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.

中文翻译：

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演化表面偏微分方程的后向差分时间离散最大正则及其在非线性问题中的应用

通过将问题拉回到初始曲面并研究固定曲面上的最大 $L^p$-正则性来显示演化曲面上线性抛物方程的最大抛物线 $L^p$-正则性。通过在固定时间冻结抛物方程中的系数并利用冻结时间周围的扰动参数，表明线性抛物方程在沿特征轨迹的演化表面上的反向差分时间离散可以保持最大 $L^p$-离散设置中的规律性。该结果用于证明非线性抛物方程的时间离散化在演化曲面上的稳定性和收敛性，对于一般的局部 Lipschitz 非线性，具有曲面的线性隐式反向微分公式特征轨迹。离散极大 $L^p$-regularity 用于证明 $L^\infty (0,T;W^{1,\infty })$ 范数中数值解的有界性和稳定性，用于有界稳定性分析中的非线性项。$L^\infty (0,T;W^{1,\infty })$ 范数中时间离散化的最优阶误差估计是通过将稳定性分析与一致性估计相结合获得的。