AI 中文总结
该研究针对外部域上的径向对称NSK方程不可渗透壁问题，证明当初值为定态解的小扰动且边界数据足够小时，存在全局时间强解并渐近收敛至定态解，所用方法结合了基本能量估计与精心设计的能量泛函。
AI 中文摘要
我们研究定义在外部域Ω={x∈ℝⁿ||x|>1}上的径向对称Navier-Stokes-Korteweg（NSK）方程初边值问题的渐近行为，具体考虑不可渗透壁问题，即边界{x∈ℝⁿ||x|=1}处的速度设为零。我们证明，若初值是定态解的小扰动且边界数据足够小，则径向对称NSK方程存在全局时间强解，且该解随时间渐近收敛至定态解。我们的方法基于基本能量估计，并结合精心设计的能量泛函。
英文摘要
We study the asymptotic behavior of the initial-boundary value problem for the radially symmetric Navier--Stokes--Korteweg (NSK) equations defined on the exterior domain $Ω= \{x\in\R^n~|~|x|> 1\}$. In particular, we consider the impermeable wall problem, where the velocity at the boundary $\{x\in\R^n~|~|x|=1\}$ is set to be zero. We show that, if the initial data is a small perturbation of the stationary solution, and the boundary data are sufficiently small, then there exists a global-in-time strong solution to the radially symmetric NSK equations, and it converges to the stationary solution time-asymptotically. Our method is based on elementary energy estimates with a combination of carefully designed energy functionals.
Comments24 pages
