The Monge-Kantorovich in Kinetic Theory

PI: Wilfrid Gangbo, Georgia Institute of technology . dss - 0200267abstract该提案使用Monge-Kantorovich理论来研究源自气体动力学理论和气象学的问题。大规模运输问题最早是由G. Monge在1781年提出的，它包括寻找将一堆具有规定分布的土移动到具有规定分布的孔的最佳方式。最优性是根据规定的成本函数来衡量的。最初的蒙日问题只处理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。我们证明可以将动力学Fokker-Planck方程（KFPE）解释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，它们的第一阶矩和标准偏差是规定的。这种对概率密度的限制是保证所构造的解的总能量守恒所必需的。这个守恒定律在非齐次方程的运动理论中很重要。我们的研究，在一个特殊的情况下，即所谓的麦克斯韦模型（KFPE），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氏距离的平方。我们打算探讨我们的结果在水动力方程研究中的意义。为了处理狄拉克质量组合等测度，1945年，Kantorovich将Monge问题推广到可能有奇异部分的测度。Kantorovich的这一概括在许多领域都得到了应用，包括形状识别，人们想要比较空间中两条曲线的相似程度。在这种情况下，很明显，曲线可以用一维度量来表示，因此，有奇异部分。与这一建议相关的其他应用是霍斯金斯在1975年引入的半营养系统。半转矩系统是关于不可压缩流体在一个系统中绕固定轴旋转的著名欧拉方程的近似。这些系统作为发展锋面的模式被引入气象学。这些模型完全缺乏分析结果，因此，有必要发展一种理论来证实或推翻气象学家的预测。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。在这个提案中，我们打算扩展合作者或研究生在以前的工作中获得的结果。Monge-Kantorovich理论成为许多数学领域的中心，包括气象学、运动理论和形状识别。在过去的几年里，人们注意到，在各个领域的一类问题，包括气体演化的研究，可以通过最小化自由能泛函来实现，而代价是人们不应该付出太多的代价来改变系统的状态。根据初步的研究，我们相信在这个建议中，我们可以使用Monge-Kantorovich理论来解决气体动力学理论中被认为重要的问题。我们从福克-普朗克方程开始我们的研究，这是一类类似于玻尔兹曼方程的方程，它们是运动理论的基础。

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.