On the unconditional long-time L2 stability of the SAV-BDF2 time-stepping schemes for the forced Navier–Stokes equations with nonsmooth initial data

In this work, the long-time $L^2$ stability of the SAV-BDF2 time-stepping scheme for the forced Navier–Stokes equations under a regularization of the initial data $u_0\in H$ using a method that lifts the solution to the space $H^1$ while preserving control over the $L^2$ norm is investigated. In this paper, the unconditional long-time $L^2$ stability of the scheme is established by utilizing a refined estimate of the $G$-norm introduced in Shiue (2025), together with recent results from Han and Wang (2024). Moreover, the uniform-in-time bound on the energy norm of the solution is derived independent on the time step size, which allows for larger time steps to speed up simulations, especially for long-term problem. This also improves the result in Han and Wang (2024) where a uniform-in-time bound on the energy norm was obtained under a condition involving the time step size and assuming the initial data is in $H^1$ directly.