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

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. When the initial data $ \boldsymbol{m}_{0}\in M_{*}^{2, 3, \infty}(\mathbb{R}^{3})\cap M_{*}^{3, 3, \infty}(\mathbb{R}^{3}) $, 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})} $ under some conditions on $ \nabla\boldsymbol{m}_{0} $ and the density of the spin-polarized current, where $ \boldsymbol{m} $ is the magnetic intensity. Moreover, when the initial data $ \boldsymbol{m}_{0}\in M_{*}^{2, 3, 1}(\mathbb{R}^{3})\cap M_{*}^{3, 3, 1}(\mathbb{R}^{3}) $, we obtain a global weak solution by using an approximation. Finally, we establish the existence and uniqueness of global smooth solutions via the regularity theory of weak solutions