动力学Bohmian方程的最优输运方法
An Optimal Transport Approach for the Kinetic Bohmian Equation
摘要：
研究了动力学Bohmian方程解的存在性理论，该方程是为Bohmian力学的相空间形式而提出的一个非线性Vlasov型方程。我们的主要思想是解释动力学Bohmian方程作为一个哈密顿系统定义在一个适当的泊松流形上建立在Wasserstein空间。首先，我们提出了一个存在性理论的动力学Bohmian方程的定态解。之后，我们发展了一个近似的版本，我们的哈密顿系统，以研究其相关的流动。然后，我们证明我们的近似版本的解决方案的存在性。最后，我们提出了一些收敛结果的近似系统，目的是建立，在极限，近似解满足动力学Bohmian方程在弱意义下。
英文摘要：
We study the existence theory of solutions of the kinetic Bohmian equation, a nonlinear Vlasov-type equation proposed for the phase-space formulation of Bohmian mechanics. Our main idea is to interpret the kinetic Bohmian equation as a Hamiltonian system defined on an appropriate Poisson manifold built on a Wasserstein space. We start by presenting an existence theory for stationary solutions of the kinetic Bohmian equation. Afterwards, we develop an approximative version of our Hamiltonian system in order to study its associated flow. We then prove existence of solutions of our approximative version. Finally, we present some convergence results for the approximative system, the aim being to establish that, in the limit, the approximative solution satisfies the kinetic Bohmian equation in a weak sense.
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