Global well-posedness of smooth solutions to the Landau-Lifshitz-Slonczewski equation

TL;DR

This paper proves the global existence and uniqueness of smooth solutions to the 3D Landau-Lifshitz-Slonczewski equation, using advanced mathematical techniques to handle nonlinearities and improve solution regularity.

Contribution

It introduces a novel approach to establish global well-posedness for the equation in Morrey space, extending local solutions and analyzing their regularity.

Findings

Global smooth solutions exist under certain initial conditions.

Boundedness of the magnetic gradient is established.

A global weak solution with improved regularity is obtained.

Abstract

In this paper, we mainly consider the global solvability of smooth solutions for the Cauchy problem of the three-dimensional Landau-Lifshitz-Slonczewski equation in the Morrey space. We derive the covariant complex Ginzburg-Landau equation by using moving frames to address the nonlinear parts. Applying the semigroup estimates and energy methods, we extend local classical solutions to global solutions and prove the boundedness of $\|\nabla\boldsymbol{m}\|_{L^{\infty}(\mathbb{R}^{3})}$, where $\boldsymbol{m}$ is the magnetic intensity. Moreover, we obtain a global weak solution by using an approximation result and improve the regularity of the obtained solution by the regularity theory. Finally, we establish the existence and uniqueness of global smooth solutions under some conditions on $\nabla\boldsymbol{m}_{0}$ and the density of the spin-polarized current.