Maximal parabolic $L^p$-regularity of linear parabolic equations on an evolving surface is shown by pulling back the problem to the initial surface and studying the maximal $L^p$-regularity on a fixed surface. By freezing the coefficients in the parabolic equations at a fixed time and utilizing a perturbation argument around the freezed time, it is shown that backward difference time discretizations of linear parabolic equations on an evolving surface along characteristic trajectories can preserve maximal $L^p$-regularity in the discrete setting. The result is applied to prove the stability and convergence of time discretizations of nonlinear parabolic equations on an evolving surface, with linearly implicit backward differentiation formulae characteristic trajectories of the surface, for general locally Lipschitz nonlinearities. The discrete maximal $L^p$-regularity is used to prove the boundedness and stability of numerical solutions in the $L^\infty(0,T;W^{1,\infty})$ norm, which is used to bound the nonlinear terms in the stability analysis. Optimal-order error estimates of time discretizations in the $L^\infty(0,T;W^{1,\infty})$ norm is obtained by combining the stability analysis with the consistency estimates.

翻译：正在变化的表面的线性抛物线方程式的偏差值 $L ⁇ p$-规律性通过将问题拉回初始表面和研究固定表面的最大值$L ⁇ p$-规律性来显示。通过在固定时间冻结抛物方方程式的系数,并在冻结时间周围使用扰动参数参数参数,可以显示在特征轨轨迹上变化的表面线性抛物方方程式的反向差异时间分解可以保持离散设置中的最大值$L ⁇ p$-规律性。结果用于证明非线性抛物方方方方程式在变化表面的时间分解的稳定性和趋同性,对普通Lipschitz非线性参数则使用直隐含的后向差异方方方方方方方程式特性。使用离差最大值 $L ⁇ p- 常规性来证明美元(0,T;W ⁇ 1,\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\