Stability of the Melting Hedgehog in the Landau-de Gennes Theory of Nematic Liquid Crystals

We investigate stability properties of the radially symmetric solution corresponding to the vortex defect (the so called "melting hedgehog") in the framework of the Landau-de Gennes model of nematic liquid crystals. We prove local stability of the melting hedgehog under arbitrary Q-tensor valued perturbations in the temperature regime near the critical supercooling temperature. As a consequence of our method, we also rediscover the loss of stability of the vortex defect in the deep nematic regime.