Vectorial Calculus of Variations in L-infinity, generalised solutions for fully nonlinear PDE systems and applications to Data Assimilation

寻找一些物理上有意义的、可量化的实体的极值是科学中普遍存在的一个非常重要的问题。从古代，当问题可能是找到包围最大面积土地的周长时，到今天最复杂的应用，这类问题的完整解决方案总是为应用打开了广阔的视野，对数学家来说本质上是有趣的，因为它通常转化为困难的，往往是技术性的问题。但是答案会影响应用程序和日常生活，就像上面的例子一样。特别是，在经典的变分学中，人们寻求最小化在一类映射上定义的泛函，通常这种泛函是积分，并模拟一些“能量”。这些泛函的极值满足一定的偏微分方程系统，即欧拉-拉格朗日方程。在20世纪60年代早期，G. Aronsson开始了泛函的研究，泛函被定义为最大值。除了与几何问题相关的内在数学兴趣外，最小化能量的“最大值”提供了更现实的模型，而不是“平均”能量的经典情况。“l -∞变分法”，这一领域今天被称为“l -∞变分法”，此后经历了巨大的发展。然而，直到最近，该理论仅限于标量情况和一阶变分问题（涉及映射及其一阶导数的最小化）。在2010年代早期，PI率先研究了在高维空间中有价值的映射的向量l无穷问题，并且可能涉及高阶导数。向量情况对大量实际应用很有意义。阻碍矢量情况发展的主要原因是缺乏适当的分析框架：新的复杂方程具有奇异“解”，需要一种理论来使其具有严格的意义并进行有效的研究。问题是，基于对偶性/分部积分或最大原则的标准PDE方法并不适用。特别是，产生的系统是非散度的，高度非线性的，退化的和具有不连续系数的。这种情况类似于20世纪10年代数学界在试图理解和严格理解量子理论中出现的“狄拉克三角洲”时所面临的情况。“广义函数”理论的发展使人们能够理解基本的物理现象。在新发现的方程的激励下，PI最近提出了一种新的“广义解”理论，适用于任何阶的完全非线性PDE系统，它允许不连续的解和系数。这种方法是无对偶性的，并且依赖于那些经典上不存在的导数的概率解释。我们的理论是一种与所有现有方法兼容的非线性替代分布。在这种情况下，PI最近开始成功地研究l无穷方程的某些情况。本研究将继续在适当的分析框架下研究l -无穷变分问题及其方程。我们有兴趣开发新的数学工具，以研究一阶和二阶变分问题和相关的PDE系统。进一步的特别重点将是将我们的结果应用于地球科学和天气预报中的变分数据同化模型。从数学上讲，数据同化面临的问题不是完全可以解决的，相反，人们试图最小化描述近似解与我们想要的精确解的偏差的“误差”。通过将目前使用的标准模型替换为“最大值”对应模型，我们可以获得更好的预测：在最小化最大值时，一开始就排除了大误差的峰值。

Finding the extremal values of some physically meaningful, quantifiable entity is a ubiquitous problem of great importance in science. From antiquity, when the problem might have been to find the perimeter that enclosed the largest area of land, to the most sophisticated application nowadays, a complete solution to such problems always opens large horizons for applications and is intrinsically interesting to mathematicians, as it translates usually into hard and often technical questions. But the answers impact applications and everyday life, as in the example above.In particular, in classical Calculus of Variations one seeks to minimise a functional defined on a class of maps, typically such functionals are integrals and model some "energy". The extrema of these functionals satisfy a certain system of PDE (Partial Differential Equations) known as the Euler-Lagrange equations. In the early 1960s G. Aronsson initiated the study of functionals which are instead defined as a maximum. Except for the intrinsic mathematical interest connected to geometric problems, minimising the "max" of an energy provides more realistic models as opposed to the classical case of the "average" energy. "Calculus of Variations in L-infinity", as this area is known today, has undergone huge development since. However, until recently the theory was restricted exclusively to the scalar case and to first order variational problems (involving minimisation of the map and its first derivatives). In the early 2010s the PI pioneered the study of vectorial L-infinity problems for maps valued in higher-dimensional spaces and involving perhaps higher order derivatives. The vectorial case is of interest to a large number of real-world applications. The main reason that hindered the development of the vector case was the absence of the appropriate analytic framework: the new complicated equations possess singular "solutions" and a theory is needed in order to make rigorous sense and to be studied effectively. The problem is that standard PDE approaches based on either duality/integration-by-parts or on the maximum principle do not apply. In particular, the systems arising are non-divergence, highly nonlinear, degenerate and with discontinuous coefficients. The situation is analogous to that the mathematical community faced in the 1910s when attempting to understand and make rigorous sense of the "Dirac Delta" which arose in Quantum Theory. The development of the theory of "generalised functions" allowed the understanding of fundamental physical phenomena.Motivated by the newly discovered equations, the PI very recently proposed a novel theory of "generalised solutions" for fully nonlinear PDE systems of any order which allows for discontinuous solutions and coefficients. This approach is duality-free and relies on the probabilistic interpretation of those derivatives which do not exist classically. Our theory is a nonlinear alternative to distributions compatible with all existing approaches. In this setting, the PI has recently begun studying successfully certain cases of the L-infinity equations. The proposed research will continue the study of L-infinity variational problems and of their equations in the proper analytic framework. We are interested in developing new mathematical tools in order to study 1st and 2nd order variational problems and the associated PDE systems. A further particular focus will be to apply our results to models of variational Data Assimilation in Earth sciences and in weather forecasting. Mathematically, Data Assimilation faces problems which are not exactly solvable and instead one tries to minimise an "error" which describes the deviation of approximate solutions from being the exact solution we would like to have. By replacing the standard models currently used with their "max" counterparts, we could obtain better predictions: spikes of large errors are at the outset excluded when minimising the maximum.