An Improved Regularity Criterion for the 3D Nonlinear Dissipative System Modeling Electro--Hydrodynamics

In this paper, we consider the regularity criterion of local smooth solution to the Cauchy problem of 3D Navier–Stokes/Poisson–Nernst–Planck system modeling electro–hydrodynamics. Precisely, we prove that if the partial horizontal derivatives of two velocity components satisfy ∫ 0 T ∥ ( ∂ 1 u 1 , ∂ 2 u 2 ) ∥ B _ p , ∞ 0 q d t < ∞ with 2 q + 3 p = 2 , 3 2 < p ≤ ∞ , then the solution ( u,v,w ) is smooth on (0 ,T ].