Global large solutions for the nonlinear dissipative system modeling electro-hydrodynamics

In this paper, we are concerned with global existence of large solutions for a dissipative model arising from electro-hydrodynamics, which is the nonlinear nonlocal 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 and fully using the algebraic structure of the system, we prove that, under some conditions imposed on the indices p, p1, q, r, α, there exist two positive constants c0, C0 such that if the initial data u0=(u0h,u03) and (v0, w0) satisfy ‖u0h‖Ḃp1,∞−1+3p1+‖u0h‖Ḃp1,∞−1+3p1α‖u03‖Ḃp1,∞−1+3p11−α+K0≤c0 with K0≔‖v0‖Ḃq,1−2+3q⁡expC0‖u0‖Ḃp,1−1+3p+C0‖w0‖Ḃr,1−2+3r+1expC0‖u0‖Ḃp,1−1+3p, then the system admits a unique global solution. Moreover, the global existence of large solution was also established in two dimensional case.