Asymptotic behavior of small-data solutions to a Keller-Segel-Navier-Stokes system with indirect signal production

Lu Yang; Xi Liu; Zhibo Hou

Czechoslovak Mathematical Journal (2023)

  • Volume: 73, Issue: 1, page 49-70
  • ISSN: 0011-4642

Abstract

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We consider the Keller-Segel-Navier-Stokes system which is considered in bounded domain with smooth boundary, where with . We show that if the initial data , , and is small enough, an associated initial-boundary value problem possesses a global classical solution which decays to the constant state exponentially with .

How to cite

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Yang, Lu, Liu, Xi, and Hou, Zhibo. "Asymptotic behavior of small-data solutions to a Keller-Segel-Navier-Stokes system with indirect signal production." Czechoslovak Mathematical Journal 73.1 (2023): 49-70. <http://eudml.org/doc/299551>.

@article{Yang2023,
abstract = {We consider the Keller-Segel-Navier-Stokes system \[ \{\left\lbrace \begin\{array\}\{ll\} n\_t+\{\bf u\}\cdot \nabla n =\Delta n - \nabla \cdot (n\nabla v ),& x\in \Omega ,\ t>0,\\ v\_t +\{\bf u\}\cdot \nabla v=\Delta v -v+w, &x\in \Omega ,\ t>0,\\ w\_t+\{\bf u\}\cdot \nabla w=\Delta w -w+n, &x\in \Omega ,\ t>0,\\ \{\bf \{u\}\}\_t + (\{\bf \{u\}\}\cdot \nabla )\{\bf \{u\}\} = \Delta \{\bf \{u\}\} + \nabla P + n\nabla \phi ,\ \nabla \cdot \{\bf u\}=0, &x\in \Omega ,\ t>0, \end\{array\}\right.\} \] which is considered in bounded domain $\Omega \subset \mathbb \{R\}^N$$(N \in \lbrace 2,3\rbrace )$ with smooth boundary, where $\phi \in C^\{1+\delta \}(\overline\{\Omega \})$ with $\delta \in (0,1)$. We show that if the initial data $\Vert n_0\Vert _\{L^\{\{N\}/\{2\}\}(\Omega )\}$, $\Vert \nabla v_0\Vert _\{L^N(\Omega )\}$, $\Vert \nabla w_0\Vert _\{L^N(\Omega )\}$ and $\Vert \{\bf u\}_0\Vert _\{L^N(\Omega )\}$ is small enough, an associated initial-boundary value problem possesses a global classical solution which decays to the constant state $(\{\bar\{n\}\}_0,\{\bar\{n\}\}_0,\{\bar\{n\}\}_0,0)$ exponentially with $\{\bar\{n\}\}_0:=(1/|\Omega |)\int _\{\Omega \}n_0(x)\{\rm d\}x$.},
author = {Yang, Lu, Liu, Xi, Hou, Zhibo},
journal = {Czechoslovak Mathematical Journal},
keywords = {Keller-Segel-Navier-Stokes; global solution; decay estimate; indirect process},
language = {eng},
number = {1},
pages = {49-70},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Asymptotic behavior of small-data solutions to a Keller-Segel-Navier-Stokes system with indirect signal production},
url = {http://eudml.org/doc/299551},
volume = {73},
year = {2023},
}

TY - JOUR
AU - Yang, Lu
AU - Liu, Xi
AU - Hou, Zhibo
TI - Asymptotic behavior of small-data solutions to a Keller-Segel-Navier-Stokes system with indirect signal production
JO - Czechoslovak Mathematical Journal
PY - 2023
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 73
IS - 1
SP - 49
EP - 70
AB - We consider the Keller-Segel-Navier-Stokes system \[ {\left\lbrace \begin{array}{ll} n_t+{\bf u}\cdot \nabla n =\Delta n - \nabla \cdot (n\nabla v ),& x\in \Omega ,\ t>0,\\ v_t +{\bf u}\cdot \nabla v=\Delta v -v+w, &x\in \Omega ,\ t>0,\\ w_t+{\bf u}\cdot \nabla w=\Delta w -w+n, &x\in \Omega ,\ t>0,\\ {\bf {u}}_t + ({\bf {u}}\cdot \nabla ){\bf {u}} = \Delta {\bf {u}} + \nabla P + n\nabla \phi ,\ \nabla \cdot {\bf u}=0, &x\in \Omega ,\ t>0, \end{array}\right.} \] which is considered in bounded domain $\Omega \subset \mathbb {R}^N$$(N \in \lbrace 2,3\rbrace )$ with smooth boundary, where $\phi \in C^{1+\delta }(\overline{\Omega })$ with $\delta \in (0,1)$. We show that if the initial data $\Vert n_0\Vert _{L^{{N}/{2}}(\Omega )}$, $\Vert \nabla v_0\Vert _{L^N(\Omega )}$, $\Vert \nabla w_0\Vert _{L^N(\Omega )}$ and $\Vert {\bf u}_0\Vert _{L^N(\Omega )}$ is small enough, an associated initial-boundary value problem possesses a global classical solution which decays to the constant state $({\bar{n}}_0,{\bar{n}}_0,{\bar{n}}_0,0)$ exponentially with ${\bar{n}}_0:=(1/|\Omega |)\int _{\Omega }n_0(x){\rm d}x$.
LA - eng
KW - Keller-Segel-Navier-Stokes; global solution; decay estimate; indirect process
UR - http://eudml.org/doc/299551
ER -

References

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  1. Bellomo, N., Bellouquid, A., Tao, Y., Winkler, M., 10.1142/S021820251550044X, Math. Models Methods Appl. Sci. 25 (2015), 1663-1763. (2015) Zbl1326.35397MR3351175DOI10.1142/S021820251550044X
  2. Cao, X., 10.3934/dcds.2015.35.1891, Discrete Contin. Dyn. Syst. 35 (2015), 1891-1904. (2015) Zbl06384058MR3294230DOI10.3934/dcds.2015.35.1891
  3. Cao, X., Lankeit, J., 10.1007/s00526-016-1027-2, Calc. Var. Partial Differ. Equ. 55 (2016), Article ID 107, 39 pages. (2016) Zbl1366.35075MR3531759DOI10.1007/s00526-016-1027-2
  4. Corrias, L., Perthame, B., 10.1016/j.mcm.2007.06.005, Math. Comput. Modelling 47 (2008), 755-764. (2008) Zbl1134.92006MR2404241DOI10.1016/j.mcm.2007.06.005
  5. Espejo, E., Suzuki, T., 10.1016/j.nonrwa.2014.07.001, Nonlinear Anal., Real World Appl. 21 (2015), 110-126. (2015) Zbl1302.35102MR3261583DOI10.1016/j.nonrwa.2014.07.001
  6. Fujie, K., Senba, T., 10.1016/j.jde.2017.02.031, J. Differ. Equations 263 (2017), 88-148. (2017) Zbl1364.35120MR3631302DOI10.1016/j.jde.2017.02.031
  7. Fujie, K., Senba, T., 10.1016/j.jde.2018.07.068, J. Differ. Equations 266 (2019), 942-976. (2019) Zbl1406.35149MR3906204DOI10.1016/j.jde.2018.07.068
  8. Hillen, T., Painter, K. J., 10.1007/s00285-008-0201-3, J. Math. Biol. 58 (2009), 183-217. (2009) Zbl1161.92003MR2448428DOI10.1007/s00285-008-0201-3
  9. Horstmann, D., From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Dtsch. Math.-Ver. 105 (2003), 103-165. (2003) Zbl1071.35001MR2013508
  10. Jin, H.-Y., 10.1016/j.jmaa.2014.09.049, J. Math. Anal. Appl. 422 (2015), 1463-1478. (2015) Zbl1307.35139MR3269523DOI10.1016/j.jmaa.2014.09.049
  11. Li, X., Xiao, Y., 10.1016/j.nonrwa.2017.02.005, Nonlinear Anal., Real World Appl. 37 (2017), 14-30. (2017) Zbl1394.35241MR3648369DOI10.1016/j.nonrwa.2017.02.005
  12. Liu, J., Wang, Y., 10.1016/j.jde.2017.01.024, J. Differ. Equations 262 (2017), 5271-5305. (2017) Zbl1377.35148MR3612542DOI10.1016/j.jde.2017.01.024
  13. Luca, M., Chavez-Ross, A., Edelstein-Keshet, L., Mogilner, A., 10.1016/S0092-8240(03)00030-2, Bull. Math. Biol. 65 (2003), 693-730. (2003) Zbl1334.92077DOI10.1016/S0092-8240(03)00030-2
  14. Nagai, T., Senba, T., Yoshida, K., Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, Ser. Int. 40 (1997), 411-433. (1997) Zbl0901.35104MR1610709
  15. Osaki, K., Yagi, A., Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkc. Ekvacioj, Ser. Int. 44 (2001), 441-469. (2001) Zbl1145.37337MR1893940
  16. Tao, Y., Wang, Z.-A., 10.1142/S0218202512500443, Math. Models Methods Appl. Sci. 23 (2013), 1-36. (2013) Zbl1403.35136MR2997466DOI10.1142/S0218202512500443
  17. Tao, Y., Winkler, M., 10.1007/s00033-015-0541-y, Z. Angew. Math. Phys. 66 (2015), 2555-2573. (2015) Zbl1328.35084MR3412312DOI10.1007/s00033-015-0541-y
  18. Tao, Y., Winkler, M., 10.1007/s00033-016-0732-1, Z. Angew. Math. Phys. 67 (2016), Article ID 138, 23 pages. (2016) Zbl1356.35054MR3562386DOI10.1007/s00033-016-0732-1
  19. Wang, Y., 10.1142/S0218202517500579, Math. Models Methods Appl. Sci. 27 (2017), 2745-2780. (2017) Zbl1378.92010MR3723735DOI10.1142/S0218202517500579
  20. Wang, Y., Winkler, M., Xiang, Z., 10.2422/2036-2145.201603_004, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 18 (2018), 421-466. (2018) Zbl1395.92024MR3801284DOI10.2422/2036-2145.201603_004
  21. Wang, Y., Yang, L., 10.1016/j.jde.2021.04.001, J. Differ. Equations 287 (2021), 460-490. (2021) Zbl1464.35052MR4242960DOI10.1016/j.jde.2021.04.001
  22. Winkler, M., 10.1016/j.jde.2010.02.008, J. Differ. Equations 248 (2010), 2889-2905. (2010) Zbl1190.92004MR2644137DOI10.1016/j.jde.2010.02.008
  23. Winkler, M., 10.1080/03605302.2011.591865, Commun. Partial Differ. Equations 37 (2012), 319-351. (2012) Zbl1236.35192MR2876834DOI10.1080/03605302.2011.591865
  24. Winkler, M., 10.1016/j.jfa.2018.12.009, J. Funct. Anal. 276 (2019), 1339-1401. (2019) Zbl1408.35132MR3912779DOI10.1016/j.jfa.2018.12.009
  25. Winkler, M., 10.1137/19M1264199, SIAM J. Math. Anal. 52 (2020), 2041-2080. (2020) Zbl1441.35079MR4091876DOI10.1137/19M1264199
  26. Winkler, M., 10.1007/s00220-021-04272-y, Commun. Math. Phys. 389 (2022), 439-489. (2022) Zbl07463712MR4365145DOI10.1007/s00220-021-04272-y
  27. Yu, H., Wang, W., Zheng, S., 10.1016/j.jmaa.2017.12.048, J. Math. Anal. Appl. 461 (2018), 1748-1770. (2018) Zbl1390.35381MR3765513DOI10.1016/j.jmaa.2017.12.048
  28. Yu, P., 10.1007/s10440-019-00307-8, Acta Appl. Math. 169 (2020), 475-497. (2020) Zbl1470.35185MR4146909DOI10.1007/s10440-019-00307-8
  29. Zhang, W., Niu, P., Liu, S., 10.1016/j.nonrwa.2019.05.002, Nonlinear Anal., Real World Appl. 50 (2019), 484-497. (2019) Zbl1435.35068MR3959244DOI10.1016/j.nonrwa.2019.05.002

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