The Monge-Kantorovich in Kinetic Theory

Pi：WilFrid Gangbo，佐治亚理工学院DMS-0200267 ABSTRACT该提案使用Monge-Kantorovich理论来研究源自气体运动论和气象学的问题。质量运输问题最早是由G.Monge在1781年提出的，它包括寻找将一堆指定分布的泥土移动到指定分布的洞中的最佳方法。最优性是根据规定的成本函数来衡量的。最初的Monge问题只涉及相对于勒贝格测度绝对连续的测度，因此，讨论分布函数是有意义的。我们证明了动力学Fokker-Planck方程(KFPE)可以解释为熵关于时间流形的梯度通量。这些流形是规定了一阶矩和标准偏差的概率密度集。为了确保构造的解的总能量守恒，需要对概率密度进行限制。这一守恒定律在非齐次方程的动力学理论中是一件大事。在一种特殊情况下，我们的研究表明，Monge-Kantorovich距离是研究这些问题的合适工具。这里使用的代价函数是欧几里得距离的平方。我们打算研究我们的结果在流体动力学方程研究中的应用。为了处理像狄拉克质量组合这样的度量，1945年，康托洛维奇将蒙格问题推广到可能有奇异部分的度量。康托洛维奇的这一推广被证明在各种领域都有应用，包括形状识别，人们想要比较生活在空间中的两条曲线看起来如何相似。在这种情况下，很明显，曲线可以用一维测量来表示，所以有奇异的部分。与这一提议相关的其他应用是霍斯金斯在1975年提出的半地转系统。半地转系统是系统中绕固定轴旋转的不可压缩流体的著名欧拉方程的近似。这些系统在气象学中被引入作为发展锋面的模型。这些模型完全缺乏分析结果，因此，有必要开发一种理论来证实或否定气象学家的预测。我们证明了这些系统是关于Monge-Kantorovich距离的无限维哈密顿系统，其代价函数是欧氏距离的平方。在这项提案中，我们打算扩展在以前与合作者或研究生的工作中获得的结果。蒙格-康托洛维奇理论成为许多数学领域的核心，包括气象学、动力学理论和形状识别。在过去的几年里，人们注意到，包括气体演化研究在内的各种领域中的一类问题，可以通过最小化自由能泛函来实现，惩罚是不应该花太多钱来改变系统的状态。在初步研究的基础上，我们认为，在这个建议中，我们可以用蒙格-康托洛维奇理论来解决气体运动论中被认为是重要的问题。我们从福克-普朗克方程开始我们的研究，福克-普朗克方程是一类类似于玻尔兹曼方程的方程，它们是动力学理论的基础。

PI: Wilfrid Gangbo, Georgia Institute of TechnologyDMS-0200267ABSTRACTThe proposal uses the Monge-Kantorovich theory to study problems that originate in the kinetic theory of gases, and meteorology. The mass transportation problem was first introduced by G. Monge in 1781 and consists into finding the optimal way for moving a pile of dirt with a prescribed distribution to holes with prescribed distributions. Optimality is measured against a prescribed cost function. The original Monge problem deals only with measures that are absolutely continuous with respect to Lebesgue measures, and so, it makes sense to talk about distribution functions. We show that one can interprete the kinetic Fokker-Planck equations (KFPE) as the gradient flux of the entropy with respect to manifolds that vary in time. These manifolds are sets of probability densities for which the first moments and the standart deviation are prescribed. That restriction on the probability densities are needed to ensure conservation of total energy for the solutions constructed. This conservation law is a big deal in kinetic theory for inhomogeneous equations. Our investigations, in a special case, the so-called Maxwellian model of (KFPE), show that the Monge-Kantorovich distance is an appropriate tool for studing these problems. The cost function used here is the square of the euclidien distance. We intend to investigate the implication of our results in the study of hydrodynamic equations. To deal with measures such as combination of dirac masses, in 1945, Kantorovich generalized the Monge problem to measures that may have singular parts. This generalization by Kantorovich turned out to find applications in various fields, including shape recognition, where one wants to compare how two curves living in the space look alike. In that case, clearly, the curves can be represented by one-dimensional measures and so, have singular parts. Other applications that are relevant to this proposal are the semigeostrophic systems, introduced by Hoskins in 1975. The semigeostrophic systems are approxamations of the celebrated Euler equations of incompressible fluids in a system, rotating around a fix axis. These system where introduced in meteorology as models which develop fronts. There is a complete lack of analytical results on these models, and so, there is a need to develop a theory that would confirm or infirm previsions made by meteorologists. We show that these systems are infinite dimensional hamiltonian systems with respect to the Monge-Kantorovich distance, whose cost function is the square of the euclidien distance. In this proposal, we intend to extend results obtained in previous works with collaborators or graduate students. The Monge-Kantorovich theory became central in many fields of mathematics including meteorology, kinetic theory, and shape recognition. Over the past few years, it has been noticed that a class of problems in various fields, including the study of evolution of gases, can be realized by minimizing a free energy functional under the penalty that one should not pay to much to change the state of the system. Based on preliminary investigations, we believe that in this proposal, we can use the Monge-Kantorovich theory to solved problems that are considered important in the kinetic theory of gases. We start our study with the Fokker-Planck equations, a class of equations similar to the Boltzmann equations, that are fundamental in kinetic theory.