Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq equations

TL;DR

This paper links Optimal Transport Theory with classical Convection Theory in geophysical flows, generalizing models like the Navier-Stokes-Boussinesq equations and relating them to various nonlinear PDEs and magnetic relaxation models.

Contribution

It establishes a unified framework connecting Optimal Transport, generalized Boussinesq equations, and magnetic relaxation models, extending previous models to new nonlinear and topological contexts.

Findings

Relates Optimal Transport models to generalized Boussinesq equations.

Connects various nonlinear PDEs like semi-geostrophic and Keller-Segel models.

Links magnetic relaxation models to stationary Euler solutions with prescribed topology.

Abstract

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 to solve some Optimal Transport problems. This model can be seen as a generalization of the Darcy-Boussinesq equations, which is a degenerate version of the Navier-Stokes-Boussinesq (NSB) equations. In a unified framework, we relate different variants of the NSB equations (in particular what we call the generalized Hydrostatic-Boussinesq equations) to various models involving Optimal Transport (and the related Monge-Ampere equation. This includes the 2D semi-geostrophic equations and some fully non-linear versions of the so-called high-field limit of the Vlasov-Poisson system and of the Keller-Segel for Chemotaxis. Finally, we show how a ``stringy'' generalization of the AHT model can be related to the magnetic relaxation model studied by Arnold and Moffatt to obtain stationary solutions of the Euler equations with prescribed topology.