arXiv subjects

Zhewen Feng

Publications and source records attributed to Zhewen Feng.

4 recordsLinked to original sources

arXiv2022

Existence of minimizers and convergence of critical points for a new Landau-de Gennes energy functional in nematic liquid crystals

The Landau-de Gennes energy in nematic liquid crystals depends on four elastic constants $L_1$, $L_2$, $L_3$, $L_4$. In the case of $L_4\neq 0$, Ball and Majumdar (Mol. Cryst. Liq. Cryst., 2010) found an example that the original Landau-de Gennes energy functional in physics does not satisfy a coercivity condition, which causes a problem in mathematics to establish existence of energy minimizers. At first, we introduce a new Landau-de Gennes energy density with $L_4\neq 0$, which is equivalent to the original Landau-de Gennes density for uniaxial tensors and satisfies the coercivity condition for all $Q$-tensors. Secondly, we prove that solutions of the Landau-de Gennes system can approach a solution of the $Q$-tensor Oseen-Frank system without using energy minimizers. Thirdly, we develop a new approach to generalize the Nguyen and Zarnescu (Calc. Var. PDEs, 2013) convergence result to the case of non-zero elastic constants $L_2$, $L_3$, $L_4$.

math.AP↗

arXiv2021

Existence and convergence of the Beris-Edwards system with general Landau-de Gennes energy

In this paper, we investigate the Beris-Edwards system for both biaxial and uniaxial $Q$-tensors with a general Landau-de Gennes energy density depending on four non-zero elastic constants. We prove existence of the strong solution of the Beris-Edwards system for uniaxial $Q$-tensors up to a maximal time. Furthermore, we prove that the strong solutions of the Beris-Edwards system for biaxial $Q$-tensors converge smoothly to the solution of the Beris-Edwards system for uniaxial $Q$-tensors up to its maximal existence time.

math.AP↗

arXiv2021

A new representation for the Landau-de Gennes energy of nematic liquid crystals

In the Landau-de Gennes theory on nematic liquid crystals, the well-known Landau-de Gennes energy depends on four elastic constants; $L_1$, $L_2$, $L_3$, $L_4$. For the general case of $L_4\neq 0$, Ball-Majumdar \cite {BM} found an example that the Landau-de Gennes energy functional from physics literature \cite{MN} does not satisfy a coercivity condition, which causes a problem in mathematics to establish existence of energy minimizers. In order to solve this problem, we observe that the original third order term on $L_4$, proposed by Schiele and Trimper \cite{ST} in physics, is a linear combination of a fourth order term and a second order term. Therefore, we can propose a new Landau-de Gennes energy, which is equal to the original for uniaxial nematic $Q$-tensors. The new Landau-de Gennes energy with general elastic constants satisfies the coercivity condition for all $Q$-tensors, which establishes a new link between mathematical and physical theory. Similarly to the work of Majumdar-Zarnescu \cite{MZ}, we prove existence and convergence of minimizers of the new Landau-de Gennes energy. Moreover, we find a new way to study the limiting problem of the Landau-de Gennes system since the cross product method \cite{Chen} on the Ginzburg-Landau equation does not work for the Landau-de Gennes system.

math.AP↗

arXiv2019

Convergence of the Ginzburg-Landau approximation for the Ericksen-Leslie system

We establish the local well-posedness of the general Ericksen-Leslie system in liquid crystals with the initial velocity and director field in $H^1 \times H_b^2$. In particular, we prove that the solutions of the Ginzburg-Landau approximation system converge smoothly to the solution of the Ericksen-Leslie system for any $t \in (0,T^\ast)$ with a maximal existence time $T^\ast$ of the Ericksen- Leslie system.

math.AP↗