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Changyou Wang

Publications and source records attributed to Changyou Wang.

At least 37 records · Page 2Linked to original sources
arXiv2020

Global weak solutions to the Stochastic Ericksen--Leslie equations in dimension two

We establish the global existence of weak martingale solutions to the simplified stochastic Ericksen--Leslie system modeling the nematic liquid crystal flow driven by Wiener-type noises on the two-dimensional bounded domains. The construction of solutions is based on the convergence of Ginzburg--Landau approximations. To achieve such a convergence, we first utilize the concentration-cancellation method for the Ericksen stress tensor fields based on a Pohozaev type argument, and second the Skorokhod compactness theorem, which is built upon a uniform energy estimate.

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arXiv2020

Isotropic-Nematic Phase Transition and Liquid Crystal Droplets

Liquid crystal droplets are of great interest from physics and applications. Rigorous mathematical analysis is challenging as the problem involves harmonic maps (and in general the Oseen-Frank model), free interfaces and topological defects which could be either inside the droplet or on its surface along with some intriguing boundary anchoring conditions for the orientation configurations. In this paper, through a study of the phase transition between the isotropic and nematic states of liquid crystal based on the Ericksen model, we can show, when the size of droplet is much larger in comparison with the ratio of the Frank constants to the surface tension, a $Γ$-convergence theorem for minimizers. This $Γ$-limit is in fact the sharp interface limit for the phase transition between the isotropic and nematic regions when the small parameter $\varepsilon$, corresponding to the transition layer width, goes to zero. This limiting process not only provides a geometric description of the shape of the droplet as one would expect, and surprisingly it also gives the anchoring conditions for the orientations of liquid crystals on the surface of the droplet depending on material constants. In particular, homeotropic, tangential, and even free boundary conditions as assumed in earlier phenomenological modelings arise naturally provided that the surface tension, Frank and Ericksen constants are in suitable ranges.

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arXiv2020

Partial regularity of a nematic liquid crystal model with kinematic transport effects

In this paper, we will establish the global existence of a suitable weak solution to the Erickson--Leslie system modeling hydrodynamics of nematic liquid crystal flows with kinematic transports for molecules of various shapes in ${\mathbb{R}^3}$, which is smooth away from a closed set of (parabolic) Hausdorff dimension at most $\frac{15}{7}$.

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arXiv2020

On static and hydrodynamic biaxial nematic liquid crystals

In the first part of this paper, we will consider minimizing configurations of the Oseen-Frank energy functional $E(n, m)$ for a biaxial nematics $(n, m):Ω\to \mathbb S^2\times \mathbb S^2$ with $n\cdot m=0$ in dimension three, and establish that it is smooth off a closed set of $1$-dimension Hausdorff measure zero. In the second part, we will consider a simplified Ericksen-Leslie system for biaxial nematics $(n, m)$ in a two dimensional domain and establish the existence of a unique global weak solution $(u, n, m)$ that is smooth off at most finitely many singular times for any initial and boundary data of finite energy. They extend to biaxial nematics of earlier results corresponding to minimizing uniaxial nematics by Hardt-Kindelerherer-Lin \cite{HKL} and a simplified hydrodynamics of uniaxial liquid crystal by Lin-Lin-Wang \cite{LLW10} respectively.

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arXiv2020

Weak compactness of simplified nematic liquid flows in 2D

For any bounded, smooth domain $Ω\subset \R^2$, %(or $Ω=\R^2$), we will establish the weak compactness property of solutions to the simplified Ericksen-Leslie system for both uniaxial and biaxial nematics, and the convergence of weak solutions of the Ginzburg-Landau type nematic liquid crystal flow to a weak solution of the simplified Ericksen-Leslie system as the parameter tends to zero. This is based on the compensated compactness property of the Ericksen stress tensors, which is obtained by the $L^p$-estimate of the Hopf differential for the Ericksen-Leslie system and the Pohozaev type argument for the Ginzburg-Landau type nematic liquid crystal flow.

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arXiv2020

Weak solutions of non-isothermal nematic liquid crystal flow in dimension three

For any smooth domain $Ω\subset \mathbb{R}^3$, we establish the existence of a global weak solution $(\mathbf{u},\mathbf{d}, θ)$ to the simplified, non-isothermal Ericksen-Leslie system modeling the hydrodynamic motion of nematic liquid crystals with variable temperature for any initial and boundary data $(\mathbf{u}_0, \mathbf{d}_0, θ_0)\in\mathbf{H}\times H^1(Ω, \mathbb{S}^2)\times L^1(Ω)$, with $ \mathbf{d}_0(Ω)\subset\mathbb{S}_+^2$ (the upper half sphere) and $\displaystyle\inf_Ωθ_0>0$.

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arXiv2019

Regularity of weak solution of variational problems modeling the Cosserat micropolar elasticity

In this paper, we consider weak solutions of the Euler-Lagrange equation to a variational energy functional modeling the geometrically nonlinear Cosserat micropolar elasticity of continua in dimension three, which is a system coupling between the Poisson equation and the equation of $p$-harmonic maps ($2\le p\le 3$). We show that if a weak solutions is stationary, then its singular set is discrete for $2<p<3$ and has zero $1$-dimensional Hausdorff measure for $p=2$. If, in addition, it is a stable-stationary weak solution, then it is regular everywhere when $p\in [2, \frac{32}{15}]$.

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arXiv2019

Finite time blow-up for the nematic liquid crystal flow in dimension two

We consider the initial-boundary value problem of a simplified nematic liquid crystal flow in a bounded, smooth domain $Ω\subset \mathbb R^2$. Given any $k$ distinct points in the domain, we develop a new {\em inner--outer gluing method} to construct solutions which blow up exactly at those $k$ points as $t$ goes to a finite time $T$. Moreover, we obtain a precise description of the blow-up.

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arXiv2019

Suitable weak solutions for the co-rotational Beris-Edwards system in dimension three

In this paper, we establish the global existence of a suitable weak solution to the co-rotational Beris-Edwards $Q$-tensor system modeling the hydrodynamic motion of nematic liquid crystals with either Landau-De Gennes bulk potential in $\mathbb R^3$ or Ball-Majumdar bulk potential in $\mathbb{T}^3$, a system coupling the forced incompressible Navier-Stokes equation with a dissipative, parabolic system of Q-tensor $Q$ in $\mathbb R^3$, which is shown to be smooth away from a closed set $Σ$ whose $1$-dimensional parabolic Hausdorff measure is zero.

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arXiv2019

Regularity of absolute minimizers for continuous convex Hamiltonians

For any $n\ge 2$, $Ω\subset\rn$, and any given convex and coercive Hamiltonian function $H\in C^{0}(\rn)$, we find an optimal sufficient condition on $H$, that is, for any $c\in\mathbb R$, the level set $H^{-1}(c)$ does not contains any line segment, such then any absolute minimizer $u\in AM_H(Ω)$ enjoys the linear approximation property. As consequences, we show that when $n=2$, if $u\in AM_H(Ω)$ then $u\in C^1$; and if $u\in AM_H(\rr^2)$ satisfies a linear growth at the infinity, then $u$ is a linear function on $\rr^2$. In particular, if $H$ is a strictly convex Banach norm $\|\cdot\|$ on $\mathbb R^2$, e.g. the $l_α$-norm for $1<α<1$, then any $u\in AM_H(Ω)$ is $C^1$. The ideas of proof are, instead of PDE approaches, purely variational and geometric.

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arXiv2018

On optimal boundary control of Ericksen-Leslie system in dimension two

In this paper, we consider the boundary value problem of a simplified Ericksen-Leslie system in dimension two with non-slip boundary condition for the velocity field $u$ and time-dependent boundary condition for the director field $d$ of unit length. For such a system, we first establish the existence of a global weak solution that is smooth away from finitely many singular times, then establish the existence of a unique global strong solution that is smooth for $t>0$ under the assumption that the image of boundary data is contained in the hemisphere $\mathbb S^2_+$. Finally, we apply these theorems to the problem of optimal boundary control of the simplified Ericksen-Leslie system and show both the existence and a necessary condition of an optimal boundary control.

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arXiv2018

Harmonic maps in connection of phase transitions with higher dimensional potential wells

This is in the sequel of authors' paper \cite{LPW} in which we had set up a program to verify rigorously some formal statements associated with the multiple component phase transitions with higher dimensional wells. The main goal here is to establish a regularity theory for minimizing maps with a rather non-standard boundary condition at the sharp interface of the transition. We also present a proof, under simplified geometric assumptions, of existence of local smooth gradient flows under such constraints on interfaces which are in the motion by the mean-curvature. In a forthcoming paper, a general theory for such gradient flows and its relation to Keller-Rubinstein-Sternberg's work \cite{KRS1, KRS2} on the fast reaction, slow diffusion and motion by the mean curvature would be addressed.

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arXiv2017

Higher dimensional Ginzburg-Landau equations under weak anchoring boundary conditions

For $n\ge 3$ and $0<ε\le 1$, let $Ω\subset\mathbb R^n$ be a bounded smooth domain and $u_ε:Ω\subset\R^n\to \mathbb R^2$ solve the Ginzburg-Landau equation under the weak anchoring boundary condition: $$\begin{cases} -Δu_ε=\frac{1}{ε^2}(1-|u_ε|^2)u_ε&\ {\rm{in}}\ \ Ω, \frac{\partial u_ε}{\partialν}+λ_ε(u_ε-g_ε)=0 & \ {\rm{on}}\ \ \partialΩ, \end{cases} $$ where the anchoring strength parameter $λ_ε=Kε^{-α}$ for some $K>0$ and $α\in [0,1)$, and $g_ε\in C^2(\partialΩ, \mathbb S^1)$. Motivated by the connection with the Landau-De Gennes model of nematic liquid crystals under weak anchoring conditions, we study the {asymptotic behavior} of $u_ε$ as $ε$ goes to zero under the condition that the total modified Ginzburg-Landau energy {satisfies} $F_ε(u_ε,Ω)\le M|\logε|$ for some $M>0$.

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arXiv2017

The harmonic map heat flow on conic manifolds

In this article, we study the the harmonic map heat flow from a manifold with conic singularities to a closed manifold. In particular, we have proved the short time existence and uniqueness of solutions as well as the existence of global solutions into manifolds with nonpositive sectional curvature. These results are established in virtue of the maximal regularity theory on manifolds with conic singularities.

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arXiv2016

Heat flow of extrinsic biharmonic maps from a four dimensional manifold with boundary

Let $(M,g)$ be a four dimensional compact Riemannian manifold with boundary and $(N,h)$ be a compact Riemannian manifold without boundary. We show the existence of a unique, global weak solution of the heat flow of extrinsic biharmonic maps from $M$ to $N$ under the Dirichlet boundary condition, which is regular with the exception of at most finitely many time slices. We also discuss the behavior of solution near the singular times. As an immediate application, we prove the existence of a smooth extrinsic biharmonic map from $M$ to $N$ under any Dirichlet boundary condition.

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