We consider ionic electrodiffusion in fluids, described by the Nernst–Planck–Navier–Stokes system. We prove that the system has global smooth solutions for arbitrary smooth data in bounded domains with a smooth boundary in three space dimensions, in the following situations. We consider: a arbitrary positive Dirichlet boundary conditions for the ionic concentrations, arbitrary Dirichlet boundary conditions for the potential, arbitrary positive initial concentrations, and arbitrary regular divergence-free initial velocities. Global regularity holds for any positive, possibly different diffusivities of the ions, in the case of two ionic species, coupled to Stokes equations for the fluid. The result also holds in the case of Navier–Stokes coupling, if the velocity is regular. The same global smoothness of solutions is proved to hold for arbitrarily many ionic species as well, but in that case we require all their diffusivities to be the same.

我们考虑由Nernst–Planck–Navier–Stokes系统描述的流体中的离子电扩散。我们证明，在以下情况下，该系统对有界空间中具有三个空间维度的光滑边界的任意光滑数据具有全局光滑解决方案。我们考虑：用于离子浓度的任意正Dirichlet边界条件，用于电势的任意Dirichlet边界条件，任意正初始浓度和任意无规则正向散度的初始速度。在两个离子种类的情况下，全局正则性适用于任何正的，可能不同的离子扩散率，并与流体的Stokes方程耦合。如果速度是规则的，则在Navier–Stokes耦合的情况下，结果也成立。