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

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