinguished Lecture Series: Prof. Endre Süli

What Seminar
When 2011-05-02
from 15:00 to 16:00
Where SUPER C, 6th floor Generali Hall
Contact Name de Haes
Contact Email
Contact Phone 0241 80 99 132
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Analytical and Computational Challenges in Navier-Stokes-Fokker-Planck Systems

Prof. Endre Süli
Mathematical Institute
University of Oxford

Analytical and Computational Challenges in Navier-Stokes-Fokker-Planck Systems
The talk will survey recent developments concerning the existence of global-in-time weak solutions to a general class of coupled microscopic-macroscopic bead-spring chain models that arise in the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains, and their numerical approximation. The class of models involves the unsteady incompressible Navier-Stokes equations in a bounded domain for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side of the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker-Planck type parabolic equation, a crucial feature of which is the presence of a center-of-mass diffusion term. With a square-integrable and divergence-free initial velocity datum for the Navier-Stokes equation and a nonnegative initial probability density function for the Fokker-Planck equation, which has finite relative entropy with respect to the Maxwellian of the spring potential, we prove the existence of global-in-time weak solutions to the coupled Navier-Stokes-Fokker-Planck system, satisfying the initial condition, such that the velocity belongs to the classical Leray space and the probability density function has bounded relative entropy and square-integrable Fisher information over any time interval. The key analytical tool in our proof is Dubinskii's compactness theorem in seminormed sets.

We also discuss computational challenges associated with the numerical approximation of the high-dimensional Fokker-Planck equation with unbounded drift in the model, by means of operator splitting techniques and greedy approximation of the high-dimensional Ornstein-Uhlenbeck operator featuring in Fokker-Planck equation.
The lecture is based on a series of papers with Professor John W. Barrett (Imperial College London), and joint work with my former doctoral student David Knezevic (now at MIT) and my final-year doctoral student Leonardo Figueroa (University of Oxford).

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