Global-in-time energy stability: a powerful analysis tool for the gradient flow problem without maximum principle or Lipschitz assumption

TL;DR

This paper introduces a new analytical approach called global-in-time energy stability that proves energy dissipation for gradient flows without requiring Lipschitz conditions or boundedness, demonstrated on the Swift-Hohenberg equation.

Contribution

The paper develops a novel method for establishing energy stability in gradient flows without strong assumptions, applicable to complex equations like Swift-Hohenberg.

Findings

Successfully proves energy stability without Lipschitz or boundedness assumptions.

Proposes a second-order accurate temporal scheme for stiff equations.

Provides optimal L^2 error estimates and numerical simulations.

Abstract

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.