In this paper, we are concerned with global existence and temporal decay rates of large solutions for a mathematical model arising from electro-hydrodynamics, which is the nonlinear dissipative system coupled by the Poisson-Nernst-Planck equations and the incompressible Navier-Stokes equations through charge transport and external forcing terms. By introducing some proper weighted functions based on carefully examining the algebraic structure of the system, we prove that there exist two positive constants c 0 , C 0 such that if the initial data ( u 0 , N 0 , P 0 ) satisfies ( ∥ u L · ∇ u L ∥ L 1 ( B _ p , 1 − 1 + 3 p ) + ∥ N 0 − P 0 ∥ B _ q , 1 − 2 + 3 q ) exp { C 0 ( ∥ N 0 + P 0 ∥ B _ r , 1 − 2 + 3 r + 1 ) exp { C 0 ∥ u 0 ∥ B _ p , 1 − 1 + 3 p } } ≤ c 0 , then the system admits a unique global solution, where u L = e t ∆ u 0 , and e t ∆ denotes the heat flow. In addition, by using the weighted Chemin-Lerner type norm and interpolation techniques, we establish the optimal temporal decay rates of these global large solutions.

This work is concerned with the global existence of large solutions to the three-dimensional dissipative fluid-dynamical model, which is a strongly coupled nonlinear nonlocal system characterized by the incompressible Navier–Stokes–Poisson–Nernst–Planck equations. Making full use of the algebraic structure of the system, we obtain the global existence of solutions without smallness assumptions imposed on the third component of the initial velocity field and the summation of initial densities of charged species. More precisely, we prove that there exist two positive constants c 0 , C 0 such that if the initial data satisfies ( ∥ u 0 h ∥ B _ p , 1 − 1 + 3 p + ∥ N 0 − P 0 ∥ B _ q , 1 − 2 + 3 q ) exp { C 0 ( ∥ u 0 3 ∥ B _ p , 1 − 1 + 3 p 2 + ( ∥ N 0 + P 0 ∥ B _ r , 1 − 2 + 3 r + 1 ) exp { C 0 ∥ u 0 3 ∥ B _ p , 1 − 1 + 3 p } + 1 ) } ≤ c 0 , then the incompressible Navier–Stokes–Poisson–Nernst–Planck equations admits a unique global solution.

We are concerned with the global existence and decay rates of large solutions for the Poisson--Nernst--Planck equations. Based on careful observation of algebraic structure of the equations and using the weighted Chemin--Lerner type norm, we obtain the global existence and optimal decay rates of large solutions without requiring the summation of initial densities of a negatively and positively charged species is small enough. Moreover, the large solution is obtained for initial data belonging to the low regularity Besov spaces with different regularity and integral indices for the different charged species, which indicates more specific coupling relations between the negatively and positively charged species.