Concentration-cancellation in the Ericksen-Leslie model (Q2217794)

The paper under consideration analyses the Ericksen-Leslie system of equations for the evolution of nematic liquid crystals on the two-dimensional torus \(\mathbb{T}^2\) and for a given time interval \([0,T]\). Subject to the incompressibility constraint, this system of equations consists of the Navier-Stokes equation describing the time evolution of the momentum/velocity \(v\in \mathbb{R}^2\) as well as an harmonic map heat flow-like equation describing the time evolution of the unitary director field \(d \in \mathbb{S}^2\) which represents the orientation or average alignment of the liquid crystals. In order to obtain a truly coupled system, the director field is transported by the velocity whereas the momentum equation takes into account, the force or stress arising from the interaction of the particles of the liquid crystal. More precisely, this latter stress term is given by \(- \mathrm{div}(\nabla d \odot \nabla d)\) where \(\mathbb{A} \odot \mathbb{B} := \mathbb{A}^T\mathbb{B}\) for tensors \(\mathbb{A}\) and \(\mathbb{B}\). \newline The main result consist of establishing the (weak star and almost everywhere) convergence \(\epsilon\rightarrow0\) of a family of weak solutions of the related Ginzburg-Landau approximate system of equations parametrised by \(\epsilon>0\), to a weak solution of the Ericksen-Leslie system globally in time. A consequence of this result is an alternative proof of the existence of a finite energy weak solution of the Ericksen-Leslie model. \newline The main idea in the proof of the main result is the use of the concentration-cancellation lemma introduced in [\textit{R. J. DiPerna} and \textit{A. Majda}, J. Am. Math. Soc. 1, No. 1, 59--95 (1988; Zbl 0707.76026)] to establish the convergence in the crucial convective term \(- \mathrm{div}(\nabla d \odot \nabla d)\) for the director field in the momentum equation. The reason for the use of this lemma is that the a priori estimates obtained for the solution to the Ginzburg-Landau approximation do not yield enough regularity for the passage in the limit of the aforementioned crucial term for the director field. More precisely, in the limit as \(\epsilon\rightarrow0\), one may only deduce that \(\nabla d_\epsilon \odot \nabla d_\epsilon \rightharpoonup^* \nabla d \odot \nabla d + \mathbb{A}\), where \(\mathbb{A}\) may be a non-zero tensor. Rather than showing that \(\mathbb{A}\) is in fact identically zero, the author illustrates that this crucial term is insensitive to concentrations and thus, satisfies the desired distributional form in the limit. It is also worth noting that unlike the more subtle Euler equation for incompressible fluids, the gradient structure of the momentum equation for the approximate Ginzburg-Landau model and thus the Ericksen-Leslie model leads to improved spatial regularity for the time derivative of the velocity. This improved regularity for \(\partial_t v\) makes the passage to the limit possible without requiring further assumptions usually needed when treating the evolutionary Euler equation.