Well-Posedness of a Fully Coupled Navier-Stokes/Q-tensor System with Inhomogeneous Boundary Data

We prove short-time well-posedness and existence of global weak solutions of the Beris--Edwards model for nematic liquid crystals in the case of a bounded domain with inhomogeneous mixed Dirichlet and Neumann boundary conditions. The system consists of the Navier--Stokes equations coupled with an evolution equation for the $Q$-tensor. The solutions possess higher regularity in time of order one compared to the class of weak solutions with finite energy. This regularity is enough to obtain Lipschitz continuity of the nonlinear terms in the corresponding function spaces. Therefore the well-posedness is shown with the aid of the contraction mapping principle using that the linearized system is an isomorphism between the associated function spaces.

[1]  R. Temam.  Navier-Stokes Equations , 1977 .

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

[3]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[4]  Fanghua Lin,et al.  Regularity and Existence of Global Solutions to the Ericksen–Leslie System in $${\mathbb{R}^2}$$R2 , 2013, 1305.5988.

[5]  Hantaek Bae,et al.  On the Navier-Stokes equations , 2009 .

[6]  FANGHUA LIN,et al.  Regularity and Existence of Global Solutions to the Ericksen–Leslie System in R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{d , 2014, Communications in Mathematical Physics.

[7]  R. Rogers,et al.  An introduction to partial differential equations , 1993 .

[8]  Helmut Abels,et al.  On a Diffuse Interface Model for Two-Phase Flows of Viscous, Incompressible Fluids with Matched Densities , 2009 .

[9]  W. McLean.  Strongly Elliptic Systems and Boundary Integral Equations , 2000 .

[10]  Arghir Zarnescu,et al.  Energy Dissipation and Regularity for a Coupled Navier–Stokes and Q-Tensor System , 2010, 1001.1032.

[11]  Mark Wilkinson,et al.  Strict Physicality of Global Weak Solutions of a Navier-Stokes Q-tensor System with Singular Potential , 2012, 1211.6083.

[12]  J. Simon.  Compact sets in the spaceLp(O,T; B) , 1986 .

[13]  G. Folland.  Introduction to Partial Differential Equations , 1976 .

[14]  M. Yoneya,et al.  Physics of Liquid Crystals , 2014 .

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

[16]  L. E. Fraenkel,et al.  NAVIER-STOKES EQUATIONS (Chicago Lectures in Mathematics) , 1990 .

[17]  Akhlesh Lakhtakia,et al.  The physics of liquid crystals, 2nd edition: P.G. De Gennes and J. Prost, Published in 1993 by Oxford University Press, Oxford, UK, pp 7,597 + xvi, ISBN: 0-19-852024 , 1995 .

[18]  Zhifei Zhang,et al.  Rigorous Derivation from Landau-de Gennes Theory to Ericksen-Leslie Theory , 2013, SIAM J. Math. Anal..

[19]  G. Galdi.  An Introduction to the Mathematical Theory of the Navier-Stokes Equations : Volume I: Linearised Steady Problems , 1994 .

[20]  Elisabetta Rocca,et al.  Evolution of non-isothermal Landau-de Gennes nematic liquid crystals flows with singular potential , 2012, 1207.1643.

[21]  Jacques Simeon,et al.  Compact Sets in the Space L~(O, , 2005 .

[22]  Xiang Xu,et al.  On the General Ericksen–Leslie System: Parodi’s Relation, Well-Posedness and Stability , 2011, 1105.2180.

[23]  Arghir Zarnescu,et al.  Global Existence and Regularity for the Full Coupled Navier-Stokes and Q-Tensor System , 2010, SIAM J. Math. Anal..

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