Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces
The Keller-Segel-Navier-Stokes system in RN{{\mathbb{R}}}^{N} is considered, where N≥3N\ge 3. We show the existence and uniqueness of local mild solutions for arbitrary initial data and gravitational potential in scaling invariant Lorentz spaces. Although such a result has already been shown by Kozo...
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De Gruyter
2025-05-01
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| Series: | Advances in Nonlinear Analysis |
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| Online Access: | https://doi.org/10.1515/anona-2025-0080 |
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| author | Takeuchi Taiki |
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| description | The Keller-Segel-Navier-Stokes system in RN{{\mathbb{R}}}^{N} is considered, where N≥3N\ge 3. We show the existence and uniqueness of local mild solutions for arbitrary initial data and gravitational potential in scaling invariant Lorentz spaces. Although such a result has already been shown by Kozono, Miura, and Sugiyama (Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid, J. Funct. Anal. 270 (2016), no. 5, 1663–1683), we reveal the precise regularities of mild solutions by showing the smoothing estimates of the heat semigroup on Lorentz spaces. The method is based on the real interpolation. In addition, we prove that the mild solutions exist globally in time, provided that the initial data are sufficiently small. Compared with the usual result, a part of the smallness conditions is reduced. We also obtain the asymptotic behavior of the global mild solutions. In the proof of the asymptotic behavior, to overcome a lack of density for the space L∞(RN){L}^{\infty }\left({{\mathbb{R}}}^{N}) to which one c0{c}_{0} of the initial data belongs, we show the decay of the global solutions without any approximation for c0{c}_{0}. |
| format | Article |
| id | doaj-art-504ddee2d64d42c588ef5b4c3e8d058c |
| institution | DOAJ |
| issn | 2191-950X |
| language | English |
| publishDate | 2025-05-01 |
| publisher | De Gruyter |
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| series | Advances in Nonlinear Analysis |
| spelling | doaj-art-504ddee2d64d42c588ef5b4c3e8d058c2025-08-20T03:08:00ZengDe GruyterAdvances in Nonlinear Analysis2191-950X2025-05-011412117212610.1515/anona-2025-0080Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spacesTakeuchi Taiki0Institute of Mathematics for Industry, Kyushu University, 744 Motooka, Nishi-ku, Fukuoka, 819-0395, JapanThe Keller-Segel-Navier-Stokes system in RN{{\mathbb{R}}}^{N} is considered, where N≥3N\ge 3. We show the existence and uniqueness of local mild solutions for arbitrary initial data and gravitational potential in scaling invariant Lorentz spaces. Although such a result has already been shown by Kozono, Miura, and Sugiyama (Existence and uniqueness theorem on mild solutions to the Keller-Segel system coupled with the Navier-Stokes fluid, J. Funct. Anal. 270 (2016), no. 5, 1663–1683), we reveal the precise regularities of mild solutions by showing the smoothing estimates of the heat semigroup on Lorentz spaces. The method is based on the real interpolation. In addition, we prove that the mild solutions exist globally in time, provided that the initial data are sufficiently small. Compared with the usual result, a part of the smallness conditions is reduced. We also obtain the asymptotic behavior of the global mild solutions. In the proof of the asymptotic behavior, to overcome a lack of density for the space L∞(RN){L}^{\infty }\left({{\mathbb{R}}}^{N}) to which one c0{c}_{0} of the initial data belongs, we show the decay of the global solutions without any approximation for c0{c}_{0}.https://doi.org/10.1515/anona-2025-0080keller-segel-navier-stokes systemasymptotic behaviorlorentz spacesscaling invariantprimary; 35b40secondary; 35a0135a2335k4535q9292c17 |
| spellingShingle | Takeuchi Taiki Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces Advances in Nonlinear Analysis keller-segel-navier-stokes system asymptotic behavior lorentz spaces scaling invariant primary; 35b40 secondary; 35a01 35a23 35k45 35q92 92c17 |
| title | Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces |
| title_full | Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces |
| title_fullStr | Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces |
| title_full_unstemmed | Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces |
| title_short | Asymptotic behavior of global mild solutions to the Keller-Segel-Navier-Stokes system in Lorentz spaces |
| title_sort | asymptotic behavior of global mild solutions to the keller segel navier stokes system in lorentz spaces |
| topic | keller-segel-navier-stokes system asymptotic behavior lorentz spaces scaling invariant primary; 35b40 secondary; 35a01 35a23 35k45 35q92 92c17 |
| url | https://doi.org/10.1515/anona-2025-0080 |
| work_keys_str_mv | AT takeuchitaiki asymptoticbehaviorofglobalmildsolutionstothekellersegelnavierstokessysteminlorentzspaces |