Numerical Analysis for Nematic Electrolytes

We consider a system of nonlinear PDEs modeling nematic electrolytes, and construct a dissipative solution with the help of its implementable, structure-inheriting space-time discretization. Computational studies are performed to study the mutual effects of electric, elastic, and viscous effects onto the molecules in a nematic electrolyte.

[1]  Andreas Prohl,et al.  CONVERGENT FINITE ELEMENT DISCRETIZATIONS OF THE NAVIER-STOKES-NERNST-PLANCK-POISSON SYSTEM , 2010 .

[2]  Matthias Hieber,et al.  Dynamics of Nematic Liquid Crystal Flows: the Quasilinear Approach , 2013, 1302.4596.

[3]  Chenhui Peng,et al.  Liquid crystals with patterned molecular orientation as an electrolytic active medium. , 2015, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Noel J. Walkington,et al.  Numerical approximation of nematic liquid crystal flows governed by the Ericksen-Leslie equations , 2011 .

[5]  L. R. Scott,et al.  The Mathematical Theory of Finite Element Methods , 1994 .

[6]  Laure Saint-Raymond,et al.  From the Vlasov–Maxwell–Boltzmann System to Incompressible Viscous Electro-magneto-hydrodynamics , 2016, 1604.01547.

[7]  Robert Lasarzik Maximally dissipative solutions for incompressible fluid dynamics , 2020, Zeitschrift für angewandte Mathematik und Physik.

[8]  Zhouping Xin,et al.  Blow-up Criteria of Strong Solutions to the Ericksen-Leslie System in ℝ3 , 2014 .

[9]  Peter Constantin,et al.  On the Nernst–Planck–Navier–Stokes system , 2018, Archive for Rational Mechanics and Analysis.

[10]  N. Meyers An $L^p$-estimate for the gradient of solutions of second order elliptic divergence equations , 1963 .

[11]  Markus Schmuck,et al.  ANALYSIS OF THE NAVIER–STOKES–NERNST–PLANCK–POISSON SYSTEM , 2009 .

[12]  E. Feireisl,et al.  Nonlinear electrokinetics in nematic electrolytes , 2019 .

[13]  Chun Liu,et al.  Approximation of Liquid Crystal Flows , 2000, SIAM J. Numer. Anal..

[14]  Wei Wang,et al.  Well-Posedness of the Ericksen–Leslie System , 2012, 1208.6107.

[15]  Dmitry Vorotnikov Dissipative solutions for equations of viscoelastic diffusion in polymers , 2008 .

[16]  Ricardo H. Nochetto,et al.  Convergence Past Singularities for a Fully Discrete Approximation of Curvature-Drive Interfaces , 1997 .

[17]  R. Rannacher,et al.  Finite element approximation of the nonstationary Navier-Stokes problem. I : Regularity of solutions and second-order error estimates for spatial discretization , 1982 .

[18]  P. Lions Mathematical topics in fluid mechanics , 1996 .

[19]  Zhouping Xin,et al.  Global existence of solutions of the liquid crystal flow for the Oseen–Frank model in R2 , 2012 .

[20]  P. G. Ciarlet,et al.  Maximum principle and uniform convergence for the finite element method , 1973 .

[21]  Andreas Prohl,et al.  Convergent discretizations for the Nernst–Planck–Poisson system , 2009, Numerische Mathematik.

[22]  F. Lin,et al.  Nonparabolic dissipative systems modeling the flow of liquid crystals , 1995 .

[23]  Dmitry Golovaty,et al.  Modeling of Nematic Electrolytes and Nonlinear Electroosmosis , 2016, SIAM J. Appl. Math..

[24]  Jacques Simon,et al.  On the Existence of the Pressure for Solutions of the Variational Navier—Stokes Equations , 1999 .

[25]  Robert Lasarzik,et al.  Dissipative solution to the Ericksen–Leslie system equipped with the Oseen–Frank energy , 2018, Zeitschrift für angewandte Mathematik und Physik.

[26]  Joseph E. Pasciak,et al.  On the stability of the L2 projection in H1(Omega) , 2002, Math. Comput..

[27]  Andreas Prohl,et al.  Computational Studies for the Stochastic Landau-Lifshitz-Gilbert Equation , 2013, SIAM J. Sci. Comput..

[28]  P. Lions,et al.  Compactness in Boltzmann’s equation via Fourier integral operators and applications. III , 1994 .

[29]  Fanghua Lin,et al.  Liquid Crystal Flows in Two Dimensions , 2010 .

[30]  Andreas Prohl,et al.  Finite Element Approximations of the Ericksen-Leslie Model for Nematic Liquid Crystal Flow , 2008, SIAM J. Numer. Anal..

[31]  Andreas Prohl,et al.  Convergent finite element discretization of the multi-fluid nonstationary incompressible magnetohydrodynamics equations , 2010, Math. Comput..

[32]  Chun Liu,et al.  Existence of Solutions for the Ericksen-Leslie System , 2000 .

[33]  Robert Lasarzik Measure-valued solutions to the Ericksen–Leslie model equipped with the Oseen–Frank energy , 2017, Nonlinear Analysis.

[34]  J. Ciavaldini Analyse Numerique d’un Probleme de Stefan a Deux Phases Par une Methode d’Elements Finis , 1975 .

[35]  Robert Lasarzik Approximation and Optimal Control of Dissipative Solutions to the Ericksen–Leslie System , 2019, Numerical Functional Analysis and Optimization.

[36]  Robert Lasarzik,et al.  Weak-strong uniqueness for measure-valued solutions to the Ericksen–Leslie model equipped with the Oseen–Frank free energy , 2017, Journal of Mathematical Analysis and Applications.