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

Abstract

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.

arXiv commentContribution to the Oberwolfach Report for the Seminar 'Stochastic Partial Differential Equations in Critical Spaces' organised by Antonio Agresti and Mark Veraar