Global existence of smooth solutions for the incompressible Landau–Lifshitz–Gilbert flow with Dzyaloshinskii–Moriya interaction in two‐dimensional torus and ℝ2

Abstract

This paper focuses on establishing the existence of smooth solutions for the incompressible Landau–Lifshitz–Gilbert equation with Dzyaloshinskii–Moriya (DM) interaction and V flow in two‐dimensional domains, including the torus and Euclidean space. The primary objective is to establish global existence results by investigating the conditions under which a smooth solution exists for all time, provided that the norm of the initial magnetization gradient is sufficiently small. Rigorous mathematical proofs for these global existence results are provided by combining analytical techniques and energy estimates. These findings enhance our understanding of solution behavior and regularity in the Landau–Lifshitz–Gilbert equation with DM interaction, shedding light on its dynamics in different spatial domains.