The 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.
