Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq equations

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.

References in corpus (1)

• A reconstruction of the initial conditions of the Universe by optimal mass transportation

Cited by in corpus (8)

• Global well-posedness for Euler-Boussinesq system with critical dissipation
• On the global regularity of axisymmetric Navier-Stokes-Boussinesq system
• Local persistence of geometric structures of the inviscid nonlinear Boussinesq system
• A refined long time asymptotic bound for 3D axially symmetric Boussinesq system with zero thermal diffusivity
• On global stability of optimal rearrangement maps
• Global well-posedness for the Euler-Boussinesq system with axisymmetric data
• A degenerate chemotaxis system with flux limitation: Finite-time blow-up
• Global existence and uniqueness for a non linear Boussinesq system in dimension two