Optimal Transport, Convection and Magnetic Relaxation

Yann Brenier
Université de Nice Sophia Antipolis

We establish a connection between Optimal Transport Theory and classical Convection Theory for geophysical flows. Our starting point is the model designed few years ago by Angenent, Haker and Tannenbaum for the numerical approximation of some Optimal Transport problems by a gradient flow approach. We interpret this geometric equation as a generalization of the Darcy-Boussinesq equations for use in Convection Theory. This suggests that the Navier-Stokes-Boussinesq equations, the basic model in Convection Theory, provide a good framework for Optimal Transport related problems, such as Hoskins' Semigeostrophic equations and some fully nonlinear version of some Chemotaxis equations. In a different direction, we introduce a "stringy" generalization of the Angenent Haker Tannenbaum model. This model is closely related to the Magnetic Relaxation model investigated by Arnold and Moffatt. It is a gradient flow for which the equilibrium states are just the
(variational) solutions of the Euler equations for incompressible flows. We also discuss a vectorial version of the Burgers equation, the 'cross-Burgers' equation. Numerical algorithms will be presented.

Presentation (PDF File)

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