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Register a new userWeak compactness of simplified nematic liquid flows in 2D
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For any bounded, smooth domain $Omegasubset R^2$, %(or $Omega=R^2$), 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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For any smooth domain $Omegasubset mathbb{R}^3$, we establish the existence of a global weak solution $(mathbf{u},mathbf{d}, theta)$ 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 $(mathbf{u}_0, mathbf{d}_0, theta_0)inmathbf{H}times H^1(Omega, mathbb{S}^2)times L^1(Omega)$, with $ mathbf{d}_0(Omega)subsetmathbb{S}_+^2$ (the upper half sphere) and $displaystyleinf_Omega theta_0>0$.
For $l$-homogeneous linear differential operators $mathcal{A}$ of constant rank, we study the implication $v_jrightharpoonup v$ in $X$ and $mathcal{A} v_jrightarrow mathcal{A} v$ in $W^{-l}Y$ implies $F(v_j)rightsquigarrow F(v)$ in $Z$, where $F$ is an $mathcal{A}$-quasiaffine function and $rightsquigarrow$ denotes an appropriate type of weak convergence. Here $Z$ is a local $L^1$-type space, either the space $mathscr{M}$ of measures, or $L^1$, or the Hardy space $mathscr{H}^1$; $X,, Y$ are $L^p$-type spaces, by which we mean Lebesgue or Zygmund spaces. Our conditions for each choice of $X,,Y,,Z$ are sharp. Analogous statements are also given in the case when $F(v)$ is not a locally integrable function and it is instead defined as a distribution. In this case, we also prove $mathscr{H}^p$-bounds for the sequence $(F(v_j))_j$, for appropriate $p<1$, and new convergence results in the dual of Holder spaces when $(v_j)$ is $mathcal{A}$-free and lies in a suitable negative order Sobolev space $W^{-beta,s}$. The choice of these Holder spaces is sharp, as is shown by the construction of explicit counterexamples. Some of these results are new even for distributional Jacobians.
We consider a full Navier-Stokes and $Q$-tensor system for incompressible liquid crystal flows of nematic type. In the two dimensional periodic case, we prove the existence and uniqueness of global strong solutions that are uniformly bounded in time. This result is obtained without any smallness assumption on the physical parameter $xi$ that measures the ratio between tumbling and aligning effects of a shear flow exerting over the liquid crystal directors. Moreover, we show the uniqueness of asymptotic limit for each global strong solution as time goes to infinity and provide an uniform estimate on the convergence rate.
We consider the simplified Ericksen-Leslie model in three dimensional bounded Lipschitz domains. Applying a semilinear approach, we prove local and global well-posedness (assuming a smallness condition on the initial data) in critical spaces for initial data in $L^3_{sigma}$ for the fluid and $W^{1,3}$ for the director field. The analysis of such models, so far, has been restricted to domains with smooth boundaries.
Topological defects are one of the most conspicuous features of liquid crystals. In two dimensional nematics, they have been shown to behave effectively as particles with both, charge and orientation, which dictate their interactions. Here, we study twisted defects that have a radially dependent orientation. We find that twist can be partially relaxed through the creation and annihilation of defect pairs. By solving the equations for defect motion and calculating the forces on defects, we identify four distinct elements that govern the relative relaxational motion of interacting topological defects, namely attraction, repulsion, co-rotation and co-translation. The interaction of these effects can lead to intricate defect trajectories, which can be controlled by setting relevant timescales.
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