二维和三维广义可压缩Navier-Stokes-Korteweg系统在任意大初始数据下强解的长时间行为
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
AI总结：
本文研究二维和三维广义可压缩Navier-Stokes-Korteweg系统，在任意大初始数据下建立强解的整体存在性与长时间行为，扩展了相关可容许参数范围并推导关键估计。
AI中文摘要：
本文针对二维和三维周期可压缩Navier-Stokes-Korteweg系统，在任意大初始数据(ρ₀,u₀)∈H³×H²下，建立了强解的整体存在性与长时间行为。其中黏性系数满足BD关系μ(ρ)=νρ^α、λ(ρ)=2ν(α-1)ρ^α，毛细系数为κ(ρ)=ε²α²ρ^(2α-3)。对于α<1的情况，我们利用密度-有效速度系统的双抛物结构，扩展了Gu-Huang-Meng-Zhou[arXiv:2603.11762 (2026)]中强解整体存在性的可容许参数范围；随后开发了时间离散策略，建立有效速度的一致可积性估计，得到密度的一致上界；进一步引入新颖的bootstrap论证逐步改进可积性估计，得到密度的一致正下界；最终推导时间全局的高阶估计，证明在不对初始数据施加任何小性假设的情况下，长时间行为满足∥ρ(t)-1/|T^N|∫_{T^N}ρ₀dx∥_{H³}+∥∇u(t)∥_{H¹}→0（t→∞）。对于临界情况α=1，我们改进了Huang-Meng-Zhang[arXiv:2602.00455 (2026)]中的可容许参数范围，建立了密度的一致上界。
英文摘要：
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.
