Gradient flow Liquid crystals Mountain Pass Theorem Least energy solution Strichartz estimates
Issue Date:
3-May-2022
Publisher:
Springer
Citation:
Barbera, D., Georgiev, V.. On standing waves and gradient-flow for the Landau–De Gennes model of nematic liquid crystals. European Journal of Mathematics, 8, Springer, 2022, ISSN:2199-675X, DOI:10.1007/s40879-022-00537-5, 672-699
Series/Report no.:
European Journal of Mathematics;8, 672–699
Abstract:
The article treats the existence of standing waves and solutions to gradient-flow equation for the Landau–De Gennes models of liquid crystals, a state of matter intermediate
between the solid state and the liquid one. The variables of the general problem are
the velocity field of the particles and the Q-tensor, a symmetric traceless matrix which
measures the anisotropy of the material. In particular, we consider the system without the velocity field and with an energy functional unbounded from below. At the
beginning we focus on the stationary problem. We outline two variational approaches
to get a critical point for the relative energy functional: by the Mountain Pass Theorem and by proving the existence of a least energy solution. Next we describe a
relationship between these solutions. Finally we consider the evolution problem and
provide some Strichartz-type estimates for the linear problem. By several applications
of these results to our problem, we prove via contraction arguments the existence of
local solutions and, moreover, global existence for initial data with small L2-norm.