An optimal result for global existence and boundedness in a three-dimensional Keller-Segel-Stokes system with nonlinear diffusion. (English) Zbl 1472.35232
Summary: This paper investigates the following Keller-Segel-Stokes system with nonlinear diffusion
\[
(KSF)
\begin{cases}
n_t+u\cdot\nabla n = \Delta n^m - \nabla\cdot(n\nabla c),\quad x\in\Omega, t > 0, \\
c_t+u\cdot\nabla c = \Delta c-c+n,\quad x\in\Omega, t > 0, \\
u_t+\nabla P = \Delta u+n\nabla\phi,\quad x\in\Omega, t > 0, \\
\nabla\cdot u = 0,\quad x\in\Omega, t > 0
\end{cases}
\]
under homogeneous boundary conditions of Neumann type for \(n\) and \(c\), and of Dirichlet type for \(u\) in a three-dimensional bounded domains \(\Omega\subseteq\mathbb{R}^3\) with smooth boundary, where \(\phi\in W^{2,\infty}(\Omega),m > 0\). It is proved that if \(m > \frac{4}{3}\), then for any sufficiently regular nonnegative initial data there exists at least one global boundedness solution for system \(KSF\), which in view of the known results for the fluid-free system mentioned below (see Introduction) is an optimal restriction on \(m\).
MSC:
| 35K65 | Degenerate parabolic equations |
| 35K51 | Initial-boundary value problems for second-order parabolic systems |
| 35K59 | Quasilinear parabolic equations |
| 35Q35 | PDEs in connection with fluid mechanics |
| 92C17 | Cell movement (chemotaxis, etc.) |
References:
| [1] | Bellomo, N.; Belloquid, 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, 9, 1663-1763 (2015) · Zbl 1326.35397 |
| [2] | Chae, M.; Kang, K.; Lee, J., Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Commun. Partial Differ. Equ., 39, 1205-1235 (2014) · Zbl 1304.35481 |
| [3] | Cieślak, T.; Stinner, C., Finite-time blowup and global-in-time unbounded solutions to a parabolic – parabolic quasilinear Keller-Segel system in higher dimensions, J. Differ. Equ., 252, 5832-5851 (2012) · Zbl 1252.35087 |
| [4] | Cieślak, T.; Stinner, C., New critical exponents in a fully parabolic quasilinear Keller-Segel system and applications to volume filling models, J. Differ. Equ., 258, 2080-2113 (2015) · Zbl 1331.35041 |
| [5] | Cieślak, T.; Winkler, M., Finite-time blow-up in a quasilinear system of chemotaxis, Nonlinearity, 21, 1057-1076 (2008) · Zbl 1136.92006 |
| [6] | Duan, R.; Lorz, A.; Markowich, P. A., Global solutions to the coupled chemotaxis-fluid equations, Commun. Partial Differ. Equ., 35, 1635-1673 (2010) · Zbl 1275.35005 |
| [7] | Giga, Y.; Sohr, H., Abstract \(l^p\) estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains, J. Funct. Anal., 102, 72-94 (1991) · Zbl 0739.35067 |
| [8] | Gu, J.; Meng, F., Some new nonlinear Volterra-Fredholm type dynamic integral inequalities on time scales, Appl. Math. Comput., 245, 235-242 (2014) · Zbl 1335.26007 |
| [9] | Hieber, M.; Prüss, J., Heat kernels and maximal \(L^p-L^q\) estimate for parabolic evolution equations, Commun. Partial Differ. Equ., 22, 1647-1669 (1997) · Zbl 0886.35030 |
| [10] | Hillen, T.; Painter, K., A user’s guide to PDE models for chemotaxis, J. Math. Biol., 58, 183-217 (2009) · Zbl 1161.92003 |
| [11] | Hillen, T.; Painter, K., Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. Appl. Math., 26, 281-301 (2001) · Zbl 0998.92006 |
| [12] | Horstmann, D., From 1970 until present: the Keller-Segel model in chemotaxis and its consequences, I, Jahresber. Dtsch. Math.-Ver., 105, 103-165 (2003) · Zbl 1071.35001 |
| [13] | Horstmann, D.; Winkler, M., Boundedness vs. blow-up in a chemotaxis system, J. Differ. Equ., 215, 52-107 (2005) · Zbl 1085.35065 |
| [14] | Keller, E.; Segel, L., Model for chemotaxis, J. Theor. Biol., 30, 225-234 (1970) · Zbl 1170.92307 |
| [15] | Li, F.; Gao, Q., Blow-up of solution for a nonlinear Petrovsky type equation with memory, Appl. Math. Comput., 274, 383-392 (2016) · Zbl 1410.35085 |
| [16] | 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, 1889-1910 (2016) · Zbl 1351.35073 |
| [17] | Liu, J.-G.; Lorz, A., A coupled chemotaxis – fluid model: global existence, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 28, 5, 643-652 (2011) · Zbl 1236.92013 |
| [18] | Liu, J.; Wang, Y., Boundedness and decay property in a three-dimensional Keller-Segel-Stokes system involving tensor-valued sensitivity with saturation, J. Differ. Equ., 261, 2, 967-999 (2016) · Zbl 1336.35191 |
| [19] | Liu, J.; Wang, Y., Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system involving a tensor-valued sensitivity with saturation, J. Differ. Equ., 262, 10, 5271-5305 (2017) · Zbl 1377.35148 |
| [20] | Lorz, A., Coupled chemotaxis fluid equations, Math. Models Methods Appl. Sci., 20, 987-1004 (2010) · Zbl 1191.92004 |
| [21] | Miller, R. L., Demonstration of sperm chemotaxis in echinodermata: Asteroidea, Holothuroidea, Ophiuroidea, J. Exp. Zool., 234, 3, 383-414 (1985) |
| [22] | Painter, K.; Hillen, T., Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10, 501-543 (2002) · Zbl 1057.92013 |
| [23] | 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, 68 (2017) · Zbl 1386.35189 |
| [24] | Simon, J., Compact sets in the space \(L^p(O, T; B)\), Ann. Mat. Pura Appl., 146, 1, 65-96 (1986) · Zbl 0629.46031 |
| [25] | Sohr, H., The Navier-Stokes Equations, An Elementary Functional Analytic Approach (2001), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 0983.35004 |
| [26] | Tao, Y.; Winkler, M., Boundedness in a quasilinear parabolic – parabolic Keller-Segel system with subcritical sensitivity, J. Differ. Equ., 252, 692-715 (2012) · Zbl 1382.35127 |
| [27] | Tao, Y.; Winkler, M., Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis – fluid system, Z. Angew. Math. Phys., 66, 2555-2573 (2015) · Zbl 1328.35084 |
| [28] | Temam, R., Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, vol. 2 (1977), North-Holland: North-Holland Amsterdam · Zbl 0383.35057 |
| [29] | Tuval, I.; Cisneros, L.; Dombrowski, C., Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, 102, 2277-2282 (2005) · Zbl 1277.35332 |
| [30] | Wang, Y., Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with subcritical sensitivity, Math. Models Methods Appl. Sci., 27, 14, 2745-2780 (2017) · Zbl 1378.92010 |
| [31] | Wang, Y.; Liu, L.; Zhang, X.; Wu, Y., Positive solutions of a fractional semipositone differential system arising from the study of HIV infection models, Appl. Math. Comput., 258, 312-324 (2015) · Zbl 1338.92143 |
| [32] | 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., XVIII, 2036-2145 (2018) |
| [33] | Wang, Y.; Xiang, Z., Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation, J. Differ. Equ., 259, 7578-7609 (2015) · Zbl 1323.35071 |
| [34] | Wang, Y.; Xiang, Z., Global existence and boundedness in a Keller-Segel-Stokes system involving a tensor-valued sensitivity with saturation: the 3D case, J. Differ. Equ., 261, 4944-4973 (2016) · Zbl 1345.35054 |
| [35] | Wang, Z.; Winkler, M.; Wrzosek, D., Global regularity vs. infinite-time singularity formation in a chemotaxis model with volume-filling effect and degenerate diffusion, SIAM J. Math. Anal., 44, 3502-3525 (2012) · Zbl 1277.35223 |
| [36] | Winkler, M., Does a volume-filling effect always prevent chemotactic collapse, Math. Methods Appl. Sci., 33, 12-24 (2010) · Zbl 1182.35220 |
| [37] | Winkler, M., Absence of collapse in a parabolic chemotaxis system with signal-dependent sensitivity, Math. Nachr., 283, 1664-1673 (2010) · Zbl 1205.35037 |
| [38] | Winkler, M., Boundedness in the higher-dimensional parabolic – parabolic chemotaxis system with logistic source, Commun. Partial Differ. Equ., 35, 1516-1537 (2010) · Zbl 1290.35139 |
| [39] | Winkler, M., Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Commun. Partial Differ. Equ., 37, 319-351 (2012) · Zbl 1236.35192 |
| [40] | Winkler, M., Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211, 455-487 (2014) · Zbl 1293.35220 |
| [41] | 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, 3789-3828 (2015) · Zbl 1333.35104 |
| [42] | Winkler, M., Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 33, 5, 1329-1352 (2016) · Zbl 1351.35239 |
| [43] | Winkler, M., How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?, Trans. Am. Math. Soc., 369, 3067-3125 (2017) · Zbl 1356.35071 |
| [44] | Winkler, M.; Djie, K. C., Boundedness and finite-time collapse in a chemotaxis system with volume-filling effect, Nonlinear Anal. TMA, 72, 1044-1064 (2010) · Zbl 1183.92012 |
| [45] | Xu, F.; Liu, L., On the well-posedness of the incompressible flow in porous media, J. Nonlinear Sci. Appl., 9, 12, 6371-6381 (2016) · Zbl 1386.35362 |
| [46] | Zhang, Q.; Li, Y., Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion, J. Differ. Equ., 259, 8, 3730-3754 (2015) · Zbl 1320.35352 |
| [47] | Zhang, Q.; Zheng, X., Global well-posedness for the two-dimensional incompressible chemotaxis-Navier-Stokes equations, SIAM J. Math. Anal., 46, 3078-3105 (2014) · Zbl 1444.35011 |
| [48] | Zheng, J., Boundedness of solutions to a quasilinear parabolic – elliptic Keller-Segel system with logistic source, J. Differ. Equ., 259, 1, 120-140 (2015) · Zbl 1331.92026 |
| [49] | Zheng, J., Boundedness of solutions to a quasilinear parabolic – parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 431, 2, 867-888 (2015) · Zbl 1333.35098 |
| [50] | Zheng, J., Boundedness in a three-dimensional chemotaxis – fluid system involving tensor-valued sensitivity with saturation, J. Math. Anal. Appl., 442, 1, 353-375 (2016) · Zbl 1342.35416 |
| [51] | Zheng, J., A note on boundedness of solutions to a higher-dimensional quasi-linear chemotaxis system with logistic source, Z. Angew. Math. Mech., 97, 4, 414-421 (2017) · Zbl 1529.92008 |
| [52] | Zheng, J., Boundedness and global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with nonlinear logistic source, J. Math. Anal. Appl., 450, 2, 1047-1061 (2017) · Zbl 1516.35103 |
| [53] | Zheng, J., Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion, J. Differ. Equ., 263, 5, 2606-2629 (2017) · Zbl 1366.35080 |
| [54] | Zheng, J., A new result for global existence and boundedness of solutions to a parabolic – parabolic Keller-Segel system with logistic source, J. Math. Anal. Appl., 462, 1, 1-25 (2018) · Zbl 1393.35087 |
| [55] | Zheng, J., An optimal result for global existence and boundedness in a three-dimensional Keller-Segel(-Navier)-Stokes system (involving a tensor-valued sensitivity with saturation) |
| [56] | J. Zheng, A new result for global existence and boundedness in a three-dimensional Keller-Segel(-Navier)-Stokes system with nonlinear diffusion, preprint.; J. Zheng, A new result for global existence and boundedness in a three-dimensional Keller-Segel(-Navier)-Stokes system with nonlinear diffusion, preprint. |
| [57] | J. Zheng, An optimal result for global existence and boundedness in a three-dimensional attraction-repulsion chemotaxis-Stokes system with nonlinear diffusion, preprint.; J. Zheng, An optimal result for global existence and boundedness in a three-dimensional attraction-repulsion chemotaxis-Stokes system with nonlinear diffusion, preprint. |
| [58] | J. Zheng, A new result for global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system involving a tensor-valued sensitivity with saturation, preprint.; J. Zheng, A new result for global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system involving a tensor-valued sensitivity with saturation, preprint. |
| [59] | Zheng, J.; Wang, Y., Boundedness and decay behavior in a higher-dimensional quasilinear chemotaxis system with nonlinear logistic source, Comput. Math. Appl., 72, 10, 2604-2619 (2016) · Zbl 1367.92022 |
| [60] | Zheng, J.; Wang, Y., A note on global existence to a higher-dimensional quasilinear chemotaxis system with consumption of chemoattractant, Discrete Contin. Dyn. Syst., Ser. B, 22, 2, 669-686 (2017) · Zbl 1360.35092 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
