Global strong solutions with large initial data for the Cauchy problem of the multi-dimensional compressible Navier-Stokes-Korteweg system
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arXiv
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917267274465280 |
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| author | Huang, Xiangdi Gu, Yongteng Lei, Muxi |
| author_facet | Huang, Xiangdi Gu, Yongteng Lei, Muxi |
| contents | In this paper, we establish global strong solutions for arbitrarily large initial data to the 2D and 3D compressible Navier-Stokes-Korteweg system, also referred to as the quantum Navier-Stokes equations, originally derived by Dunn and Serrin [Arch. Ration. Mech. Anal. 88(2):95-133, 1985]. Specifically, we prove the existence of global strong solutions for arbitrarily large initial data in the case $N=2$ when $γ\ge 1$, and $N=3$ with $1 \le γ< 8/3$ for the associated Cauchy problem. By employing techniques from Littlewood-Paley theory, range truncation analysis, refined Nash-Moser and De Giorgi iteration methods, we derive positive upper and lower bounds for the density. As a consequence, we are able to treat the whole-space case with strictly positive far-field density. To the best of our knowledge, this is the first result that establishes global strong solutions for physically relevant compressible Navier-Stokes equations in the whole space, without imposing any symmetry or special geometric assumptions on the initial data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_10700 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Global strong solutions with large initial data for the Cauchy problem of the multi-dimensional compressible Navier-Stokes-Korteweg system Huang, Xiangdi Gu, Yongteng Lei, Muxi Analysis of PDEs 35D35, 35Q30, 35Q35, 35Q40, 76N10 In this paper, we establish global strong solutions for arbitrarily large initial data to the 2D and 3D compressible Navier-Stokes-Korteweg system, also referred to as the quantum Navier-Stokes equations, originally derived by Dunn and Serrin [Arch. Ration. Mech. Anal. 88(2):95-133, 1985]. Specifically, we prove the existence of global strong solutions for arbitrarily large initial data in the case $N=2$ when $γ\ge 1$, and $N=3$ with $1 \le γ< 8/3$ for the associated Cauchy problem. By employing techniques from Littlewood-Paley theory, range truncation analysis, refined Nash-Moser and De Giorgi iteration methods, we derive positive upper and lower bounds for the density. As a consequence, we are able to treat the whole-space case with strictly positive far-field density. To the best of our knowledge, this is the first result that establishes global strong solutions for physically relevant compressible Navier-Stokes equations in the whole space, without imposing any symmetry or special geometric assumptions on the initial data. |
| title | Global strong solutions with large initial data for the Cauchy problem of the multi-dimensional compressible Navier-Stokes-Korteweg system |
| topic | Analysis of PDEs 35D35, 35Q30, 35Q35, 35Q40, 76N10 |
| url | https://arxiv.org/abs/2602.10700 |