Global-in-time strong solutions for the 2D and 3D generalized compressible Navier-Stokes-Korteweg system with arbitrarily large initial data
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| Format: | Preprint |
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2026
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| _version_ | 1866917335021912064 |
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| author | Gu, Yongteng Huang, Xiangdi Meng, Weili Zhou, Huitao |
| author_facet | Gu, Yongteng Huang, Xiangdi Meng, Weili Zhou, Huitao |
| contents | In 1901, Korteweg formulated a constitutive equation for the Cauchy stress tensor to provide a continuum mechanical model for capillarity within fluids. Dunn and Serrin [Arch. Ration. Mech. Anal. 88(2):95-133,1985] in 1985 further modified the system of compressible fluids based on the Korteweg theory of capillarity. Since then, for the 2D and 3D compressible Navier-Stokes-Korteweg system, the global existence of strong solutions with arbitrarily large initial data have remained a challenging open problem. In this paper, we provide an affirmative answer to this longstanding open problem. Specifically, under the assumption that the viscosity coefficients satisfy a BD-type algebraic relation of the form $μ(ρ)=νρ^α$ and $λ(ρ)=2ν(α-1)ρ^α$, and that the Korteweg stress tensor complies with a generalized Bohm identity of the form $κ(ρ)=\varepsilon^2α^2ρ^{2α-3}$, we establish the global existence of strong solutions for the 2D and 3D systems in torus with arbitrarily large regular initial data. The analysis is carried out in the intermediary non-dispersive regime, characterized by the condition that the capillarity coefficient constant $\varepsilon$ does not exceed the viscosity constant $ν$. This result provides the first proof of the global-in-time existence of strong solutions for the 3D general Navier-Stokes-Korteweg system with arbitrarily large initial data in the non-dispersive regime. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_11762 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Global-in-time strong solutions for the 2D and 3D generalized compressible Navier-Stokes-Korteweg system with arbitrarily large initial data Gu, Yongteng Huang, Xiangdi Meng, Weili Zhou, Huitao Analysis of PDEs 35D35, 35Q30, 35Q35, 35Q40, 76N10 In 1901, Korteweg formulated a constitutive equation for the Cauchy stress tensor to provide a continuum mechanical model for capillarity within fluids. Dunn and Serrin [Arch. Ration. Mech. Anal. 88(2):95-133,1985] in 1985 further modified the system of compressible fluids based on the Korteweg theory of capillarity. Since then, for the 2D and 3D compressible Navier-Stokes-Korteweg system, the global existence of strong solutions with arbitrarily large initial data have remained a challenging open problem. In this paper, we provide an affirmative answer to this longstanding open problem. Specifically, under the assumption that the viscosity coefficients satisfy a BD-type algebraic relation of the form $μ(ρ)=νρ^α$ and $λ(ρ)=2ν(α-1)ρ^α$, and that the Korteweg stress tensor complies with a generalized Bohm identity of the form $κ(ρ)=\varepsilon^2α^2ρ^{2α-3}$, we establish the global existence of strong solutions for the 2D and 3D systems in torus with arbitrarily large regular initial data. The analysis is carried out in the intermediary non-dispersive regime, characterized by the condition that the capillarity coefficient constant $\varepsilon$ does not exceed the viscosity constant $ν$. This result provides the first proof of the global-in-time existence of strong solutions for the 3D general Navier-Stokes-Korteweg system with arbitrarily large initial data in the non-dispersive regime. |
| title | Global-in-time strong solutions for the 2D and 3D generalized compressible Navier-Stokes-Korteweg system with arbitrarily large initial data |
| topic | Analysis of PDEs 35D35, 35Q30, 35Q35, 35Q40, 76N10 |
| url | https://arxiv.org/abs/2603.11762 |