On the large-time behavior of strong solutions to the generalized compressible Navier-Stokes-Korteweg system in 2D and 3D for arbitrarily large initial data

Abstract

In this paper, we establish the global existence and large-time behavior of strong solutions for the two- and three-dimensional periodic compressible Navier-Stokes-Korteweg system with arbitrarily large initial data $(ρ_0,u_0)\in H^3\times H^2$. The viscosity coefficients satisfy the BD relation $μ(ρ)=νρ^α$ and $λ(ρ)=2ν(α-1)ρ^α$, while the capillarity coefficient is given by $κ(ρ)=\varepsilon^2α^2ρ^{2α-3}$. For the case $α<1$, we first enlarge the admissible parameter range for the global existence of strong solutions established in Gu-Huang-Meng-Zhou [arXiv:2603.11762 (2026)] by exploiting the doubly parabolic structure of the density-effective velocity system. We then develop a time-discretization strategy to establish uniform integrability estimates for the effective velocity, yielding a uniform upper bound for the density. Furthermore, we introduce a novel bootstrap argument to successively improve these integrability estimates, which leads to a uniform positive lower bound for the density. Finally, we derive global-in-time higher-order estimates and prove the large-time behavior \[ \left\|ρ(t)-\frac{1}{|\mathbb{T}^N|}\int_{\mathbb T^N}ρ_0 dx\right\|_{H^3} +\|\nabla u(t)\|_{H^1} \longrightarrow0, \qquad t\to\infty, \] without imposing any smallness assumption on the initial data. For the critical case $α=1$, we improve the admissible parameter range established in Huang-Meng-Zhang [arXiv:2602.00455 (2026)] and establish a uniform upper bound for the density.