Geodesic
Global classical solutions in a self-consistent chemotaxis(-Navier)-Stokes system
Czechoslovak Mathematical Journal, Tome 74 (2024) no. 1, pp. 153-175
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

The self-consistent chemotaxis-fluid system $$ \begin{cases} n_t+u\cdot \nabla n =\Delta n - \nabla \cdot (n\nabla c )+\nabla \cdot (n\nabla \phi ), \in \Omega ,\ t>0,\\ c_t +u\cdot \nabla c=\Delta c -nc,\quad \in \Omega ,\ t>0,\\ u_t+\kappa (u\cdot \nabla ) u+\nabla P=\Delta u - n\nabla \phi +n \nabla c,\qquad \in \Omega ,\ t>0,\\ \nabla \cdot u=0,\quad \in \Omega ,\ t>0, \end{cases} $$ is considered under no-flux boundary conditions for $n, c$ and the Dirichlet boundary condition for $u$ on a bounded smooth domain $ \Omega \subset \mathbb {R}^N$ $(N=2,3)$, $\kappa \in \lbrace 0,1 \rbrace $. The existence of global bounded classical solutions is proved under a smallness assumption on $\|c_{0}\|_{L^{\infty }(\Omega )}$. \endgraf Both the effect of gravity (potential force) on cells and the effect of the chemotactic force on fluid are considered here, and thus the coupling is stronger than the most studied chemotaxis-fluid systems. The literature on self-consistent chemotaxis-fluid systems of this type so far concentrates on the nonlinear cell diffusion as an additional dissipative mechanism. To the best of our knowledge, this is the first result on the boundedness of a self-consistent chemotaxis-fluid system with linear cell diffusion.
The self-consistent chemotaxis-fluid system $$ \begin{cases} n_t+u\cdot \nabla n =\Delta n - \nabla \cdot (n\nabla c )+\nabla \cdot (n\nabla \phi ), \in \Omega ,\ t>0,\\ c_t +u\cdot \nabla c=\Delta c -nc,\quad \in \Omega ,\ t>0,\\ u_t+\kappa (u\cdot \nabla ) u+\nabla P=\Delta u - n\nabla \phi +n \nabla c,\qquad \in \Omega ,\ t>0,\\ \nabla \cdot u=0,\quad \in \Omega ,\ t>0, \end{cases} $$ is considered under no-flux boundary conditions for $n, c$ and the Dirichlet boundary condition for $u$ on a bounded smooth domain $ \Omega \subset \mathbb {R}^N$ $(N=2,3)$, $\kappa \in \lbrace 0,1 \rbrace $. The existence of global bounded classical solutions is proved under a smallness assumption on $\|c_{0}\|_{L^{\infty }(\Omega )}$. \endgraf Both the effect of gravity (potential force) on cells and the effect of the chemotactic force on fluid are considered here, and thus the coupling is stronger than the most studied chemotaxis-fluid systems. The literature on self-consistent chemotaxis-fluid systems of this type so far concentrates on the nonlinear cell diffusion as an additional dissipative mechanism. To the best of our knowledge, this is the first result on the boundedness of a self-consistent chemotaxis-fluid system with linear cell diffusion.
DOI : 10.21136/CMJ.2023.0570-22
Classification : 35K55, 35Q35, 35Q92, 92C17
Keywords: chemotaxis; Navier-Stokes system; self-consistent; global existence; boundedness 
@article{10_21136_CMJ_2023_0570_22,
     author = {Li, Yanjiang and Yu, Zhongqing and Huang, Yumei},
     title = {Global classical solutions in a self-consistent {chemotaxis(-Navier)-Stokes} system},
     journal = {Czechoslovak Mathematical Journal},
     pages = {153--175},
     year = {2024},
     volume = {74},
     number = {1},
     doi = {10.21136/CMJ.2023.0570-22},
     mrnumber = {4717827},
     zbl = {07893372},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2023.0570-22/}
}
TY  - JOUR
AU  - Li, Yanjiang
AU  - Yu, Zhongqing
AU  - Huang, Yumei
TI  - Global classical solutions in a self-consistent chemotaxis(-Navier)-Stokes system
JO  - Czechoslovak Mathematical Journal
PY  - 2024
SP  - 153
EP  - 175
VL  - 74
IS  - 1
UR  - http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2023.0570-22/
DO  - 10.21136/CMJ.2023.0570-22
LA  - en
ID  - 10_21136_CMJ_2023_0570_22
ER  - 
%0 Journal Article
%A Li, Yanjiang
%A Yu, Zhongqing
%A Huang, Yumei
%T Global classical solutions in a self-consistent chemotaxis(-Navier)-Stokes system
%J Czechoslovak Mathematical Journal
%D 2024
%P 153-175
%V 74
%N 1
%U http://geodesic.mathdoc.fr/articles/10.21136/CMJ.2023.0570-22/
%R 10.21136/CMJ.2023.0570-22
%G en
%F 10_21136_CMJ_2023_0570_22
Li, Yanjiang; Yu, Zhongqing; Huang, Yumei. Global classical solutions in a self-consistent chemotaxis(-Navier)-Stokes system. Czechoslovak Mathematical Journal, Tome 74 (2024) no. 1, pp. 153-175. doi: 10.21136/CMJ.2023.0570-22

[1] Bellomo, N., Bellouquid, A., Tao, Y., Winkler, M.: Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues. Math. Models Methods Appl. Sci. 25 (2015), 1663-1763. | DOI | MR | JFM

[2] Bellomo, N., Outada, N., Soler, J., Tao, Y., Winkler, M.: Chemotaxis and cross-diffusion models in complex environments: Models and analytic problems toward a multiscale vision. Math. Models Methods Appl. Sci. 32 (2022), 713-792. | DOI | MR | JFM

[3] Cao, X.: Global classical solutions in chemotaxis(-Navier)-Stokes system with rotational flux term. J. Differ. Equations 261 (2016), 6883-6914. | DOI | MR | JFM

[4] Carrillo, J. A., Peng, Y., Xiang, Z.: Global existence and decay rates to self-consistent chemotaxis-fluid system. Available at , 36 pages. | arXiv | MR

[5] Chae, M., Kang, K., Lee, J.: Existence of smooth solutions to coupled chemotaxis-fluid equations. Discrete Contin. Dyn. Syst. 33 (2013), 2271-2297. | DOI | MR | JFM

[6] Chae, M., Kang, K., Lee, J.: Global existence and temporal decay in Keller-Segel models coupled to fluid equations. Commun. Partial Differ. Equations 39 (2014), 1205-1235. | DOI | MR | JFM

[7] Francesco, M. Di, Lorz, A., Markowich, P. A.: Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: Global existence and asymptotic behavior. Discrete Contin. Dyn. Syst. 28 (2010), 1437-1453. | DOI | MR | JFM

[8] Duan, R., Li, X., Xiang, Z.: Global existence and large time behavior for a two-dimensional chemotaxis-Navier-Stokes system. J. Differ. Equations 263 (2017), 6284-6316. | DOI | MR | JFM

[9] Duan, R., Lorz, A., Markowich, P. A.: Global solutions to the coupled chemotaxis-fluid equations. Commun. Partial Differ. Equations 35 (2010), 1635-1673. | DOI | MR | JFM

[10] Galdi, G. P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Vol. I: Linearized Steady Problems. Springer Tracts in Natural Philosophy 38. Springer, Berlin (1994). | DOI | MR | JFM

[11] He, P., Wang, Y., Zhao, L.: A further study on a 3D chemotaxis-Stokes system with tensor-valued sensitivity. Appl. Math. Lett. 90 (2019), 23-29. | DOI | MR | JFM

[12] Horstmann, D.: From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I. Jahresber. Dtsch. Math.-Ver. 105 (2003), 103-165. | MR | JFM

[13] Ishida, S.: Global existence and boundedness for chemotaxis-Navier-Stokes systems with position-dependent sensitivity in 2D bounded domains. Discrete Contin. Dyn. Syst. 35 (2015), 3463-3482. | DOI | MR | JFM

[14] Jin, C.: Global bounded solution in three-dimensional chemotaxis-Stokes model with arbitrary porous medium slow diffusion. Available at , 24 pages. | arXiv | MR

[15] Jin, C., Wang, Y., Yin, J.: Global solvability and stability to a nutrient-taxis model with porous medium slow diffusion. Available at , 35 pages. | arXiv | MR

[16] Keller, E. F., Segel, L. A.: Initiation of slime mold aggregation viewed as an instability. J. Theor. Biol. 26 (1970), 399-415. | DOI | MR | JFM

[17] Kozono, H., Yanagisawa, T.: Leray's problem on the stationary Navier-Stokes equations with inhomogeneous boundary data. Math. Z. 262 (2009), 27-39. | DOI | MR | JFM

[18] Li, X., Wang, Y., Xiang, Z.: Global existence and boundedness in a 2D Keller-Segel-Stokes system with nonlinear diffusion and rotational flux. Commun. Math. Sci. 14 (2016), 1889-1910. | DOI | MR | JFM

[19] Liu, J.-G., Lorz, A.: A coupled chemotaxis-fluid model: Global existence. Ann. Inst. H. Poincaré, Anal. Non Linéaire 28 (2011), 643-652. | DOI | MR | JFM

[20] Lorz, A.: Coupled chemotaxis fluid model. Math. Models Methods Appl. Sci. 20 (2010), 987-1004. | DOI | MR | JFM

[21] Patlak, C. S.: Random walk with persistence and external bias. Bull. Math. Biophys. 15 (1953), 311-338. | DOI | MR | JFM

[22] Peng, Y., Xiang, Z.: Global existence and boundedness in a 3D Keller-Segel-Stokes system with nonlinear diffusion and rotational flux. Z. Angew. Math. Phys. 68 (2017), Article ID 68, 26 pages. | DOI | MR | JFM

[23] Tan, Z., Zhang, X.: Decay estimates of the coupled chemotaxis-fluid equations in $\Bbb{R}^3$. J. Math. Anal. Appl. 410 (2014), 27-38. | DOI | MR | JFM

[24] Tao, Y.: Boundedness in a chemotaxis model with oxygen consumption by bacteria. J. Math. Anal. Appl. 381 (2011), 521-529. | DOI | MR | JFM

[25] Tao, Y., Winkler, M.: Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion. Discrete Contin. Dyn. Syst. 32 (2012), 1901-1914. | DOI | MR | JFM

[26] Tuval, I., Cisneros, L., Dombrowski, C., Wolgemuth, C. W., Kessler, J. O., Goldstein, R. E.: Bacterial swimming and oxygen transport near contact lines. Proc. Natl. Acad. Sci. USA 102 (2005), 2277-2282. | DOI | JFM

[27] Wang, Y.: Global solvability in a two-dimensional self-consistent chemotaxis-Navier- Stokes system. Discrete Contin. Dyn. Syst., Ser. S 13 (2020), 329-349. | DOI | MR | JFM

[28] Wang, Y., Winkler, M., Xiang, Z.: Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity. Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 18 (2018), 421-466. | DOI | MR | JFM

[29] Wang, Y., Winkler, M., Xiang, Z.: The small-convection limit in a two-dimensional chemotaxis-Navier-Stokes system. Math. Z. 289 (2018), 71-108. | DOI | MR | JFM

[30] Wang, Y., Zhao, L.: A 3D self-consistent chemotaxis-fluid system with nonlinear diffusion. J. Differ. Equations 269 (2020), 148-179. | DOI | MR | JFM

[31] Winkler, M.: Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity. Math. Nachr. 283 (2010), 1664-1673. | DOI | MR | JFM

[32] Winkler, M.: Aggregation vs. global diffusive behavior in the higher-dimensional Keller- Segel model. J. Differ. Equations 248 (2010), 2889-2905. | DOI | MR | JFM

[33] Winkler, M.: Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops. Commun. Partial Differ. Equations 37 (2012), 319-351. | DOI | MR | JFM

[34] Winkler, M.: Stabilization in a two-dimensional chemotaxis-Navier-Stokes system. Arch. Ration. Mech. Anal. 211 (2014), 455-487. | DOI | MR | JFM

[35] Winkler, M.: Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity. Calc. Var. Partial Differ. Equ. 54 (2015), 3789-3828. | DOI | MR | JFM

[36] Winkler, M.: Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system. Ann. Inst. H. Poincaré, Anal. Non Linéaire 33 (2016), 1329-1352. | DOI | MR | JFM

[37] Winkler, M.: How far do chemotaxis-driven forces influence regularity in the Navier- Stokes system?. Trans. Am. Math. Soc. 369 (2017), 3067-3125. | DOI | MR | JFM

[38] Winkler, M.: Global existence and stabilization in a degenerate chemotaxis-Stokes system with mildly strong diffusion enhancement. J. Differ. Equations 264 (2018), 6109-6151. | DOI | MR | JFM

[39] Winkler, M.: A three-dimensional Keller-Segel-Navier-Stokes system with logistic source: Global weak solutions and asymptotic stabilization. J. Funct. Anal. 276 (2019), 1339-1401. | DOI | MR | JFM

[40] Yu, P.: Global existence and boundedness in a chemotaxis-Stokes system with arbitrary porous medium diffusion. Math. Methods Appl. Sci. 43 (2020), 639-657. | DOI | MR | JFM

[41] Zhang, Q., Li, Y.: Convergence rates of solutions for a two-dimensional chemotaxis- Navier-Stokes system. Discrete Contin. Dyn. Syst., Ser. B 20 (2015), 2751-2759. | DOI | MR | JFM

[42] Zhang, Q., Li, Y.: Global weak solutions for the three-dimensional chemotaxis-Navier- Stokes system with nonlinear diffusion. J. Differ. Equations 259 (2015), 3730-3754. | DOI | MR | JFM

[43] Zhang, Q., Zheng, X.: Global well-posedness for the two-dimensional incompressible chemotaxis-Navier-Stokes equations. SIAM J. Math. Anal. 46 (2014), 3078-3105. | DOI | MR | JFM

[44] Zheng, J.: Global existence and boundedness in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity. Ann. Mat. Pura Appl. (4) 201 (2022), 243-288. | DOI | MR | JFM

Cité par Sources :