中文摘要：

利用Galerkin方法得到了周期边值问题的局部光滑解，然后在小初值的条件下对光滑解做关于时间的整体先验估计，得到了二维Landau－Lifshitz－Darwin方程组在小初值条件下的整体光滑解。

英文摘要：

The local existence of smooth solution with the periodic initial value condition is firstly obtained by usingGalerkin method. Based on it, the global existence of the smooth solution for the 2 - dimensional Landau - Lifshitz-Darwin coupled system with small initial data is further derived by making a priori estimate globally in time.