Global smooth solutions for the Landau-Lifshitz-Bloch equation with helicity term (Q2161435)

The authors show the global existence and uniqueness of smooth solutions for the Cauchy problem of the Landau-Lifshitz-Bloch equations with a helicity term: \[ \dfrac{\partial \mathbf{m}}{\partial t} = \lambda ( \mathbf{m}\times \mathbf{H}_{\mathrm{eff}}) +L_1 \dfrac{(\mathbf{m}\cdot \mathbf{H}_{\mathrm{eff}})\mathbf{m}}{M^2}-L_2 \dfrac{( \mathbf{m}\times (\mathbf{m}\times \mathbf{H}_{\mathrm{eff}}))}{M^2} \] where \(M=|\mathbf{m}|\), the \(L\)'s are positive constants, and the effective field \(\mathbf{H}_{\mathrm{eff}}\) is given by the variational derivative of the total energy. Here the authors consider the effective field given by \[ \mathbf{H}_{\mathrm{eff}} = \Delta \mathbf{m} +\nabla \times \mathbf{m} -\dfrac{1}{\chi_1}\mathbf{m}^1 -\dfrac{1}{\chi_2} \mathbf{m}^2-\dfrac{1}{2\chi_{\parallel}}(|\mathbf{m}|^2+1)\mathbf{m} \] where the \(\chi\)'s denote constants, \(\mathbf{m}_1 = (m^1, 0,0)\) and \(\mathbf{m}^2=(0, m^2, 0)\). The authors develop \textit{a priori} estimates and use the continuity method to obtain global smooth solutions.