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arXiv · 2608.05654

On dynamic stability of energetically stable equilibria of the Navier-Stokes-Korteweg flows

Yoshikazu Giga·Naoto Kajiwara·Kazuyuki Tsuda
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Abstract

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. Pr\"uss, M. Wilke and G. Simonett (2013).

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Yoshikazu Giga, Naoto Kajiwara, Kazuyuki Tsuda. 2026-08-06. On dynamic stability of energetically stable equilibria of the Navier-Stokes-Korteweg flows. https://arxiv.org/abs/2608.05654

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