Well-posedness for the Navier–Stokes–Nernst–Planck–Poisson system in Triebel–Lizorkin space and Besov space with negative indices

Abstract

This paper is concerned with the well-posedness of the Navier–Stokes–Nerst–Planck–Poisson system (NSNPP). Let sp=−2+n/p. We prove that the NSNPP has a unique local solution (u→,v,w)∈EuT⁎×EvT⁎×EvT⁎ for (u→0,v0,w0) in a subspace, i.e., Vu1×Vv1×Vv1, of F∞−1,2×Bpsp,∞×Bpsp,∞ with ∇⋅u→0=0. We also prove that there exists a unique small global solution (u→,v,w)∈Eu∞×Ev∞×Ev∞ for any small initial data (u→0,v0,w0)∈F˙∞−1,2×B˙psp,∞×B˙psp,∞ with ∇⋅u→0=0.