Global existence of weak solutions to a nematic electrolyte model in 2D

In this paper, we study a system of partial differential equations modeling the dynamics of nematic electrolytes on a two-dimensional torus ${\mathbb{T}}^2$. The model couples the Poisson–Nernst–Planck equations for the evolution of ion concentrations and electrostatic potential with the Ericksen-Leslie equations for the flow and orientation of nematic liquid crystals. We prove the global existence of weak solutions to this system. The proof relies on a Ginzburg-Landau approximation scheme. Key steps include deriving uniform energy estimates and establishing the strong convergence of the director field by utilizing a Pohozaev-type identity to handle the lack of compactness in the Ericksen stress tensor.