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

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 $\mathbb{T}<sup>N$ ($N=2,3$), provided that the viscosity coefficients satisfy a BD-type algebraic relation ($μ(ρ) = νρ^α, λ(ρ) = 2ν(α-1)ρ<sup>α$) and the Korteweg stress tensor complies with a generalized Bohm identity ($$κ(ρ) = \varepsilon² α² ρ<sup>{2α-3}$$). 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 ($\mathbb{R}<sup>N$, $N=2,3$) 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 $\mathbb{T}<sup>N$ to unbounded whole space $\mathbb{R}<sup>N$, thus applicable to more general physical settings.

• The paper establishes the global existence and uniqueness of strong solutions to the multi-dimensional compressible NSK system, handling arbitrarily large initial perturbations.
• It employs refined truncation, weighted Nash–Moser iteration, and BD-entropy techniques to control density bounds and achieve higher-order regularity in unbounded domains.
• The analysis extends results from periodic domains to ℝⁿ, providing robust tools for further studies on vacuum dynamics and numerical stability in capillary fluid models.

Global Strong Solutions for the Multi-Dimensional Compressible Navier-Stokes-Korteweg System with Arbitrarily Large Initial Data

Introduction and Historical Overview

The paper "On the Cauchy problem for the multi-dimensional compressible Navier-Stokes-Korteweg system: Global strong solutions with arbitrarily large initial data" (2604.23652) addresses a core, longstanding open problem in the mathematical theory of compressible capillary fluids: the global well-posedness of strong solutions for the multi-dimensional compressible Navier-Stokes-Korteweg (NSK) system with arbitrarily large initial data in the unbounded spatial domain $\mathbb{R}^N$ ($N=2,3$).

The NSK system generalizes the compressible Navier-Stokes equations by including Korteweg-type capillarity effects, modeling internal forces associated with density gradients. Since the foundational works of Korteweg and later Dunn-Serrin, understanding the global dynamics of capillary fluids under general, large perturbations has required new techniques, especially due to the difficulties posed by nontrivial far-field conditions and domain noncompactness.

Early progress was made on periodic domains $\mathbb{T}^N$, where global strong solutions for large initial data were established under specialized algebraic relations between the viscosity and capillarity coefficients (Gu et al., 12 Mar 2026). However, the Cauchy problem in $\mathbb{R}^N$ remained unresolved, hampered by the lack of density integrability, difficulties in controlling far-field behavior, and the absence of compactness.

Problem Setting and Main Results

The paper rigorously studies the isentropic compressible NSK system in dimension $N=2$ or $N=3$: $$\begin{cases} \rho_t + \operatorname{div}(\rho u) = 0, \ (\rho u)_t + \operatorname{div}(\rho u \otimes u) + \nabla \rho^\gamma = \operatorname{div}(2\mu(\rho)\mathbb{D}u) + \nabla(\lambda(\rho) \operatorname{div} u) + \operatorname{div} \mathbb{K}, \end{cases}$$ with general adiabatic exponent $\gamma \geq 1$, density-dependent viscosity and capillarity coefficients, and the Korteweg stress tensor $\mathbb{K}$ embodying higher-order density derivatives.

The study focuses on the algebraic structure:

• $\mu(\rho) = \nu\,\rho^\alpha$, $N=2,3$0 (Bresch-Desjardins (BD) entropy structure)
• $N=2,3$1 (generalized Bohm identity) with $N=2,3$2 and detailed constraints on $N=2,3$3 to ensure physical admissibility and mathematical coercivity.

The initial data are only required to be regular perturbations of the far-field equilibrium state with arbitrary amplitude, i.e., $N=2,3$4, $N=2,3$5, and $N=2,3$6.

Main theorem: Under the stated coefficient structure and parameter constraints, the Cauchy problem for the NSK system in $N=2,3$7 ($N=2,3$8) admits a unique global strong solution for all $N=2,3$9, with

$\mathbb{T}^N$0

and full higher-order regularity for $\mathbb{T}^N$1 on any interval $\mathbb{T}^N$2.

This result extends the large-data strong solution theory from the periodic case to the unbounded spatial domain, without any smallness condition, for a broad class of nonlinear viscosity and capillarity structures.

Analytical Innovations and Technical Approach

The global theory in unbounded domains must overcome several severe challenges absent in the periodic case. Among the most technically demanding are:

• Integrability of the density and far-field behavior: Standard arguments relying on the finite measure of the domain are inapplicable.
• Uniform upper and lower bounds for the density: Vacuum formation and blowup cannot be excluded by compactness or maximum principles in the whole space.
• Propagation of regularity with general large data: Classic Picard/Fujita-Kato fixed-point iterations break down far from equilibrium.

The analysis employs several major methodological advances:

1. Piecewise Analysis of the Adiabatic Exponent $\mathbb{T}^N$3 (for $\mathbb{T}^N$4)

By decomposing the analysis according to intervals in $\mathbb{T}^N$5, the authors extract the optimal pressure contribution to the weighted momentum integrability estimates. This yields sharp ranges for allowable adiabatic exponents and permits extension to the critical regime as $\mathbb{T}^N$6.

2. Refined Truncation and Weighted Nash-Moser Iteration

To address the lack of integrability at infinity, the key a priori estimates are performed on truncated high/low density regions, with the supports dynamically controlled by energy and entropy inequalities. The main progression:

• Uniform density upper bound by Nash-Moser iteration on truncated density, leveraging critical $\mathbb{T}^N$7-momentum integrability.
• Density lower bound via a new Nash-Moser scheme for the truncated inverse density, tailored to the singular structure of the vacuum regime, exploiting improved control from the previous step.

This approach crucially relies on the relative entropy and effective velocity method, which ensures that the support of large/small density sets remains uniformly bounded in time, a nontrivial fact in unbounded domains.

3. Decoupling and Higher-Order Regularity

For $\mathbb{T}^N$8, the first-level higher-order energy estimates reveal a strongly coupled nonlinear structure, preventing direct closure. The authors introduce a Gronwall functional involving weighted $\mathbb{T}^N$9-norms of $\mathbb{R}^N$0, exploiting the dissipativity intrinsic to the BD structure and using dissipation from auxiliary derivatives as "damping" to extract higher regularity.

4. Iterative Closure and Blowup Criterion

All steps work via a contradiction argument: assuming finite maximal existence time, energy, entropy, and the Nash-Moser arguments show that all necessary quantities remain uniformly bounded, enabling a continuation procedure and thereby establishing global well-posedness.

Strong Analytical Estimates and Claims

The paper delivers quantitative results for several fundamental quantities. For all $\mathbb{R}^N$1, there exist constants depending on the system parameters and norms of the initial data such that:

• The density $\mathbb{R}^N$2 remains uniformly positive and bounded globally in space-time; no vacuum or concentration occurs.
• The solution $\mathbb{R}^N$3 exhibits the regularity

$\mathbb{R}^N$4

and analogously for the time derivatives.

• Solutions persist for arbitrarily large data under the only assumption of an $\mathbb{R}^N$5 initial perturbation from equilibrium, without any smallness condition.

The admissible ranges of $\mathbb{R}^N$6 precisely match the best known or expected in the periodic setting. Notably, the critical value of $\mathbb{R}^N$7 in the quantum case ($\mathbb{R}^N$8) is recovered as the limiting case.

Implications and Perspective

The extension of the global strong solution theory to the unbounded Cauchy problem for the multi-dimensional compressible NSK system is significant on several mathematical and physical grounds:

• It removes the bounded domain restriction and shows that strongly nonlinear density-dependent viscosity and capillarity, together with BD-structure, suffice to prevent both blowup and vacuum even for large perturbations.
• The structural conditions on the coefficients are optimal in the sense of compatibility with energy and BD-entropy methods, indicating a deep linkage between modern entropy-based PDE analysis and physically relevant models.
• The Nash-Moser methodology in combination with careful truncation emerges as a highly robust framework for addressing global problems in unbounded domains; its adaptability may influence subsequent work on related degenerate diffusive/hyperbolic systems.
• The result closes a major gap in the regularity theory for compressible quantum fluids and capillary fluids, and may have implications for numerical analysis, stability of sharp interface models, and asymptotic limits (e.g., vanishing capillarity or viscosity).

Future Directions

Several open avenues are highlighted by this work:

• Vacuum and phase boundary dynamics: Analysis outside the strictly positive density regime, particularly for initial data allowing vacuum, remains a challenging direction.
• Weak-Strong uniqueness for the NSK system with general coefficients on the whole space, exploiting the established strong theory.
• Extension to less regular initial data and other nonlinearities in constitutive relations.
• Numerical implications: Development of robust schemes that inherit global-in-time stability properties suggested by the analytical theory.

Conclusion

This work establishes, for the first time, the global existence and uniqueness of strong solutions to the multi-dimensional compressible Navier-Stokes-Korteweg system in the unbounded domain for arbitrarily large initial data, under a broad and physically relevant class of density-dependent nonlinearities. The proof hinges on new techniques in truncation, weighted iteration, and entropy methods, providing a rigorous foundation for further mathematical and physical investigation of compressible capillary flows in realistic settings.