The Monge-Kantorovich in Kinetic Theory

主要研究者：Wilfrid Gangbo，格鲁吉亚理工学院DMS-0200267摘要该提案使用Monge-Kantorovich理论来研究起源于气体动力学理论和气象学的问题。大众运输问题最早是由G. Monge在1781年提出了一种新的方法，即寻找将一堆具有指定分布的泥土移动到具有指定分布的孔中的最佳方法。最优性是根据规定的成本函数来衡量的。最初的蒙日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。我们表明，可以将动力学Fokker-Planck方程（KFPE）表示为熵相对于随时间变化的流形的梯度通量。这些流形是概率密度的集合，其中规定了一阶矩和标准偏差。需要对概率密度进行限制，以确保所构造的解的总能量守恒。这个守恒定律在非齐次方程的动力学理论中很重要。我们的调查，在一个特殊的情况下，所谓的麦克斯韦模型（KFPE），表明Monge-Kantorovich距离是一个适当的工具，研究这些问题。这里使用的成本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的研究中的意义。为了处理措施，如结合狄拉克群众，在1945年，康托洛维奇广义的蒙赫问题的措施，可能有奇异的部分。Kantorovich的这一概括在各个领域都有应用，包括形状识别，人们想比较空间中的两条曲线看起来是如何相似的。在这种情况下，很明显，曲线可以用一维测度表示，因此具有奇异部分。与这个建议有关的其他应用是霍斯金斯在1975年提出的半地转系统。半地转系统是不可压缩流体著名的欧拉方程在绕固定轴旋转系统中的近似。这些系统在气象学中作为发展锋面的模型引入。这些模型完全缺乏分析结果，因此，需要发展一种理论来证实或削弱气象学家的预测。我们证明了这些系统是关于Monge-Kantorovich距离的无穷维Hamilton系统，其代价函数是欧氏距离的平方。在这个建议中，我们打算扩展与合作者或研究生在以前的作品中获得的结果。 蒙格-康托洛维奇理论成为许多数学领域的中心，包括气象学、动力学理论和形状识别。在过去的几年里，人们已经注意到，在各种领域中的一类问题，包括气体演化的研究，可以通过最小化自由能泛函来实现，在惩罚下，人们不应该付出太多来改变系统的状态。根据初步的研究，我们认为，在这个建议中，我们可以使用Monge-Kantorovich理论来解决被认为是重要的气体动力学理论中的问题。我们从Fokker-Planck方程开始研究，这是一类类似于Boltzmann方程的方程，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.