Keller-Segel-Navier-Stokes systems involving general sensitivities with Signal-Dependent Power-Law Decay
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| Format: | Preprint |
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2026
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| _version_ | 1866910046037737472 |
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| author | Ahn, Jaewook Hwang, Sukjung |
| author_facet | Ahn, Jaewook Hwang, Sukjung |
| contents | This paper investigates a two-dimensional Keller--Segel--Navier--Stokes system with a tensor-valued chemotactic sensitivity $S(x,n,c)$. Under a signal-dependent power-decay condition $|S(x,n,c)| \le s_0 (s_1+c)^{-γ}$, we establish the global existence and uniform-in-time boundedness of classical solutions for both fluid-coupled ($γ> 1/2$) and fluid-free ($γ> 0$) systems. The proof relies on a sequence of localized energy estimates, including the $L^{2}_{\rm loc}$-smallness of the weighted gradient of the signal concentration, to overcome the mathematical difficulties arising from signal production and fluid transport. Furthermore, under specific structural assumptions on the sensitivity tensor, we prove that solutions of the fluid-free system converge exponentially to the spatially homogeneous steady state. To this end, we establish an interpolation inequality involving the Hölder norm, which is of independent interest and seems to have broad applications. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_07870 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Keller-Segel-Navier-Stokes systems involving general sensitivities with Signal-Dependent Power-Law Decay Ahn, Jaewook Hwang, Sukjung Analysis of PDEs 35B45, 35A09, 35K57, 35Q35, 35Q92 This paper investigates a two-dimensional Keller--Segel--Navier--Stokes system with a tensor-valued chemotactic sensitivity $S(x,n,c)$. Under a signal-dependent power-decay condition $|S(x,n,c)| \le s_0 (s_1+c)^{-γ}$, we establish the global existence and uniform-in-time boundedness of classical solutions for both fluid-coupled ($γ> 1/2$) and fluid-free ($γ> 0$) systems. The proof relies on a sequence of localized energy estimates, including the $L^{2}_{\rm loc}$-smallness of the weighted gradient of the signal concentration, to overcome the mathematical difficulties arising from signal production and fluid transport. Furthermore, under specific structural assumptions on the sensitivity tensor, we prove that solutions of the fluid-free system converge exponentially to the spatially homogeneous steady state. To this end, we establish an interpolation inequality involving the Hölder norm, which is of independent interest and seems to have broad applications. |
| title | Keller-Segel-Navier-Stokes systems involving general sensitivities with Signal-Dependent Power-Law Decay |
| topic | Analysis of PDEs 35B45, 35A09, 35K57, 35Q35, 35Q92 |
| url | https://arxiv.org/abs/2603.07870 |