article published Balázs Kovács author 1004410000-0001-9872-3474 Buyang Li author 841 department <jats:title>Abstract</jats:title> <jats:p>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.</jats:p> Oxford University Press (OUP)2022 eng Applied MathematicsComputational MathematicsGeneral Mathematics 0272-4979 1464-364210.1093/imanum/drac033 Kovács B, Li B. Maximal regularity of backward difference time discretization for evolving surface PDEs and its application to nonlinear problems. <i>IMA Journal of Numerical Analysis</i>. Published online 2022. doi:<a href="https://doi.org/10.1093/imanum/drac033">10.1093/imanum/drac033</a> B. Kovács and B. Li, “Maximal regularity of backward difference time discretization for evolving surface PDEs and its application to nonlinear problems,” <i>IMA Journal of Numerical Analysis</i>, 2022, doi: <a href="https://doi.org/10.1093/imanum/drac033">10.1093/imanum/drac033</a>. Kovács, Balázs, and Buyang Li. “Maximal Regularity of Backward Difference Time Discretization for Evolving Surface PDEs and Its Application to Nonlinear Problems.” <i>IMA Journal of Numerical Analysis</i>, 2022. <a href="https://doi.org/10.1093/imanum/drac033">https://doi.org/10.1093/imanum/drac033</a>. @article{Kovács_Li_2022, title={Maximal regularity of backward difference time discretization for evolving surface PDEs and its application to nonlinear problems}, DOI={<a href="https://doi.org/10.1093/imanum/drac033">10.1093/imanum/drac033</a>}, journal={IMA Journal of Numerical Analysis}, publisher={Oxford University Press (OUP)}, author={Kovács, Balázs and Li, Buyang}, year={2022} } B. Kovács, B. Li, IMA Journal of Numerical Analysis (2022). Kovács, Balázs, and Buyang Li. “Maximal Regularity of Backward Difference Time Discretization for Evolving Surface PDEs and Its Application to Nonlinear Problems.” <i>IMA Journal of Numerical Analysis</i>, Oxford University Press (OUP), 2022, doi:<a href="https://doi.org/10.1093/imanum/drac033">10.1093/imanum/drac033</a>. Kovács, B., &#38; Li, B. (2022). Maximal regularity of backward difference time discretization for evolving surface PDEs and its application to nonlinear problems. <i>IMA Journal of Numerical Analysis</i>. <a href="https://doi.org/10.1093/imanum/drac033">https://doi.org/10.1093/imanum/drac033</a> 459642023-07-10T11:45:14Z2024-04-03T09:17:59Z