Weak sequential stability for a nonlinear model of nematic electrolytes

Eduard Feireisl; Elisabetta Rocca; Giulio Schimperna; Arghir Zarnescu

doi:10.3934/dcdss.2020366

Article Contents

2021, Volume 14, Issue 1: 219-241. Doi: 10.3934/dcdss.2020366

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Weak sequential stability for a nonlinear model of nematic electrolytes

1.
Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, CZ-115 67 Praha 1, Czech Republic
2.
Università degli Studi di Pavia, Dipartimento di Matematica and IMATI-C.N.R, Via Ferrata 5, 27100, Pavia, Italy
3.
IKERBASQUE, Basque Foundation for Science, Maria Diaz de Haro 3, 48013, Bilbao, Bizkaia, Spain
4.
BCAM, Basque Center for Applied Mathematics, Mazarredo 14, E48009 Bilbao, Bizkaia, Spain
5.
"Simion Stoilow" Institute of the Romanian Academy, 21 Calea Griviţei, 010702 Bucharest, Romania

^* Corresponding author: Elisabetta Rocca
^* Corresponding author: Elisabetta Rocca

To Alex Mielke, with friendship and admiration

Received: September 2019

Revised: February 2020

Early access: May 2020

Published: January 2021

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Abstract

In this article we study a system of nonlinear PDEs modelling the electrokinetics of a nematic electrolyte material consisting of various ions species contained in a nematic liquid crystal.

The evolution is described by a system coupling a Nernst-Planck system for the ions concentrations with a Maxwell's equation of electrostatics governing the evolution of the electrostatic potential, a Navier-Stokes equation for the velocity field, and a non-smooth Allen-Cahn type equation for the nematic director field.

We focus on the two-species case and prove apriori estimates that provide a weak sequential stability result, the main step towards proving the existence of weak solutions.

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Mathematics Subject Classification: Primary: 35Q60, 76D05; Secondary: 35B45.

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