Analysis of models for large-scale geophysical flows

严格分析模拟大气和海洋流动的高度非线性方程是一项非常困难的任务，通常超出了当前数学工具的能力。对于完整的控制方程，即著名的可压缩Navier-Stokes方程（当忽略粘性时称为欧拉方程），这当然是正确的，其解太复杂而无法计算，即使是数值计算。事实上，在实践中，为天气预报等应用提供信息的建模是基于控制方程的平均版本或简化约简。虽然这些方程被广泛用于模拟复杂的物理现象和进行数值近似，但模型的可解性和计算近似的有效性主要取决于数学方法，而不是严格的数学基础。这一建议涉及一个特殊的方程系统，半地转系统，模拟大尺度的无粘地球物理流的动力学，这个特殊的模型的重要性在于，作为一个渐近约化，该系统预计是一个更准确的近似的完整的模型比在实践中使用的其他约化。此外，当某些参数，例如地球自转系数，被认为是可变的，这种减少的有效性也仍然存在。由于这个原因，该模型可以近似的大规模的流动动力学更准确地比模型的解决方案被假定为接近一个统一的参考状态。在数学上，半地转系统支持奇异解，因此它可以精确地捕捉到锋面形成等现象。这一点很重要，因为这个系统的物理推导是由模拟大气锋形成的需要精确地指导的。半地转模式的数学兴趣已经恢复，因为发现了一个特定的变量变化，众所周知的从业者，将其转换为一个系统，可以通过使用变分分析和最优传输理论的现代技术进行严格分析。在过去的二十年里，这些领域的活动取得了非常重要的成果和进展，这取决于精密和复杂的数学工具。这个项目的首要目标是适应和翻译这些技术，并使用最近的新见解，以获得越来越现实的情况下的半地转系统的解决方案的存在性和唯一性的结果。本文的研究还旨在证明该系统作为Euler方程简化的有效性和渐近阶，从而为基于这些方程的数值和物理建模奠定严格的基础。

The rigorous analysis of the highly nonlinear equations that model atmospheric and oceanic flows is a very difficult task, generally beyond the power of current mathematical tools. This is certainly true of the full governing equations, the famous compressible Navier-Stokes equations (called Euler equations when viscosity is neglected), whose solution is too complicated to compute, even numerically. Indeed, in practice the modelling that informs applications, such as forecasting the weather, is based on averaged versions or simplified reductions of the governing equations. While such equations are used ubiquitously to model complex physical phenomena and to perform numerical approximations, both the solvability of the models and the validity of the approximations computed rests largely on heuristics rather than on rigorous mathematical ground. This proposal concerns a particular system of equations, the semi-geostrophic system, that models the large-scale dynamics of inviscid geophysical flows.The importance of this particular model rests on the fact that, as an asymptotic reduction, the system is expected to be a more accurate approximation to the full model than other reductions used in practice. In addition the validity of this reduction persists also when certain parameters, for example the earth rotation coefficient, are taken to be variable. For this reason, the model can approximate the large-scale dynamics of the flow more accurately than models whose solutions are assumed close to a uniform reference state. Mathematically, the semi-geostrophic system supports singular solutions, thus it can capture rigorously phenomena such as front formation. This is important in view of the fact that the physical derivation of this system was guided precisely by the need to model the formation of atmospheric fronts.The mathematical interest in the semi-geostrophic model has been revived by the discovery that a specific change of variables, well known to practitioners, transforms it into a system that can be analysed rigorously by using modern techniques of variational analysis and optimal transport theory. Activity in these areas in the past twenty years has seen very important results and advances, depending on delicate and sophisticated mathematical tools. The overarching aim of this project is to adapt and translate these techniques and, using new recent insights, to obtain results on the existence and uniqueness of solutions of the semigeostrophic system in increasingly realistic cases. The research proposed also aims at proving the validity and asymptotic order of the system as a reduction of the Euler equations, thus putting on rigorous foundations the numerical and physical modelling based on these equations.