On the Cauchy problem for the multi-dimensional compressible Navier-Stokes-Korteweg system: Global strong solutions with arbitrarily large initial data

TL;DR

This paper proves the global existence of strong solutions for the multi-dimensional compressible Navier-Stokes-Korteweg system with large initial data in the whole space, extending previous results from periodic domains.

Contribution

It establishes the first global strong solution existence for the Cauchy problem with large data, using a refined truncation and Nash-Moser iteration scheme.

Findings

Proved global strong solutions for the Cauchy problem in  and 3 dimensions.

Extended large-data theory from bounded domains to the whole space.

Overcame integrability issues with a novel analytical approach.

Abstract

Since the pioneering work of Korteweg (1901) and the subsequent refinement of capillary fluid models by Dunn and Serrin (1985), the global existence of strong solutions to the multi-dimensional compressible Navier-Stokes-Korteweg (NSK) system with arbitrarily large initial data has stood as a formidable open problem in fluid mechanics. This challenge was recently addressed by [Gu-Huang-Meng-Zhou, arXiv:2603.11762], who established the global existence of strong solutions for arbitrarily large initial data on the periodic domain $\mathbb{T}^N$ ($N=2,3$), provided that the viscosity coefficients satisfy a BD-type algebraic relation ($\mu(\rho) = \nu \rho^\alpha, \lambda(\rho) = 2\nu(\alpha-1)\rho^\alpha$) and the Korteweg stress tensor complies with a generalized Bohm identity ($\kappa(\rho) = \varepsilon^2 \alpha^2 \rho^{2\alpha-3}$). However, the existence of global strong solutions for the Cauchy problem under these conditions has remained an open question. In this paper, we resolve this problem by proving the global existence of strong solutions for the Cauchy problem ($\mathbb{R}^N$, $N=2,3$) with arbitrarily large initial data and non-vacuum far-field density. By employing a refined truncation analysis combined with an original modified Nash-Moser type iteration scheme, we overcome the difficulties arising from the lack of integrability for the density in the whole space. This result extends the large-data theory of compressible Navier-Stokes-Korteweg equations from bounded torus $\mathbb{T}^N$ to unbounded whole space $\mathbb{R}^N$, thus applicable to more general physical settings.