On the regularity of the \(Q\)-tensor depending on the data (Q2827621)

This is a report of the thirteenth International Conference Zaragoza-Pau on Mathematics and its Applications (Jaca, September 15--18, 2014).NEWLINENEWLINEThe authors consider the Landau-De Gennes theory, which is a model to study the behavior of nematic liquid crystals filling a bounded domain \(\Omega\subset\mathbb{R}^3\). Let \(u : (0; T)\times \mathbb{R}^3\longrightarrow \mathbb{R}^3\) be the flow velocity, \(p : (0; T)\times \mathbb{R}^3\longrightarrow \mathbb{R}\) be the pressure, and \(Q : (0; T)\times \mathbb{R}^3\longrightarrow \mathbb{R}^{3\times 3}\) be a symmetric and trace-zero tensor describing the orientation of the liquid molecules. One considers the following system: NEWLINE\[NEWLINE D_t u-\mu\Delta u +\nabla p=\nabla\cdot\tau (Q)+\nabla\cdot\sigma (Q), \nabla\cdot u=0, D_t Q-S(\nabla u,Q)=-\gamma H(Q). NEWLINE\]NEWLINE Here, \(D_t=\partial_t +u\cdot\nabla\) is the material derivative, \(\mu >0\) is the viscosity coefficient, and \(\gamma > 0\) is an elastic constant. Furthermore, \(\tau (Q)\), \(\sigma (Q)\), and \(S(\nabla u,Q)\) are some tensors. The authors consider an initial-boundary value problem for this system, for which only a few regularity results are known. The authors give several new types of the regularity results of the solutions, containing particular results for the Dirichlet problem and the Neumann problem.NEWLINENEWLINEFor the entire collection see [Zbl 1344.00012].