Stability of electrokinetic Couette states in the Poisson--Nernst--Planck--Navier--Stokes system
Abstract
In this paper, we study a two-dimensional Poisson--Nernst--Planck--Navier--Stokes system in a periodic channel driven by wall motion and an imposed tangential electric field, with two ionic species that may have unequal diffusivities. At the electroneutral Couette state, a diffusivity-weighted ionic energy combines with the triangular structure of the linearized operator to yield exponential linear stability for fixed (A,E0) and all positive diffusivities. Small-data nonlinear exponential stability is then obtained on a complex interpolation space adapted to the graph domain of the generator by combining analytic-semigroup smoothing with quadratic estimates. For unequal diffusivities, the nonzero streamwise modes of the linearized ionic subsystem satisfy a bounded-channel enhanced-dissipation estimate at rate Dmin1/3|A|2/3 under an explicit strong-shear condition. When D+=D-, a Fourier-mode factorization separates the common Couette advection--diffusion operator from a contractive drift--reaction semigroup, so the same scalar mixing rate is retained without any smallness condition on the electrostatic coupling. We also construct an exact Poisson--Boltzmann/electroosmotic Couette family for prescribed wall potentials. For sufficiently small wall-potential amplitude, its linearized generator is a small graph-domain perturbation of the electroneutral generator, which yields linear and nonlinear exponential stability on the same interpolation scale.
Related
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- Vanishing transport noise enhances stability of stratified Couette flow in the 3D Boussinesq system
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- The stability threshold for Boussinesq equations around stratified Couette flow with unequal viscosity and thermal diffusivity on 𝕋×ℝ2
BibTeX
@misc{pan2026stability,
title = {{Stability of electrokinetic Couette states in the Poisson--Nernst--Planck--Navier--Stokes system}},
author = {J. Pan and L. Wang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29339},
eprint = {2609.29339},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29339},
}
RIS
TY - UNPB
TI - Stability of electrokinetic Couette states in the Poisson--Nernst--Planck--Navier--Stokes system
AU - J. Pan
AU - L. Wang
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29339
UR - https://arxiv.org/abs/2609.29339
AB - In this paper, we study a two-dimensional Poisson--Nernst--Planck--Navier--Stokes system in a periodic channel driven by wall motion and an imposed tangential electric field, with two ionic species that may have unequal diffusivities. At the electroneutral Couette state, a diffusivity-weighted ionic energy combines with the triangular structure of the linearized operator to yield exponential linear stability for fixed (A,E_0) and all positive diffusivities. Small-data nonlinear exponential stability is then obtained on a complex interpolation space adapted to the graph domain of the generator by combining analytic-semigroup smoothing with quadratic estimates. For unequal diffusivities, the nonzero streamwise modes of the linearized ionic subsystem satisfy a bounded-channel enhanced-dissipation estimate at rate D_min^1/3|A|^2/3 under an explicit strong-shear condition. When D_+=D_-, a Fourier-mode factorization separates the common Couette advection--diffusion operator from a contractive drift--reaction semigroup, so the same scalar mixing rate is retained without any smallness condition on the electrostatic coupling. We also construct an exact Poisson--Boltzmann/electroosmotic Couette family for prescribed wall potentials. For sufficiently small wall-potential amplitude, its linearized generator is a small graph-domain perturbation of the electroneutral generator, which yields linear and nonlinear exponential stability on the same interpolation scale.
ER -