Well-Posedness and Regularity for a Modified Liquid Crystal Model with Transport Noise

We prove well-posedness for a stochastic Lin-Liu model of modified liquid crystals, coupling an incompressible fluid to a director field under transport and multiplicative noise. On smooth bounded domains in two dimensions, we prove global well-posedness for energy-class initial data, provided the leading coefficient of the polynomial director potential is positive. On the torus, we establish local well-posedness in time-weighted $L^p$-$L^q$ frameworks with Besov initial data, including regimes of low and negative regularity.