Blow-up Criteria for the Three Dimensional Nonlinear Dissipative System Modeling Electro-hydrodynamics

TL;DR

This paper establishes criteria for the breakdown of smooth solutions in a complex electro-hydrodynamics model, highlighting the dominant role of vorticity and providing new blow-up conditions.

Contribution

It introduces new blow-up criteria for a coupled Navier-Stokes and Poisson-Nernst-Planck system, emphasizing the vorticity's influence on solution breakdown.

Findings

Maximum vorticity controls solution blow-up

Prodi-Serrin type blow-up criteria are established

Velocity field is more influential than charge density in blow-up

Abstract

In this paper, we investigate some sufficient conditions for the breakdown of local smooth solutions to the three dimensional nonlinear nonlocal dissipative system modeling electro-hydrodynamics. This model is a strongly coupled system by the well-known incompressible Navier-Stokes equations and the classical Poisson-Nernst-Planck equations. We show that the maximum of the vorticity field alone controls the breakdown of smooth solutions, which reveals that the velocity field plays a more dominant role than the density functions of charged particles in the blow-up theory of the system. Moreover, some Prodi-Serrin type blow-up criteria are also established.