AI 中文总结
该研究针对空间维数≤3的等温纳维-斯托克斯-科特韦格流，证明能量稳定且非退化的孤立平衡态指数稳定，非孤立平衡态附近解指数收敛，应用了Prüss等人2013年的稳定性原理。
AI 中文摘要
本文研究有界区域或周期单元中的纳维-斯托克斯-科特韦格方程，考虑的压力相对于密度可能不单调，因此存在允许两相的非常数平衡态。采用简单希尔伯特空间框架，证明当空间维数不超过3时，若孤立平衡态是能量稳定且非退化的，则在等温纳维-斯托克斯-科特韦格流下指数稳定；对于非孤立情况，证明靠近能量稳定平衡态的全局时间解会指数快速收敛到可能的另一平衡态，且未对平衡态施加小性假设。证明中应用了J. Prüss、M. Wilke和G. Simonett（2013）提出的（广义）稳定性原理。
英文摘要
We consider the Navier-Stokes-Korteweg equations in a bounded domain or a periodic cell. The pressure considered in this paper may not be monotone with respect to the density so that there exist non-constant equilibria allowing two-phases. Using a simple Hilbert space framework, we prove that if an isolated equilibrium is energetically stable and non- degenerate, it is exponentially stable under the isothermal Navier-Stokes-Korteweg flows when the space dimension is less than or equal to three. For non-isolated case, we prove that a global-in-time solution near an energetically stable equilibrium converges to possibly another equilibrium exponentially fast. No smallness assumptions on equilibria are imposed. For the proof we apply a (generalized) stability principle due to J. Prüss, M. Wilke and G. Simonett (2013).
