Global existence of unique solutions to equations for pattern formation in active mixtures

Abstract

In this paper, we deal with two models for pattern formation in active system on the d-dimensional torus Td=−ππd, d ≥ 2, with the periodic boundary conditions.(1)We first consider the model in (Lee and Kardar, 2001) describing the density and the tubule orientation field. After perturbing the orientation field around (1,0), we show that there is a unique global-in-time solution to the perturbed model when initial data is sufficiently small in the energy space H2.(2)The second model under consideration is Active model C in (Maryshev et al., 2020). In this case, we perturb the density around 12, which generates a damping term, and we prove that there is a unique global-in-time solution when initial data is sufficiently small in Wiener space A0.Although these two models are treated in different ways, both models are presented in this paper from the perspective of dealing with pattern formation phenomenon.