Global Existence and Regularity for the Active Liquid Crystal System

Zhang, Rongfang (2019) Global Existence and Regularity for the Active Liquid Crystal System. Doctoral Dissertation, University of Pittsburgh. (Unpublished)

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Abstract

We study the hydrodynamics of active liquid crystals in the Beris-Edwards hydrodynamic framework with the Landau-de Gennes Q-tensor order parameter to describe liquid crystalline ordering. For the incompressible case, the existence of global weak solutions in two and three spatial dimensions is established. The higher regularity of the weak solutions and the weak-strong uniqueness are also obtained by using the Littlewood-Paley decomposition in two space dimensions.
For the inhomogeneous incompressible case, Faedo-Galerkin's method is adopted to construct the solutions for the initial-boundary value problem. Two levels of approximations are used and the weak convergence is obtained through compactness estimates by new techniques due to the active terms. The existence of global weak solutions in dimension two and three is established in a bounded domain. For the compressible case where the concentration of the active particles in the system is not constant, we prove the existence of global weak solutions for this active system in three space dimensions by the three-level approximations and weak convergence argument. New techniques and estimates are developed to overcome the difficulties caused by the active terms.

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Details

Item Type:	University of Pittsburgh ETD
Status:	Unpublished
Creators/Authors:	Creators Email Pitt Username ORCID Zhang, Rongfang roz14@pitt.edu roz14
ETD Committee:	Title Member Email Address Pitt Username ORCID Committee Chair Wang, Dehua dhwang@pitt.edu dhwang Committee Member Chen, Ming MINGCHEN@pitt.edu Mingchen Committee Member Jiang, Huiqiang hqjiang@pitt.edu hqjiang Committee Member Iyer, Gautam gautam@math.cmu.edu
Date:	31 January 2019
Date Type:	Publication
Defense Date:	29 November 2018
Approval Date:	31 January 2019
Submission Date:	12 November 2018
Access Restriction:	No restriction; Release the ETD for access worldwide immediately.
Number of Pages:	136
Institution:	University of Pittsburgh
Schools and Programs:	Dietrich School of Arts and Sciences > Mathematics
Degree:	PhD - Doctor of Philosophy
Thesis Type:	Doctoral Dissertation
Refereed:	Yes
Uncontrolled Keywords:	Active hydrodynamics, active liquid crystals, Navier-Stokes equations, incompressible flow, inhomogeneous incompressible flow, compressible flow, weak solutions, strong solutions, global well-posedness, regularity, weak-strong uniqueness, weak convergence.
Date Deposited:	31 Jan 2019 17:40
Last Modified:	31 Jan 2019 17:40
URI:	http://d-scholarship.pitt.edu/id/eprint/35701

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