Le comportement d'une suspension bactérienne dans une goutte de liquide incompressible est décrit par les équations de chemotaxis-Navier–Stokes. Cet article introduit un échange d'oxygène entre la goutte et son environnement et une croissance logistique de la population bactérienne. Le système généralise le prototype
In biology, the behaviour of a bacterial suspension in an incompressible fluid drop is modelled by the chemotaxis-Navier–Stokes equations. This paper introduces an exchange of oxygen between the drop and its environment and an additionally logistic growth of the bacteria population. A prototype system is given by
Requiring sufficiently smooth initial data, the existence of a unique global classical solution for is proved, where is bounded in time for all , as well as the existence of a global weak solution for .
@article{AIHPC_2017__34_4_1013_0, author = {Braukhoff, Marcel}, title = {Global (weak) solution of the {chemotaxis-Navier{\textendash}Stokes} equations with non-homogeneous boundary conditions and logistic growth}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, pages = {1013--1039}, publisher = {Elsevier}, volume = {34}, number = {4}, year = {2017}, doi = {10.1016/j.anihpc.2016.08.003}, mrnumber = {3661869}, zbl = {1417.92028}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.anihpc.2016.08.003/} }
TY - JOUR AU - Braukhoff, Marcel TI - Global (weak) solution of the chemotaxis-Navier–Stokes equations with non-homogeneous boundary conditions and logistic growth JO - Annales de l'I.H.P. Analyse non linéaire PY - 2017 SP - 1013 EP - 1039 VL - 34 IS - 4 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.anihpc.2016.08.003/ DO - 10.1016/j.anihpc.2016.08.003 LA - en ID - AIHPC_2017__34_4_1013_0 ER -
%0 Journal Article %A Braukhoff, Marcel %T Global (weak) solution of the chemotaxis-Navier–Stokes equations with non-homogeneous boundary conditions and logistic growth %J Annales de l'I.H.P. Analyse non linéaire %D 2017 %P 1013-1039 %V 34 %N 4 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.anihpc.2016.08.003/ %R 10.1016/j.anihpc.2016.08.003 %G en %F AIHPC_2017__34_4_1013_0
Braukhoff, Marcel. Global (weak) solution of the chemotaxis-Navier–Stokes equations with non-homogeneous boundary conditions and logistic growth. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 4, pp. 1013-1039. doi : 10.1016/j.anihpc.2016.08.003. http://archive.numdam.org/articles/10.1016/j.anihpc.2016.08.003/
[1] Global attractor for approximate system of chemotaxis and growth, Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal., Volume 10 (2003) no. 1–3, pp. 309–315 (1918–2538/e) | MR | Zbl
[2] Physical Chemistry, Oxford University Press, 2006
[3] Volume effects in the Keller–Segel model: energy estimates preventing blow-up, J. Math. Pures Appl. (9), Volume 86 (2006) no. 2, pp. 155–175 | DOI | MR | Zbl
[4] Global existence and temporal decay in Keller–Segel models coupled to fluid equations, Commun. Partial Differ. Equ., Volume 39 (2014) no. 7, pp. 1205–1235 (1532-4133/e) | DOI | MR | Zbl
[5] Navier–Stokes Equations, University of Chicago Press, Chicago, IL etc., 1988 | DOI | MR | Zbl
[6] On a fourth-order degenerate parabolic equation: global entropy estimates, existence, and qualitative behavior of solutions, SIAM J. Math. Anal., Volume 29 (1998) no. 2, pp. 321–342 (1095-7154/e) | DOI | MR | Zbl
[7] Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior, Discrete Contin. Dyn. Syst., Volume 28 (2010) no. 4, pp. 1437–1453 (1553-5231/e) | DOI | MR | Zbl
[8] A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, Int. Math. Res. Not., Volume 2014 (2014) no. 7, pp. 1833–1852 (1687-0247/e) | DOI | MR | Zbl
[9] Global solutions to the coupled chemotaxis-fluid equations, Commun. Partial Differ. Equ., Volume 35 (2010) no. 9, pp. 1635–1673 (1532-4133/e) | DOI | MR | Zbl
[10] Reaction terms avoiding aggregation in slow fluids, Nonlinear Anal., Real World Appl., Volume 21 (2015), pp. 110–126 | DOI | MR | Zbl
[11] Partial Differential Equations, American Mathematical Society, Providence, RI, 2010 | MR | Zbl
[12] Partial Differential Equations, Holt, Rinehart and Winston, Inc., New York etc., 1969 (262 pp) | MR | Zbl
[13] On the Navier–Stokes initial value problem. I, Arch. Ration. Mech. Anal., Volume 16 (1964), pp. 269–315 (1432-0673/e) | DOI | MR | Zbl
[14] Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag. IV, Berlin–Heidelberg–New York, 1981 | DOI | MR | Zbl
[15] Linear and Quasi-Linear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23, American Mathematical Society (AMS). XI, Providence, RI, 1968 (translated from the Russian by S. Smith 648 pp) | MR | Zbl
[16] Evolution equations associated to contraction semigroups in spaces, J. Funct. Anal., Volume 72 (1987), pp. 252–262 | DOI | MR | Zbl
[17] Problèmes aux limites non homogenes et applications, vol. 1, 1968 | Zbl
[18] Probability and Potentials, Blaisdell Publishing Co., 1966 | MR | Zbl
[19] Aggregating pattern dynamics in a chemotaxis model including growth, Phys. A, Stat. Mech. Appl., Volume 230 (1996) no. 3–4, pp. 499–543 | DOI
[20] Semilinear parabolic problems define semiflows on spaces, Trans. Am. Math. Soc., Volume 278 (1983), pp. 21–55 (1088-6850/e) | DOI | MR | Zbl
[21] Application of the Trudinger–Moser inequality to a parabolic system of chemotaxis, Funkc. Ekvacioj, Ser. Int., Volume 40 (1997) no. 3, pp. 411–433 | MR | Zbl
[22] Global existence for a chemotaxis-growth system in , Adv. Math. Sci. Appl., Volume 12 (2002) no. 2, pp. 587–606 | MR | Zbl
[23] Exponential attractor for a chemotaxis-growth system of equations, Nonlinear Anal., Theory Methods Appl., Volume 51 (2002) no. 1, pp. 119–144 | DOI | MR | Zbl
[24] Compact sets in the space , Ann. Mat. Pura Appl. (4), Volume 146 (1987), pp. 65–96 (1618-1891/e) | DOI | MR | Zbl
[25] The Navier–Stokes Equations. An Elementary Functional Analytic Approach, Birkhäuser, Basel, 2001 | MR | Zbl
[26] Global existence and boundedness in a Keller–Segel–Stokes model with arbitrary porous medium diffusion, Discrete Contin. Dyn. Syst., Volume 32 (2012) no. 5, pp. 1901–1914 (1553-5231/e) | DOI | MR | Zbl
[27] 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 | DOI | Numdam | MR | Zbl
[28] Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, Volume 102 (2005) no. 7, pp. 2277–2282 (1091-6490/e) | DOI | Zbl
[29] Funktionalanalysis, Springer, Berlin, 2011 (978-3-642-21017-4/ebook) | DOI | Zbl
[30] Aggregation vs. global diffusive behavior in the higher-dimensional Keller–Segel model, J. Differ. Equ., Volume 248 (2010) no. 12, pp. 2889–2905 | DOI | MR | Zbl
[31] Boundedness in the higher-dimensional parabolic–parabolic chemotaxis system with logistic source, Commun. Partial Differ. Equ., Volume 35 (2010) no. 8, pp. 1516–1537 (1532-4133/e) | DOI | MR | Zbl
[32] Global large-data solutions in a chemotaxis-(Navier–)Stokes system modeling cellular swimming in fluid drops, Commun. Partial Differ. Equ., Volume 37 (2012) no. 1–3, pp. 319–351 (1532-4133/e) | DOI | MR | Zbl
[33] Global weak solutions in a three-dimensional chemotaxis-Navier–Stokes system, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, Volume 33 (2016) no. 5, pp. 1329–1352 | DOI | Numdam | MR | Zbl
[34] Boundedness and large time behavior in a three-dimensional chemotaxis-Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differ. Equ., Volume 54 (2015) no. 4, pp. 3789–3828 (1432-0835/e) | DOI | MR | Zbl
[35] Michael Winkler, A two-dimensional chemotaxis-Stokes system with rotational flux: global solvability, eventual smoothness and stabilization, preprint.
[36] Michael Winkler, Youshan Tao, Blow-up prevention by quadratic degradation in a two-dimensional Keller–Segel–Navier–Stokes system, preprint. | MR
Cité par Sources :