Mathematical problems of nematic liquid crystals: between dynamical and stationary problems

Mathematical studies of nematic liquid crystals address in general two rather different perspectives: That of fluid mechanics and that of calculus of variations. The former focuses on dynamical problems while the latter focuses on stationary ones. The two are usually studied with different mathematical tools and address different questions. The aim of this brief review is to give the practitioners in each area an introduction to some of the results and problems in the other area. Also, aiming to bridge the gap between the two communities, we will present a couple of research topics that generate natural connections between the two areas. This article is part of the theme issue 'Topics in mathematical design of complex materials'

Uniform profile near the point defect of Landau-de Gennes model

For the Landau-de Gennes functional on 3D domains, $$I_\varepsilon(Q,\Omega):=\int_{\Omega}\left\{\frac12|\nabla Q|^2+\frac{1}{\varepsilon^2}\left( -\frac{a^2}{2}\mathrm{tr}(Q^2)-\frac{b^2}{3}\mathrm{tr}(Q^3)+\frac{c^2}{4}[\mathrm{tr}(Q^2)]^2 \right) \right\}\,dx,$$ it is well-known that under suitable boundary conditions, the global minimizer $Q_\varepsilon$ converges strongly in $H^1(\Omega)$ to a uniaxial minimizer $Q_*=s_+(n_*\otimes n_*-\frac13\mathrm{Id})$ up to some subsequence \e_n\rightarrow\infty , where $n_*\in H^1(\Omega,\mathbb{S}^2)$ is a minimizing harmonic map. In this paper we further investigate the structure of $Q_\varepsilon$ near the core of a point defect $x_0$ which is a singular point of the map $n_*$. The main strategy is to study the blow-up profile of $Q_{\varepsilon_n}(x_n+\varepsilon_n y)$ where $\{x_n\}$ are carefully chosen and converge to $x_0$. We prove that $Q_{\varepsilon_n}(x_n+\varepsilon_n y)$ converges in $C^2_{loc}(\mathbb{R}^n)$ to a tangent map $Q(x)$ which at infinity behaves like a ``hedgehog" solution that coincides with the asymptotic profile of $n_*$ near $x_0$. Moreover, such convergence result implies that the minimizer $Q_{\varepsilon_n}$ can be well approximated by the Oseen-Frank minimizer $n_*$ outside the $O(\varepsilon_n)$ neighborhood of the point defect

Sphere-valued harmonic maps with surface energy and the K13 problem

We consider an energy functional motivated by the celebrated K13 problem in the Oseen-Frank theory of nematic liquid crystals. It is defined for sphere-valued functions and appears as the usual Dirichlet energy with an additional surface term. It is known that this energy is unbounded from below and our aim has been to study the local minimizers. We show that even having a critical point in a suitable energy space imposes severe restrictions on the boundary conditions. Having suitable boundary conditions makes the energy functional bounded and in this case we study the partial regularity of the minimizers.Romanian National Authority for Scientific Research and Innovation, CNCSUEFISCDI, project number PN-II-RU-TE-2014-4-065

Asymptotic stability of ground states in 2D nonlinear Schr\"odinger equation including subcritical cases

We consider a class of nonlinear Schr\"odinger equation in two space dimensions with an attractive potential. The nonlinearity is local but rather general encompassing for the first time both subcritical and supercritical (in $L^2$) nonlinearities. We study the asymptotic stability of the nonlinear bound states, i.e. periodic in time localized in space solutions. Our result shows that all solutions with small initial data, converge to a nonlinear bound state. Therefore, the nonlinear bound states are asymptotically stable. The proof hinges on dispersive estimates that we obtain for the time dependent, Hamiltonian, linearized dynamics around a careful chosen one parameter family of bound states that "shadows" the nonlinear evolution of the system. Due to the generality of the methods we develop we expect them to extend to the case of perturbations of large bound states and to other nonlinear dispersive wave type equations

Design of effective bulk potentials for nematic liquid crystals via colloidal homogenisation

We consider a Landau–de Gennes model for a suspension of small colloidal inclusions in a nematic host. We impose suitable anchoring conditions at the boundary of the inclusions, and we work in the dilute regime — i.e. the size of the inclusions is much smaller than the typical separation distance between them, so that the total volume occupied by the inclusions is small. By studying the homogenised limit, and proving rigorous convergence results for local minimisers, we compute the effective free energy for the doped material. In particular, we show that not only the phase transition temperature, but also any coefficient of the quartic Landau–de Gennes bulk potential can be tuned, by suitably choosing the surface anchoring energy density

Asymptotic behavior of the interface for entire vector minimizers in phase transitions

We study globally bounded entire minimizers $u:\mathbb{R}^n\rightarrow\mathbb{R}^m$ of Allen-Cahn systems for potentials $W\geq 0$ with $\{W=0\}=\{a_1,...,a_N\}$ and $W(u)\sim |u-a_i|^\alpha$ near $u=a_i$, $0<\alpha<2$. Such solutions are, over large regions, identically equal to some zeroes of the potential $a_i$'s. We establish the estimates $$\mathcal{L}^n(I_0\cap B_r(x_0))\leq c_1r^{n-1},\quad \mathcal{H}^{n-1}(\partial^* I_0\cap B_r(x_0))\geq c_2r^{n-1}, \quad r\geq r_0(x_0)$$ for the diffuse interface $I_0:=\{x\in\mathbb{R}^n: \min_{1\leq i\leq N}|u(x)-a_i|>0\}$ and the free boundary $\partial I_0$. Furthermore, if $\alpha=1$ we establish the upper bound $$ \mathcal{H}^{n-1}(\partial^* I_0\cap B_r(x_0))\leq c_3r^{n-1}, \quad r\geq r_0(x_0). $

Global well-posedness and twist-wave solutions for the inertial Qian-Sheng model of liquid crystals

We consider the inertial Qian-Sheng model of liquid crystals which couples a hyperbolic-type equation involving a second-order material derivative with a forced incompressible Navier-Stokes system. We study the energy law and prove a global well-posedness result. We further provide an example of twist-wave solutions, that is solutions of the coupled system for which the flow vanishes for all times.Marie Sklodowska-Curie grant agreement No 642768, Romanian National Authority for Scientific Research and Innovation, CNCS-UEFISCDI, project number PN-II-RU-TE-2014-4-0657

Entire Minimizers of Allen–Cahn Systems with Sub-Quadratic Potentials

We study entire minimizers of the Allen–Cahn systems. The specific feature of our systems are potentials having a finite number of global minima, with sub-quadratic behaviour locally near their minima. The corresponding formal Euler–Lagrange equations are supplemented with free boundaries. We do not study regularity issues but focus on qualitative aspects. We show the existence of entire solutions in an equivariant setting connecting the minima of W at infinity, thus modeling many coexisting phases, possessing free boundaries and minimizing energy in the symmetry class. We also present a very modest result of existence of free boundaries under no symmetry hypotheses. The existence of a free boundary can be related to the existence of a specific sub-quadratic feature, a dead core, whose size is also quantified

Shear flow dynamics in the Beris-Edwards model of nematic liquid crystals

We consider the Beris-Edwards model describing nematic liquid crystal dynamics and restrict to a shear flow and spatially homogeneous situation. We analyze the dynamics focusing on the effect of the flow. We show that in the co-rotational case one has gradient dynamics, up to a periodic eigenframe rotation, while in the non-co-rotational case we identify the short and long time regime of the dynamics. We express these in terms of the physical variables and compare with the predictions of other models of liquid crystal dynamics

Effective surface energies in nematic liquid crystals as homogenised rugosity effects

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