最优输运、对流、磁弛豫和广义Boussinesq方程
Optimal Transport, Convection, Magnetic Relaxation and Generalized Boussinesq equations
摘要：
我们建立了地球物理流动的最优输运理论和经典对流理论之间的联系。我们的出发点是几年前由Angenent，Haker和Tannenbaum设计的模型，用于解决一些最优输运问题。该模型可以看作是Darcy-Boussinesq方程的推广，Darcy-Boussinesq方程是Navier-Stokes-Boussinesq（NSB）方程的退化版本。在一个统一的框架中，我们将NSB方程的不同变体（特别是我们称之为广义流体静力学-Boussinesq方程）与涉及最优输运的各种模型（以及相关的Monge-Ampere方程）联系起来。这包括二维半地转方程和一些完全非线性版本的所谓的高场极限的Vlasov-Poisson系统和Keller-Segel的趋化性。最后，我们展示了如何“弦”的AHT模型的推广可以与阿诺德和莫法特研究的磁弛豫模型，以获得固定的解决方案的欧拉方程与规定的拓扑结构。
英文摘要：
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.
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