From the Q-Tensor Flow for the Liquid Crystal to the Harmonic Map Flow

Abstract

In this paper, we consider the solutions of the relaxed Q-tensor flow in $${\mathbb{R}^3}$$ with small parameter $${\epsilon}$$ . We show that the limiting map is the so-called harmonic map flow. As a consequence, we present a new proof for the global existence of a weak solution for the harmonic map flow in three dimensions as in [18, 23], where the Ginzburg–Landau approximation approach was used.