Maximal regularity is a kind of a priori estimates for parabolic-type equations and it plays an important role in the theory of nonlinear differential equations. The aim of this paper is to investigate the temporally discrete counterpart of maximal regularity for the discontinuous Galerkin (DG) time-stepping method. We will establish such an estimate without logarithmic factor over a quasi-uniform temporal mesh. To show the main result, we introduce the temporally regularized Green's function and then reduce the discrete maximal regularity to a weighted error estimate for its DG approximation. Our results would be useful for investigation of DG approximation of nonlinear parabolic problems.

翻译：最大正则性是一类抛物型方程的先验估计，在非线性微分方程理论中具有重要作用。本文旨在研究间断Galerkin（DG）时间步进方法的时间离散最大正则性对应形式。我们将在拟均匀时间网格上建立无对数因子的此类估计。为证明主要结论，我们引入时间正则化格林函数，并将其DG近似的离散最大正则性转化为加权误差估计。本文结果对非线性抛物问题的DG近似研究具有重要应用价值。