Regularity and existence of global solutions to the Ericksen-Leslie system in ${\mathbb{R}}^2$

Abstract

In this paper, we first establish the regularity theorem for suitable weak solutions to the Ericksen-Leslie system in dimensions two. Building on such a regularity, we then establish the existence of a global weak solution to the Ericksen-Leslie system in ${\mathbb{R}}^2$ for any initial data in the energy space, under the physical constraint conditions on the Leslie coefficients ensuring the dissipation of energy of the system, which is smooth away from at most finitely many times. This extends earlier works by Lin,Lin, and Wang on a simplified nematic liquid crystal flow in dimensions two.

33 pages

Cited by in corpus (41)

• Recent developments of analysis for hydrodynamic flow of nematic liquid crystals
• Mathematics and liquid crystals
• Finite time singularity of the nematic liquid crystal flow in dimension three
• On well-posedness of Ericksen-Leslie's hyperbolic incompressible liquid crystal model
• On well-posedness of Ericksen-Leslie's paraboloc-hyperbolic liquid crystal model
• Global strong solutions of the full Navier-Stokes and $Q$-tensor system for nematic liquid crystal flows in $2D$: existence and long-time behavior
• Global well-posedness for the dynamical Q-tensor model of liquid crystals
• Non-isothermal general Ericksen-Leslie system: derivation, analysis and thermodynamics-consistency
• Suitable weak solutions for the co-rotational Beris-Edwards system in dimension three
• Small data global regularity for 3-D Ericksen-Leslie's hyperbolic liquid crystal model without kinematic transport
• Global Well-posedness for the Three Dimensional Simplified Inertial Ericksen-Leslie Systems Near Equilibrium
• Optimal boundary control of a simplified Ericksen--Leslie system for nematic liquid crystal flows in $2D$
• Poiseuille flow of nematic liquid crystals via the full Ericksen-Leslie model
• Global solution to the three-dimensional liquid crystal flows of Q-tensor model
• Global Weak Solutions to Non-isothermal Nematic Liquid Crystals in 2D
• The zero inertia limit of Ericksen-Leslie's model for liquid crystals
• Well-posedness of the Ericksen-Leslie System for the Oseen-Frank Model in L^3_{uloc}(\mathbb{R}^3)
• Small data global regularity for simplified 3-D Ericksen-Leslie's compressible hyperbolic liquid crystal model
• On the uniqueness of weak solutions to the Ericksen-Leslie liquid crystal model in ${\mathbb{R}}^2$
• A note on the Stochastic Ericksen-Leslie equations for nematic liquid crystals
• Convergence of the Ginzburg-Landau approximation for the Ericksen-Leslie system
• Finite time blow-up for the nematic liquid crystal flow in dimension two
• On static and hydrodynamic biaxial nematic liquid crystals
• Hydrodynamic interactions in polar liquid crystals on evolving surfaces
• Thermodynamical Consistent Modeling and Analysis of Nematic Liquid Crystal Flows
• From the Landau-de Gennes theory to the Ericksen-Leslie theory in dimension two
• Uniqueness of weak solutions for the general Ericksen-Leslie system with Ginzburg-Landau penalization in T^2
• On optimal boundary control of Ericksen-Leslie system in dimension two
• Partial regularity of a nematic liquid crystal model with kinematic transport effects
• Local well-posedness of strong solutions to density-dependent liquid crystal system
• On well-posedness of Ericksen-Leslie's parabolic-hyperbolic liquid crystal model in compressible flow
• Local well-posedness and small Deborah limit of a molecule-based $Q$-tensor system
• Well-posedness of a fully-coupled Navier-Stokes/Q-tensor system with inhomogeneous boundary data
• Rigorous derivation from Landau-de Gennes theory to Ericksen-Leslie theory
• Global weak solutions to the Stochastic Ericksen--Leslie equations in dimension two
• Weak solutions of non-isothermal nematic liquid crystal flow in dimension three
• Existence and large time behavior to the nematic liquid crystal equations in Besov-Morrey spaces
• Partial regularity of suitable weak solutions of the Navier-Stokes-Planck-Nernst-Poisson equation
• Rigorous justification of the uniaxial limit from Qian-Sheng's inertial $Q$-tensor theory to the Ericksen-Leslie theory
• Global solution to the nematic liquid crystal flows with heat effect
• Time decay rate of global strong solutions to nematic liquid crystal flows in ${\mathbb{R}}_{+}^3$