The chemotaxis–Navier–Stokes system

$$\{\begin{array}{ccc}\hfill {n}_{t}+u\cdot \mathrm{\nabla}n\phantom{\rule{0.2em}{0ex}}& \hfill =\phantom{\rule{0.2em}{0ex}}\hfill & \mathrm{\Delta}n-\mathrm{\nabla}\cdot (n\chi (c)\mathrm{\nabla}c),\phantom{\rule{0.2em}{0ex}}\hfill \\ \hfill {c}_{t}+u\cdot \mathrm{\nabla}c\phantom{\rule{0.2em}{0ex}}& \hfill =\phantom{\rule{0.2em}{0ex}}\hfill & \mathrm{\Delta}c-nf(c),\phantom{\rule{0.2em}{0ex}}\hfill \\ \hfill {u}_{t}+(u\cdot \mathrm{\nabla})u\phantom{\rule{0.2em}{0ex}}& \hfill =\phantom{\rule{0.2em}{0ex}}\hfill & \mathrm{\Delta}u+\mathrm{\nabla}P+n\mathrm{\nabla}\mathrm{\Phi},\phantom{\rule{0.2em}{0ex}}\hfill \\ \hfill \mathrm{\nabla}\cdot u\phantom{\rule{0.2em}{0ex}}& \hfill =\phantom{\rule{0.2em}{0ex}}\hfill & 0,\phantom{\rule{0.2em}{0ex}}\hfill \end{array}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}(\star )$$ |

The present work gives an affirmative answer to the question of global solvability for (⋆) in the following sense: Under mild assumptions on the initial data, and under modest structural assumptions on f and χ, inter alia allowing for the prototypical case when

$$f(s)=s\phantom{\rule{1em}{0ex}}\text{for all}s\ge 0\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\chi \equiv \mathit{const}.,$$ |

This solution is obtained as the limit of smooth solutions to suitably regularized problems, where appropriate compactness properties are derived on the basis of a priori estimates gained from an energy-type inequality for (⋆) which in an apparently novel manner combines the standard ${L}^{2}$ dissipation property of the fluid evolution with a quasi-dissipative structure associated with the chemotaxis subsystem in (⋆).

Keywords: Chemotaxis, Navier–Stokes, Global existence

@article{AIHPC_2016__33_5_1329_0, author = {Winkler, Michael}, title = {Global weak solutions in a three-dimensional {chemotaxis{\textendash}Navier{\textendash}Stokes} system}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {1329--1352}, publisher = {Elsevier}, volume = {33}, number = {5}, year = {2016}, doi = {10.1016/j.anihpc.2015.05.002}, zbl = {1351.35239}, language = {en}, url = {http://www.numdam.org/articles/10.1016/j.anihpc.2015.05.002/} }

TY - JOUR AU - Winkler, Michael TI - Global weak solutions in a three-dimensional chemotaxis–Navier–Stokes system JO - Annales de l'I.H.P. Analyse non linéaire PY - 2016 SP - 1329 EP - 1352 VL - 33 IS - 5 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.anihpc.2015.05.002/ DO - 10.1016/j.anihpc.2015.05.002 LA - en ID - AIHPC_2016__33_5_1329_0 ER -

%0 Journal Article %A Winkler, Michael %T Global weak solutions in a three-dimensional chemotaxis–Navier–Stokes system %J Annales de l'I.H.P. Analyse non linéaire %D 2016 %P 1329-1352 %V 33 %N 5 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.anihpc.2015.05.002/ %R 10.1016/j.anihpc.2015.05.002 %G en %F AIHPC_2016__33_5_1329_0

Winkler, Michael. Global weak solutions in a three-dimensional chemotaxis–Navier–Stokes system. Annales de l'I.H.P. Analyse non linéaire, Volume 33 (2016) no. 5, pp. 1329-1352. doi : 10.1016/j.anihpc.2015.05.002. http://www.numdam.org/articles/10.1016/j.anihpc.2015.05.002/

[1] X. Cao, S. Ishida, Global-in-time bounded weak solutions to a degenerate quasilinear Keller–Segel system with rotation, preprint. | Zbl

[2] X. Cao, Y. Wang, Global classical solutions of a 3D chemotaxis–Stokes system with rotation, preprint.

[3] Existence of smooth solutions to coupled chemotaxis–fluid equations, Discrete Contin. Dyn. Syst., Ser. A, Volume 33 (2013) no. 6, pp. 2271–2297 | DOI | Zbl

[4] Global existence and temporal decay in Keller–Segel models coupled to fluid equations, Commun. Partial Differ. Equ., Volume 39 (2014), pp. 1205–1235 | DOI | Zbl

[5] Sinking, merging and stationary plumes in a coupled chemotaxis–fluid model: a high-resolution numerical approach, J. Fluid Mech., Volume 694 (2012), pp. 155–190 | DOI | Zbl

[6] Chemotaxis–fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior, Discrete Contin. Dyn. Syst., Ser. A, Volume 28 (2010), pp. 1437–1453 | Zbl

[7] Reaction terms avoiding aggregation in slow fluids, Nonlinear Anal., Real World Appl., Volume 21 (2015), pp. 110–126 | DOI | Zbl

[8] Self-concentration and large-scale coherence in bacterial dynamics, Phys. Rev. Lett., Volume 93 (2004) (098103-1-4) | DOI

[9] Global solutions to the coupled chemotaxis–fluid equations, Commun. Partial Differ. Equ., Volume 35 (2010), pp. 1635–1673 | Zbl

[10] A note on global existence for the chemotaxis–Stokes model with nonlinear diffusion, Int. Math. Res. Not., Volume 2012 (2012) (rns270, 20 pages) | DOI

[11] The Stokes operator in ${L}_{r}$ spaces, Proc. Jpn. Acad., Volume 2 (1981), pp. 85–89 | Zbl

[12] Solutions for semilinear parabolic equations in ${L}_{p}$ and regularity of weak solutions of the Navier–Stokes system, J. Differ. Equ., Volume 61 (1986), pp. 186–212 | Zbl

[13] Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer, Berlin, Heidelberg, New York, 1981 | DOI | Zbl

[14] A user's guide to PDE models for chemotaxis, J. Math. Biol., Volume 58 (2009), pp. 183–217 | DOI | Zbl

[15] S. Ishida, Global existence for chemotaxis–Navier–Stokes systems with rotation in 2D bounded domains, preprint.

[16] Biomixing by chemotaxis and enhancement of biological reactions, Commun. Partial Differ. Equ., Volume 37 (2012) no. 1–3, pp. 298–318 | Zbl

[17] Linear and Quasi-Linear Equations of Parabolic Type, Amer. Math. Soc. Transl., vol. 23, Amer. Math. Soc., Providence, RI, 1968 | Zbl

[18] Sur le mouvement d'un liquide visqueus amplissant l'espace, Acta Math., Volume 63 (1934), pp. 193–248 | DOI | JFM

[19] Résolution de problèmes elliptiques quasilinéaires, Arch. Ration. Mech. Anal., Volume 74 (1980), pp. 335–353 | DOI | Zbl

[20] A coupled chemotaxis–fluid model: global existence, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 28 (2011) no. 5, pp. 643–652 | Numdam | Zbl

[21] Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., Volume 20 (2010), pp. 987–1004 | DOI | Zbl

[22] A coupled Keller–Segel–Stokes model: global existence for small initial data and blow-up delay, Commun. Math. Sci., Volume 10 (2012), pp. 555–574 | DOI | Zbl

[23] On energy inequality, smoothness and large time behaviour in ${L}^{2}$ for weak solutions of the Navier–Stokes equations, Math. Z., Volume 199 (1988), pp. 465–478 | DOI | Zbl

[24] N. Mizoguchi, M. Winkler, Blow-up in the two-dimensional parabolic Keller–Segel system, preprint.

[25] Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, Ser. Int., Volume 40 (1997), pp. 411–433 | Zbl

[26] Finite dimensional attractor for one-dimensional Keller–Segel equations, Funkc. Ekvacioj, Volume 44 (2001), pp. 441–469 | Zbl

[27] Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Birkhäuser, Basel/Boston/Berlin, 2007 | Zbl

[28] The Navier–Stokes Equations. An Elementary Functional Analytic Approach, Birkhäuser, Basel, 2001 | Zbl

[29] Schauder estimates for the evolutionary generalized Stokes problem, Nonlinear Equations and Spectral Theory, Amer. Math. Soc. Transl., Series 2, vol. 220, Amer. Math. Soc., Providence, RI, 2007, pp. 165–200 | DOI | Zbl

[30] Y. Tao, Boundedness in a Keller–Segel–Stokes system modeling the process of coral fertilization, preprint.

[31] Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differ. Equ., Volume 252 (2012), pp. 2520–2543 | Zbl

[32] Global existence and boundedness in a Keller–Segel–Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst., Ser. A, Volume 32 (2012) no. 5, pp. 1901–1914 | Zbl

[33] Locally bounded global solutions in a three-dimensional chemotaxis–Stokes system with nonlinear diffusion, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 30 (2013) no. 1, pp. 157–178 | Numdam | Zbl

[34] Navier–Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, vol. 2, North-Holland, Amsterdam, 1977 | Zbl

[35] Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, Volume 102 (2005), pp. 2277–2282 | DOI | Zbl

[36] Weak solutions for a bioconvection model related to Bacillus subtilis, Commun. Math. Sci., Volume 12 (2014), pp. 545–563 | DOI | Zbl

[37] The Navier–Stokes equations – a neverending challenge?, Jahresber. Dtsch. Math.-Ver., Volume 101 (1999), pp. 1–25 | Zbl

[38] Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model, J. Differ. Equ., Volume 248 (2010), pp. 2889–2905 | DOI | Zbl

[39] Global large-data solutions in a chemotaxis–(Navier–)Stokes system modeling cellular swimming in fluid drops, Commun. Partial Differ. Equ., Volume 37 (2012), pp. 319–351 | DOI | Zbl

[40] Finite-time blow-up in the higher-dimensional parabolic–parabolic Keller–Segel system, J. Math. Pures Appl., Volume 100 (2013), pp. 748–767 | arXiv | DOI | Zbl

[41] Stabilization in a two-dimensional chemotaxis–Navier–Stokes system, Arch. Ration. Mech. Anal., Volume 211 (2014) no. 2, pp. 455–487 | DOI | Zbl

[42] M. Winkler, Boundedness and large time behavior in a three-dimensional chemotaxis–Stokes system with nonlinear diffusion and general sensitivity, preprint.

*Cited by Sources: *