Communications in Mathematical Sciences

Volume 17 (2019)

Number 1

The 3D nonlinear dissipative system modeling electro-diffusion with blow-up in one direction

Pages: 131 – 147



Qiao Liu (Key Laboratory of High Performance Computing and Stochastic Information Processing, Ministry of Education of China; and College of Mathematics and Statistics, Hunan Normal University, Changsha, Hunan, China)


This paper establishes a sufficient condition for the breakdown of local smooth solutions, to the Cauchy problem of the 3D Navier–Stokes/Poisson–Nernst–Planck system modeling electro-diffusion, via one directional derivative of the horizontal component of the velocity field (i.e., $(\partial_i u_1 , \partial_j u_2 , 0)$ where $i, j \in \lbrace 1,2,3 \rbrace)$ in the framework of the anisotropic Lebesgue spaces.


Navier–Stokes/Poisson–Nernst–Planck system, blow-up, anisotropic Lebesgue spaces

2010 Mathematics Subject Classification

35B44, 35K55, 35Q35, 76W05

This paper is partially supported by the National Natural Science Foundation of China (11401202), and by the China Postdoctoral Science Foundation (2015M570053, 2016T90063).

Received 7 May 2018

Accepted 19 October 2018

Published 30 May 2019