Global well-posedness for Ericksen-Leslie system with zero viscosity

Abstract

We study the global well-posedness of the simplified Ericksen-Leslie system with zero viscosity on the periodic domain T 2 \mathbb {T}^2 . Precisely, we prove the global existence and uniqueness of smooth solution to Ericksen-Leslie system if the initial data ( u 0 , ∇ d 0 ) (u_0,\nabla d_0) is small in H 4 ( T 2 ) × H 4 ( T 2 ) H^4(\mathbb {T}^2)\times H^4(\mathbb {T}^2) . Furthermore, we derive the time decay estimate of ∇ d \nabla d in H 1 ( T 2 ) H^1(\mathbb {T}^2) .