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Compactness, existence, and partial regularity in hydrodynamics of liquid crystals

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posted on 04.08.2021, 15:49 by Hengrong DuHengrong Du
This thesis mainly focuses on the PDE theories that arise from the study of hydrodynamics of nematic liquid crystals.

In Chapter 1, we give a brief introduction of the Ericksen--Leslie director theory and Beris--Edwards Q-tensor theory to the PDE modeling of dynamic continuum description of nematic liquid crystals. In the isothermal case, we derive the simplified Ericksen--Leslie equations with general targets via the energy variation approach. Following this, we introduce a simplified, non-isothermal Ericksen--Leslie system and justify its thermodynamic consistency.

In Chapter 2, we study the weak compactness property of solutions to the Ginzburg--Landau approximation of the simplified Ericksen--Leslie system. In 2-D, we apply the Pohozaev type argument to show a kind of concentration cancellation occurs in the weak sequence of Ginzburg--Landau system. Furthermore, we establish the same compactness for non-isothermal equations with approximated director fields staying on the upper semi-sphere in 3-D. These compactness results imply the global existence of weak solutions to the limit equations as the small parameter tends to zero.

In Chapter 3, we establish the global existence of a suitable weak solution to the co-rotational Beris–Edwards system for both the Landau–De Gennes and Ball–Majumdar bulk potentials in 3-D, and then study its partial regularity by proving that the 1-D parabolic Hausdorff measure of the singular set is 0.

In Chapter 4, motivated by the study of un-corotational Beris--Edwards system, we construct a suitable weak solution to the full Ericksen--Leslie system with Ginzburg--Landau potential in 3-D, and we show it enjoys a (slightly weaker) partial regularity, which asserts that it is smooth away from a closed set of parabolic Hausdorff dimension at most 15/7.

History

Degree Type

Doctor of Philosophy

Department

Mathematics

Campus location

West Lafayette

Advisor/Supervisor/Committee Chair

Changyou Wang

Additional Committee Member 2

Jie Shen

Additional Committee Member 3

Nung Kwan Yip

Additional Committee Member 4

Daniel Phillips

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