A relative energy inequality for an anisotropic Navier--Stokes--Nernst--Planck--Poisson system --- Weak-strong uniqueness and a posteriori error estimates

Abstract

In this work, we build upon the framework of suitable weak solutions to the anisotropic Navier--Stokes--Nernst--Planck--Poisson (NSNPP) system, as developed in [HPL25], and establish a relative energy inequality for these solutions. This inequality serves as the basis for proving the weak-strong uniqueness property. Additionally, we exploit the relative energy inequality as a tool to obtain a posteriori error estimates. We are interested in the high-viscosity-low-Reynolds number limit of the NSNPP, which leads to the anisotropic Stokes--Nernst--Planck--Poisson (SNPP) system. Utilizing the relative energy framework, we derive an error estimate for the distance between solutions to the NSNPP and SNPP model in natural energy and dissipation norms. Moreover, we prove the existence of regular local-in-time solutions to the NSNPP as well as the SNPP system, possessing sufficient regularity to be admissible in the relative energy framework.