Maximal regularity of backward difference time discretization for evolving surface PDEs and its application to nonlinear problems

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