Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model
|Author||Guillén González, Francisco Manuel
Rodríguez Bellido, María Ángeles
Rojas Medar, Marko Antonio
|Department||Universidad de Sevilla. Departamento de Ecuaciones Diferenciales y Análisis Numérico|
|Published in||Mathematische Nachrichten, 282 (6), 846-867.|
|Abstract||In  L. C. Berselli, On a Regularity Criterion for the Solutions to the 3D Navier-Stokes Equations, Diff. and Integral Eq., Vol. 15, Number 9, 1129-1137 (2002). , L. Berselli showed that the additional regularity hypothesis ...
In  L. C. Berselli, On a Regularity Criterion for the Solutions to the 3D Navier-Stokes Equations, Diff. and Integral Eq., Vol. 15, Number 9, 1129-1137 (2002). , L. Berselli showed that the additional regularity hypothesis for the velocity gradient ∇u ∈ L 2q 2q−3 (0, T;L q (Ω)), for some q ∈ (3/2, +∞], implies the strong regularity for the weak solutions of the Navier-Stokes equations. In this work, we prove that such hypothesis is also sufficient in order to obtain the strong solution for a nematic Liquid Crystal model (a coupled system of velocity u and orientation crystals vector d) when periodic boundary conditions for d are considered. For Neumann and Dirichlet boundary conditions, we obtain the same result only for the cases of q ∈ [2, 3], whereas in the cases q ∈ (3/2, 2) ∪ (3, +∞], we also need to impose an additional regularity hypothesis for d (either on ∇d or ∆d). On the other hand, when the following hypothesis for u of Serrin’s type is imposed ( J. Serrin, On the interior regularity of weak solutions of the Navier-Stokes equations, Arch. Rat. Mech. Anal. 9 (3), 187-195 (1962): u ∈ L 2p p−3 (0, T;L p (Ω)) for some p ∈ (3, +∞], we can obtain strong regularity only in the case of periodic boundary conditions for d. For Neumann or Dirichlet boundary conditions, additional regularity for d must be imposed in all cases.
|Cite||Guillén González, F.M., Rodríguez Bellido, M.Á. y Rojas Medar, M.A. (2009). Sufficient conditions for regularity and uniqueness of a 3D nematic liquid crystal model. Mathematische Nachrichten, 282 (6), 846-867.|
Editor´s version: http://dx.doi.org/10.1002/mana.200610776