Global well-posedness and decay rates for the three dimensional incompressible active liquid crystals
School of Mathematics and Statistics, Nantong University（南通大学数学与统计学院）；School of Mathematical Sciences, MOE-LSC and CMA-Shanghai, Shanghai Jiao Tong University（上海交通大学数学科学学院）
This paper investigates the global well-posedness and large-time behavior of 3D incompressible active liquid crystals under constant activity, modeled by a coupled system of forced incompressible Navier-Stokes equations for the velocity and a parabolic system for the $Q$-tensor order parameter. By employing refined commutator estimates, the existence and uniqueness of global strong solutions are proved for small initial data $(Q_0,u_0)\in H^{s+1}\times H^s$ $(s\geq 2)$ with activity $c>c_\star$, which improves a previous result in \cite{active-limit}. In addition, if the initial data further belong to $L^1$ and $s\geq 4$, we obtain a mixing decay estimate on $\|\partial^kQ(t)\|_{L^2}$ that combines both an extra exponential decay factor at a rate proportional to $(c-c_\star)Γ$ and the optimal algebraic decay rate that coincides with that of the heat kernel, where $k\leq s-1$. This result reveals that, in the high activity regime, active nematics become isotropic with an activity-dependent exponential convergence rate, and the estimate is stable in the infinite rotational viscosity limit, as $Γ\rightarrow 0$. Meanwhile, the sharp decay estimate on $\|\partial^ku(t)\|_{L^2}$ is also derived for $k\leq s-2$ with an additional initial assumption. The proof is established via a combination of the Green's function method and the time-weighted energy method. To the best of our knowledge, these results are the first reported for active/passive nematic liquid crystals within the Beris-Edwards framework, and the enhanced decay effect of the orientational field is essentially derived from the free energy. Furthermore, in the passive setting, our result implies the phase transition of thermotropic liquid crystals at high temperatures.
The present paper is dedicated to the global large solutions and incompressible limit for the compressible flow of liquid crystals under the assumption on almost constant density and large volume viscosity. The result is based on Fourier analysis and involved so-called critical Besov norm.
Global existence and incompressible limit in critical spaces for compressible flow of liquid crystals
在临界空间框架下，研究了双折射液晶可压缩流动的柯西问题。我们首先建立了整体解的存在性和唯一性，只要初始数据接近某些平衡态。这一结果改进了Hu和Wu的工作[SIAM J. Math. Anal.，45（2013），pp. 2678-2699]通过放松初始数据在指向矢场方面的规则性要求。然后，我们考虑的不可压缩极限问题，准备不良的初始数据。我们证明了当马赫数趋于零时，液晶可压缩流动的整体解在某些函数空间收敛于相应的不可压缩模型的解。此外，还得到了精确的收敛速度。
We study the Ericksen-Leslie's hyperbolic incompressible liquid crystal model. Under some constraints on the Leslie coefficients which ensure the basic energy law is dissipative, we prove the local-in-time existence and uniqueness of the classical solution to the system with finite initial energy. Furthermore, with an additional assumption on the coefficients which provides a damping effect, and the smallness of the initial energy, the unique global classical solution can be established.
In this paper, we prove a new type of energy estimates for the compressible Euler's equation with free boundary, with a boundary part and an interior part. These can be thought of as a generalization of the energies in Christodoulou and Lindblad [CL] to the compressible case and do not require the fluid to be irrotational. In addition, we show that our estimates are in fact uniform in the sound speed. As a consequence, we obtain convergence of solutions of compressible Euler equations with a free boundary to solutions of the incompressible equations, generalizing the result of Ebin [Eb] to whe…
In this paper we establish the incompressible limit for the compressible free-boundary Euler equations with surface tension in the case of a liquid. Compared to the case without surface tension treated recently, the presence of surface tension introduces severe new technical challenges, in that several boundary terms that automatically vanish when surface tension is absent now contribute at top order. Combined with the necessity of producing estimates uniform in the sound speed in order to pass to the limit, such difficulties imply that neither the techniques employed for the case without surf…
In this paper, we consider the short time classical solution to a simplified hydrodynamic flow modeling incompressible, nematic liquid crystal materials in dimension three. We establish a criterion for possible breakdown of such solutions at a finite time. More precisely, if $(u,d)$ is smooth up to time $T$ provided that $\int_0^T|\nabla\times u(t,\cdot)|_{BMO(\mathbb R^3)}+|\nabla d(t,\cdot)|^8_{L^4(\mathbb R^3)}dt<\infty $
In this article, we consider the Ericksen-Leslie's hyperbolic system for incompressible liquid crystal model without kinematic transport in three spatial dimensions, which is a nonlinear coupling of incompressible Navier-Stokes equations with wave map to $\mathbb{S}^2$. Global regularity for small and smooth initial data near the equilibrium is proved. The proof relies on the idea of space-time resonance.
In this paper, we investigate the uniform regularity and vanishing limit for the incompressible nematic liquid crystal flows in three dimensional bounded domain. It is shown that there exists a unique strong solution for the incompressible nematic liquid crystal flows with boundary condition in a finite time interval which is independent of the viscosity. The solution is uniformly bounded in a conormal Sobolev space. Finally, we also study the convergence rate of the viscous solutions to the inviscid ones.
We are concerned with the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible nematic liquid crystal flows on the whole space $\mathbb{R}^{2}$ with vacuum as far field density. It is proved that the 2D nonhomogeneous incompressible nematic liquid crystal flows admits a unique global strong solution provided the initial data density and the gradient of orientation decay not too slow at infinity, and the basic energy $\|\sqrt{ρ_0}\mathbf{u}_0\|_{L^2}^2+\|\nabla\mathbf{d}_0\|_{L^2}^2$ is small. In particular, the initial density may contain vacuum states and even have compact…
We prove a continuation critereon for incompressible liquids with free surface boundary. We combine the energy estimates of Christodoulou and Lindblad with an analog of the estimate due to Beale, Kato, and Majda for the gradient of the velocity in terms of the vorticity, and use this to show solution can be continued so long as the second fundamental form and injectivity radius of the free boundary, the vorticity, and one derivative of the velocity on the free boundary remain bounded, assuming that the Taylor sign condition holds.
In this paper, we prove the stability of half-degree point defect profiles in $\mathbb{R}^2$ for the nematic liquid crystal within Landau-de Gennes model.
In this paper we provide a sufficient condition, in terms of the horizontal gradient of two horizontal velocity components and the gradient of liquid crystal molecular orientation field, for the breakdown of local in time strong solutions to the three-dimensional incompressible nematic liquid crystal flows. More precisely, let $T_{*}$ be the maximal existence time of the local strong solution $(u, d)$, then $T_{*}<+\infty$ if and only if \begin{align*} \int_{0}^{T_*} \big( \|\nabla_h u^h\|_{\dot{B}^0_{p,\frac{2p}{3}}}^q + \|\nabla d\|_{\dot{B}^0_{\infty,\infty}}^2 \big)dt = \infty\ \ \text{wit…
For suitable initial and boundary data, we construct infinitely many weak solutions to the nematic liquid crystal flows in dimension three. These solutions are in the axisymmetric class with bounded energy and backward bubbling at a large time.
In this paper, we establish the uniquely existence of the global mild solution to the nematic liquid crystal equations in Besov-Morrey spaces. Some self-similarity and large time behavior of the global mild solution are also investigated.
In this note we prove the existence and uniqueness of local maximal smooth solution of the stochastic simplified Ericksen-Leslie systems modelling the dynamics of nematic liquid crystals under stochastic perturbations.
A complex non-Newtonian fluid models the nematic liquid crystal flows confined in a bounded domain in $\mathbb{R}^3$ is considered. The system is a forced incompressible Navier-Stokes equation coupled with a parabolic type Q-tensor flows. Under suitable condition on the coefficients of the Landau-de Gennes energy functional, global strong solution for small initial data is proved.
We study the hydrodynamics of compressible flows of active liquid crystals in the Beris-Edwards hydrodynamics framework, using the Landau-de Gennes $Q$-tensor order parameter to describe liquid crystalline ordering. 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.
In this paper, we obtain optimal time-decay rates in $L^r(\mathbb R^3_+)$ for $r\ge 1$ of global strong solutions to the nematic liquid crystal flows in $\mathbb R^3_+$, provided the initial data has small $L^3(\mathbb R^3_+)$-norm.
We study the effect of boundary rugosity in nematic liquid crystalline systems. We consider a highly general formulation of the problem, able to simultaneously deal with several liquid crystal theories. We use techniques of Gamma convergence and demonstrate that the effect of fine-scale surface oscillations may be replaced by an effective homogenised surface energy on a simpler domain. The homogenisation limit is then quantitatively studied in a simplified setting, obtaining convergence rates.
