Low Mach number limit for the Navier--Stokes--Korteweg equations with a stationary force
Abstract
In this paper, we investigate the low Mach number limit for the three-dimensional compressible Navier--Stokes--Korteweg equations in the whole space under a small stationary external force. We first construct a family of small stationary solutions uniformly with respect to the Mach number ε and prove that both the stationary density fluctuation and the compressible component of the stationary velocity are of order ε2. For ill-prepared non-stationary perturbations around these stationary solutions, we establish the existence and uniqueness of global strong solution by combining uniform high-order energy estimates with a low-frequency Besov estimate and a Kawashima-type compensating functional. The main difficulty is that Korteweg tensor not only changes the elliptic structure of the stationary problem, but also modifies the dispersive mechanism of the acoustic modes. In Korteweg-symmetric variables, the associated spectral projections are uniformly bounded zero-order Fourier multipliers, while the acoustic-capillary phase is wave-like at low frequencies and Schrödinger-like at high frequencies. Since the source terms generated by the stationary coefficients are generally not integrable in time, we decompose the Duhamel source according to its time-integrability and frequency behavior. Dyadic dispersive estimates, high-frequency damping estimates, and maximal regularity for the heat equation yield the global-in-time convergence rate εmin{1/r, 1/2−1/p} in the mixed Besov norms Lr(0,∞;Ḃsp,1). As a consequence, Besov embeddings also yield quantitative convergence in the mixed Lebesgue norms Lr(0,∞;Lp).
Related
- Invariant measures for the one-dimensional stochastic Navier-Stokes-Korteweg equations
- Liouville theorems for the stationary Navier-Stokes equations under partial component conditions
- uniqueness and asymptotic behavior of the stationary Navier-Stokes equations in a slab
- Liouville Theorems for Stationary Navier-Stokes Equations via the Radial Velocity Component
- On the effects of rotation on stationary Navier--Stokes equations in the half spaces in scale-critical spaces
BibTeX
@misc{ni2026mach,
title = {{Low Mach number limit for the Navier--Stokes--Korteweg equations with a stationary force}},
author = {Jinkai Ni and Luqi Wang and Yichi Zhang and Zhipeng Zhang},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.00727},
eprint = {2608.00727},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.00727},
}
RIS
TY - UNPB
TI - Low Mach number limit for the Navier--Stokes--Korteweg equations with a stationary force
AU - Jinkai Ni
AU - Luqi Wang
AU - Yichi Zhang
AU - Zhipeng Zhang
PY - 2026
DA - 2026/08/30
PB - arXiv
DO - 10.48550/arxiv.2608.00727
UR - https://arxiv.org/abs/2608.00727
AB - In this paper, we investigate the low Mach number limit for the three-dimensional compressible Navier--Stokes--Korteweg equations in the whole space under a small stationary external force. We first construct a family of small stationary solutions uniformly with respect to the Mach number ε and prove that both the stationary density fluctuation and the compressible component of the stationary velocity are of order ε^2. For ill-prepared non-stationary perturbations around these stationary solutions, we establish the existence and uniqueness of global strong solution by combining uniform high-order energy estimates with a low-frequency Besov estimate and a Kawashima-type compensating functional. The main difficulty is that Korteweg tensor not only changes the elliptic structure of the stationary problem, but also modifies the dispersive mechanism of the acoustic modes. In Korteweg-symmetric variables, the associated spectral projections are uniformly bounded zero-order Fourier multipliers, while the acoustic-capillary phase is wave-like at low frequencies and Schrödinger-like at high frequencies. Since the source terms generated by the stationary coefficients are generally not integrable in time, we decompose the Duhamel source according to its time-integrability and frequency behavior. Dyadic dispersive estimates, high-frequency damping estimates, and maximal regularity for the heat equation yield the global-in-time convergence rate ε^min{1/r, 1/2−1/p} in the mixed Besov norms L^r(0,∞;Ḃ^s_p,1). As a consequence, Besov embeddings also yield quantitative convergence in the mixed Lebesgue norms L^r(0,∞;L^p).
ER -