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Register a new userTwisted solutions to a simplified Ericksen-Leslie equation
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In this article we construct global solutions to a simplified Ericksen-Leslie system on $mathbb{R}^3$. The constructed solutions are twisted and periodic along the $x_3$-axis with period $d = 2pi big/ mu$. Here $mu > 0$ is the twist rate. $d$ is the distance between two planes which are parallel to the $x_1x_2$-plane. Liquid crystal material is placed in the region enclosed by these two planes. Given a well-prepared initial data, our solutions exist classically for all $t in [0, infty)$. However these solutions become singular at all points on the $x_3$-axis and escape into third dimension exponentially while $t rightarrow infty$. An optimal blow up rate is also obtained.
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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.
We consider a general family of regularized systems for the full Ericksen-Leslie model for the hydrodynamics of liquid crystals in $n$-dimensional compact Riemannian manifolds, $n$=2,3. The system we consider consists of a regularized family of Navier-Stokes equations (including the Navier Stokes-$alpha $-like equation, the Leray-$alpha $ equation, the Modified Leray-$alpha $ equation, the Simplified Bardina model, the Navier Stokes-Voigt model and the Navier-Stokes equation) for the fluid velocity $u$ suitably coupled with a parabolic equation for the director field $d$. We establish existence, stability and regularity results for this family. We also show the existence of a finite dimensional global attractor for our general model, and then establish sufficiently general conditions under which each trajectory converges to a single equilibrium by means of a Lojasiewicz-Simon inequality.
In this paper, we study the connection between the Ericksen-Leslie equations and the Beris-Edwards equations in dimension two. It is shown that the weak solutions to the Beris-Edwards equations converge to the one to the Ericksen-Leslie equations as the elastic coefficient tends to zero. Moreover, the limiting weak solutions to the Ericksen-Leslie equations may have singular points.
In this paper, we study the Cauchy problem of the Poiseuille flow of full Ericksen-Leslie model for nematic liquid crystals. The model is a coupled system of a parabolic equation for the velocity and a quasilinear wave equation for the director. For a particular choice of several physical parameter values, we construct solutions with smooth initial data and finite energy that produce, in finite time, cusp singularities - blowups of gradients. The formation of cusp singularity is due to local interactions of wave-like characteristics of solutions, which is different from the mechanism of finite time singularity formations for the parabolic Ericksen-Leslie system. The finite time singularity formation for the physical model might raise some concerns for purposes of applications. This is, however, resolved satisfactorily; more precisely, we are able to establish the global existence of weak solutions that are Holder continuous and have bounded energy. One major contribution of this paper is our identification of the effect of the flux density of the velocity on the director and the reveal of a singularity cancellation - the flux density remains uniformly bounded while its two components approach infinity at formations of cusp singularities.
We prove the existence and uniqueness of measure solutions to the conservative renewal equation and analyze their long time behavior. The solutions are built by using a duality approach. This construction is well suited to apply the Doeblins argument which ensures the exponential relaxation of the solutions to the equilibrium.
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