Global-in-time energy stability: a powerful analysis tool for the gradient flow problem without maximum principle or Lipschitz assumption
<|reference_start|>Global-in-time energy stability: a powerful analysis tool for the gradient flow problem without maximum principle or Lipschitz assumption: Before proving (unconditional) energy stability for gradient flows, most existing studies either require a strong Lipschitz condition regarding the non-linearity or certain $L^{\infty}$ bounds on the numerical solutions (the maximum principle). However, proving energy stability without such premises is a very challenging task. In this paper, we aim to develop a novel analytical tool, namely global-in-time energy stability, to demonstrate energy dissipation without assuming any strong Lipschitz condition or $L^{\infty}$ boundedness. The fourth-order-in-space Swift-Hohenberg equation is used to elucidate the theoretical results in detail. We also propose a temporal second-order accurate scheme for efficiently solving such a strongly stiff equation. Furthermore, we present the corresponding optimal $L^2$ error estimate and provide several numerical simulations to demonstrate the dynamics.<|reference_end|>
@article{sun2024global-in-time,
    title={Global-in-time energy stability: a powerful analysis tool for the
  gradient flow problem without maximum principle or Lipschitz assumption},
    author={J.Sun, H.Wang, H.Zhang, X.Qian, and S.Song},
    journal={arXiv preprint arXiv:2406.07941},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.07941},
    primaryClass={math.NA cs.NA}
}