Invariant measures for the one-dimensional stochastic Navier-Stokes-Korteweg equations
Abstract
We investigate the long-time behaviour of a one-dimensional compressible viscous fluid with general capillarity and density dependent viscosity, driven by a stochastic force. In particular, we prove the existence of invariant measures by applying the Krylov-Bogoliubov method in a setting where the dynamics is supported on a non-complete phase space. This analysis is further enhanced by the derivation of a refined stability result determining the continuous dependence with respect to the initial data. The present paper exhibits some properties and results for Korteweg fluids which are not known in absence of the capillarity tensor. In particular, we prove that the Markov semigroup associated with strong solutions is Feller and we can consider ranges of the adiabatic and viscosity exponents γ and α larger than those available in the current ergodic literature for compressible fluids. Also the interplay between the choice of the physical domain and the use of a damping term is discussed.
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BibTeX
@misc{donatelli2026invariant,
title = {{Invariant measures for the one-dimensional stochastic Navier-Stokes-Korteweg equations}},
author = {Donatella Donatelli and Lorenzo Pescatore and Stefano Spirito},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2606.07423},
eprint = {2606.07423},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.07423},
}
RIS
TY - UNPB
TI - Invariant measures for the one-dimensional stochastic Navier-Stokes-Korteweg equations
AU - Donatella Donatelli
AU - Lorenzo Pescatore
AU - Stefano Spirito
PY - 2026
DA - 2026/10/26
PB - arXiv
DO - 10.48550/arxiv.2606.07423
UR - https://arxiv.org/abs/2606.07423
AB - We investigate the long-time behaviour of a one-dimensional compressible viscous fluid with general capillarity and density dependent viscosity, driven by a stochastic force. In particular, we prove the existence of invariant measures by applying the Krylov-Bogoliubov method in a setting where the dynamics is supported on a non-complete phase space. This analysis is further enhanced by the derivation of a refined stability result determining the continuous dependence with respect to the initial data. The present paper exhibits some properties and results for Korteweg fluids which are not known in absence of the capillarity tensor. In particular, we prove that the Markov semigroup associated with strong solutions is Feller and we can consider ranges of the adiabatic and viscosity exponents γ and α larger than those available in the current ergodic literature for compressible fluids. Also the interplay between the choice of the physical domain and the use of a damping term is discussed.
ER -