2023/01/16 by Huang, Jiaxi, Jiang, Ning, Zhao, Lifeng
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2301.06235
This article is devoted to the two dimensional simplified Ericksen-Leslie's hyperbolic system for incompressible liquid crystal model, where the direction d of liquid crystal molecules satisfies a wave map equation with an acoustical metric. We established the almost global well-posedness for small and smooth initial data near the constant equilibrium. Our proof relies on the idea of vector-field method and ghost weight method. There are two key ingredients in our proof: (i) Inspired by the gauge theory in Tataru \citeTataru,Tataru05, we reformulate the wave map equation into a free wave equation with acoustical metric, where the nonlinearity is annihilated due to the geometry of \mathbb S1; (ii) Motivated by the ghost weight method in Alinhac \citeA01, we introduce a new and important ``good unknown", the velocity u, which provides the additional dissipation u/⟨ t-r⟩∈ L2tL2x. These new observations turn out to be extremely crucial in resolving the system in low dimensions.