论文标题
良好的动态密度功能理论的适合性和平衡行为
Well-Posedness and Equilibrium Behaviour of Overdamped Dynamic Density Functional Theory
论文作者
论文摘要
我们建立了过度阻尼动态密度功能理论（DDFT）的全局良好性：一种非线性的，非局部的非局部非局部微分方程，用于胶体流体的统计机械模型，以及其他应用，包括非线性反应 - 散热系统和意见动力学。在非线性的无升边界条件下，我们确定弱密度和通量的存在和独特性，受到两体水动力相互作用（HI）。我们还表明，相对于通常的（Helmholtz）自由能函数，密度是Lyapunov稳定的。主要是，这是通过以隐式梯度流形式重写密度的动力学来完成的，类似于经典的Smoluchowski方程，但在空间上不均匀扩散和对流张量。我们还严格地表明，固定密度与HI张量无关，并证明了成倍的快速收敛到平衡。
We establish the global well-posedness of overdamped dynamic density functional theory (DDFT): a nonlinear, nonlocal integro-partial differential equation used in statistical mechanical models of colloidal fluids, and other applications including nonlinear reaction-diffusion systems and opinion dynamics. With nonlinear no-flux boundary conditions, we determine the existence and uniqueness of the weak density and flux, subject to two-body hydrodynamic interactions (HI). We also show that the density is Lyapunov stable with respect to the usual (Helmholtz) free energy functional. Principally, this is done by rewriting the dynamics for the density in an implicit gradient flow form, resembling the classical Smoluchowski equation but with spatially inhomogeneous diffusion and advection tensors. We also rigorously show that the stationary density is independent of the HI tensors, and prove exponentially fast convergence to equilibrium.
