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Convergence rates of solutions for a two-species chemotaxis-Navier-Stokes sytstem with competitive kinetics
1. | Department of Mathematics, South China University of Technology, Guangzhou 510640, China |
2. | Institute for Mathematical Sciences, Renmin University of China, Beijing 100872, China |
$\begin{equation*} \begin{cases} & (n_1)_t + u\cdot\nabla n_1 = \Delta n_1 - \chi_1\nabla\cdot(n_1\nabla c) + \mu_1n_1(1- n_1 - a_1n_2), \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in \Omega,\ t>0, \\ & (n_2)_t + u\cdot\nabla n_2 = \Delta n_2 - \chi_2\nabla\cdot(n_2\nabla c) + \mu_2n_2(1- a_2n_1 - n_2), \\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x\in \Omega,\ t>0, \\ & c_t + u\cdot\nabla c = \Delta c -(\alpha n_1 + \beta n_2)c, x \in \Omega,\ t>0, \\ & \ u_t + \kappa (u\cdot\nabla) u = \Delta u + \nabla P + (\gamma n_1 + \delta n_2)\nabla\phi, \quad \nabla\cdot u = 0, \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ x \in \Omega,\ t>0 \end{cases} \end{equation*}$ |
$n_1,n_2,c$ |
$u$ |
$\Omega \subset \mathbb{R}^d(d\in\{2,3\})$ |
$2$ |
$3$ |
$\kappa = 0$ |
$\frac{\max\{\chi_1,\chi_2\}}{\min\{\mu_1,\mu_2\}}\|c_0\|_{L^\infty(\Omega)} $ |
$(n_1, n_2, c, u)$ |
$L^\infty$ |
$(n_1(\cdot,t), n_2(\cdot,t), u(\cdot,t))\overset{t\rightarrow\infty}\rightarrow \begin{cases} (\frac{1 - a_1}{1 - a_1a_2},\frac{1 - a_2}{1 - a_1a_2},0) \text{ exponentially,}\\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{ if } a_1, a_2 \in (0, 1), \\ (0,1,0) \text{ exponentially, if } a_1>1> a_2, \\ (0,1,0) \text{ algebraically, if } a_1 = 1> a_2, \\ (1,,0,0) \text{ exponentially, if } a_2>1> a_1, \\ (1,0,0) \text{ algebraically, if } a_2 = 1> a_1. \end{cases}$ |
$c$ |
$0$ |
$u$ |
$-\Delta$ |
$\Omega$ |
$a_i, \mu_i, \alpha$ |
$\beta$ |
$d$ |
References:
[1] |
X. Bai and M. Winkler,
Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.
doi: 10.1512/iumj.2016.65.5776. |
[2] |
T. Black, J. Lankeit and M. Mizukami,
On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876.
doi: 10.1093/imamat/hxw036. |
[3] |
X. Cao,
Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.
doi: 10.3934/dcds.2015.35.1891. |
[4] |
X. Cao, S. Kurima and M. Mizukami, Global existence and asymptotic behavior of classical solutions for a 3D two-species chemotaxis-Stokes system with competitive kinetics, arXiv: 1703.01794, Math Meth Appl Sci., 41 (2018), 3138–3154.
doi: 10.1002/mma.4807. |
[5] |
E. Conway, D. Hoff and J. Smoller,
Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math., 35 (1978), 1-16.
doi: 10.1137/0135001. |
[6] |
P. De Mottoni and F. Rothe,
Convergence to homogeneous equilibrium state for generalized Volterra-Lotka systems with diffusion, SIAM J. Appl. Math., 37 (1979), 648-663.
doi: 10.1137/0137048. |
[7] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
[8] |
M. Hirata, S. Kurima, M. Mizukami and T. Yokota,
Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, J. Differential Equations, 263 (2017), 470-490.
doi: 10.1016/j.jde.2017.02.045. |
[9] |
M. Hirata, S. Kurima, M. Mizukami and T. Yokota,
Boundedness and stabilization in a three-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, Proceedings of EQUADIFF 2017 Conference, (2017), 11-20.
doi: 10.1016/j.jde.2017.02.045. |
[10] |
H. Jin and Z. Wang,
Global stability of prey-taxis systems, J. Differential Equations, 262 (2017), 1257-1290.
doi: 10.1016/j.jde.2016.10.010. |
[11] |
Y. Kan-on and E. Yanagida,
Existence of nonconstant stable equilibria in competition-diffusion equations, Hiroshima Math. J., 23 (1993), 193-221.
|
[12] |
O. Ladyzhenskaya, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, RI, 1968. |
[13] |
J. Lankeit,
Long-term behaviour in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci., 26 (2016), 2071-2109.
doi: 10.1142/S021820251640008X. |
[14] |
J. Lankeit and Y. Wang,
Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption, Discrete Contin. Dyn. Syst., 37 (2017), 6099-6121.
doi: 10.3934/dcds.2017262. |
[15] |
Y. Lou and W.-M. Ni,
Diffusion, self-diffusion and cross-diffusion, J. Differential Equaations, 131 (1996), 79-131.
doi: 10.1006/jdeq.1996.0157. |
[16] |
M. Mimura, S. I. Ei and Q. Fang,
Effect of domain-shape on coexistence problems in a competition-diffusion system, J. Math. Biol., 29 (1991), 219-237.
doi: 10.1007/BF00160536. |
[17] |
M. Mizukami,
Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2301-2319.
doi: 10.3934/dcdsb.2017097. |
[18] |
M. Mizukami and T. Yokota,
Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669.
doi: 10.1016/j.jde.2016.05.008. |
[19] |
M. Negreanu and J. I. Tello,
Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617.
doi: 10.1016/j.jde.2014.11.009. |
[20] |
M. Negreanu and J. I. Tello,
On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781.
doi: 10.1137/140971853. |
[21] |
M. Porzio and V. Vespri,
Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.
doi: 10.1006/jdeq.1993.1045. |
[22] |
C. Stinner, J. I. Tello and M. Winkler,
Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626.
doi: 10.1007/s00285-013-0681-7. |
[23] |
Y. Tao,
Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529.
doi: 10.1016/j.jmaa.2011.02.041. |
[24] |
Y. Tao and M. Winkler,
Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543.
doi: 10.1016/j.jde.2011.07.010. |
[25] |
Y. Tao and M. Winkler,
Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.
doi: 10.1007/s00033-015-0541-y. |
[26] |
Y. Tao and M. Winkler,
Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.
doi: 10.1137/15M1014115. |
[27] |
Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system, Z. Angew. Math. Phys., 67 (2016), Art. 138, 23 pp.
doi: 10.1007/s00033-016-0732-1. |
[28] |
I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein,
Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. U.S.A., 102 (2005), 2277-2282.
doi: 10.1073/pnas.0406724102. |
[29] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[30] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[31] |
M. Winkler,
Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.
doi: 10.1080/03605302.2011.591865. |
[32] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[33] |
M. Winkler,
Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211 (2014), 455-487.
doi: 10.1007/s00205-013-0678-9. |
[34] |
M. Winkler,
Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.
doi: 10.1016/j.jde.2014.04.023. |
[35] |
M. Winkler,
How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?, Trans. Amer. Math. Soc., 369 (2017), 3067-3125.
doi: 10.1090/tran/6733. |
[36] |
T. Xiang,
How strong a logistic damping can prevent blow-up for the minimal Keller-Segel chemotaxis system?, J. Math. Anal. Appl., 459 (2018), 1172-1200.
doi: 10.1016/j.jmaa.2017.11.022. |
[37] |
Q. Zhang and Y. Li,
Convergence rates of solutions for a two-dimensional chemotaxis-Navier-Stokes system, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2751-2759.
doi: 10.3934/dcdsb.2015.20.2751. |
show all references
References:
[1] |
X. Bai and M. Winkler,
Equilibration in a fully parabolic two-species chemotaxis system with competitive kinetics, Indiana Univ. Math. J., 65 (2016), 553-583.
doi: 10.1512/iumj.2016.65.5776. |
[2] |
T. Black, J. Lankeit and M. Mizukami,
On the weakly competitive case in a two-species chemotaxis model, IMA J. Appl. Math., 81 (2016), 860-876.
doi: 10.1093/imamat/hxw036. |
[3] |
X. Cao,
Global bounded solutions of the higher-dimensional Keller-Segel system under smallness conditions in optimal spaces, Discrete Contin. Dyn. Syst., 35 (2015), 1891-1904.
doi: 10.3934/dcds.2015.35.1891. |
[4] |
X. Cao, S. Kurima and M. Mizukami, Global existence and asymptotic behavior of classical solutions for a 3D two-species chemotaxis-Stokes system with competitive kinetics, arXiv: 1703.01794, Math Meth Appl Sci., 41 (2018), 3138–3154.
doi: 10.1002/mma.4807. |
[5] |
E. Conway, D. Hoff and J. Smoller,
Large time behavior of solutions of systems of nonlinear reaction-diffusion equations, SIAM J. Appl. Math., 35 (1978), 1-16.
doi: 10.1137/0135001. |
[6] |
P. De Mottoni and F. Rothe,
Convergence to homogeneous equilibrium state for generalized Volterra-Lotka systems with diffusion, SIAM J. Appl. Math., 37 (1979), 648-663.
doi: 10.1137/0137048. |
[7] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, 840, Springer-Verlag, Berlin-New York, 1981. |
[8] |
M. Hirata, S. Kurima, M. Mizukami and T. Yokota,
Boundedness and stabilization in a two-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, J. Differential Equations, 263 (2017), 470-490.
doi: 10.1016/j.jde.2017.02.045. |
[9] |
M. Hirata, S. Kurima, M. Mizukami and T. Yokota,
Boundedness and stabilization in a three-dimensional two-species chemotaxis-Navier-Stokes system with competitive kinetics, Proceedings of EQUADIFF 2017 Conference, (2017), 11-20.
doi: 10.1016/j.jde.2017.02.045. |
[10] |
H. Jin and Z. Wang,
Global stability of prey-taxis systems, J. Differential Equations, 262 (2017), 1257-1290.
doi: 10.1016/j.jde.2016.10.010. |
[11] |
Y. Kan-on and E. Yanagida,
Existence of nonconstant stable equilibria in competition-diffusion equations, Hiroshima Math. J., 23 (1993), 193-221.
|
[12] |
O. Ladyzhenskaya, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, RI, 1968. |
[13] |
J. Lankeit,
Long-term behaviour in a chemotaxis-fluid system with logistic source, Math. Models Methods Appl. Sci., 26 (2016), 2071-2109.
doi: 10.1142/S021820251640008X. |
[14] |
J. Lankeit and Y. Wang,
Global existence, boundedness and stabilization in a high-dimensional chemotaxis system with consumption, Discrete Contin. Dyn. Syst., 37 (2017), 6099-6121.
doi: 10.3934/dcds.2017262. |
[15] |
Y. Lou and W.-M. Ni,
Diffusion, self-diffusion and cross-diffusion, J. Differential Equaations, 131 (1996), 79-131.
doi: 10.1006/jdeq.1996.0157. |
[16] |
M. Mimura, S. I. Ei and Q. Fang,
Effect of domain-shape on coexistence problems in a competition-diffusion system, J. Math. Biol., 29 (1991), 219-237.
doi: 10.1007/BF00160536. |
[17] |
M. Mizukami,
Boundedness and asymptotic stability in a two-species chemotaxis-competition model with signal-dependent sensitivity, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 2301-2319.
doi: 10.3934/dcdsb.2017097. |
[18] |
M. Mizukami and T. Yokota,
Global existence and asymptotic stability of solutions to a two-species chemotaxis system with any chemical diffusion, J. Differential Equations, 261 (2016), 2650-2669.
doi: 10.1016/j.jde.2016.05.008. |
[19] |
M. Negreanu and J. I. Tello,
Asymptotic stability of a two species chemotaxis system with non-diffusive chemoattractant, J. Differential Equations, 258 (2015), 1592-1617.
doi: 10.1016/j.jde.2014.11.009. |
[20] |
M. Negreanu and J. I. Tello,
On a two species chemotaxis model with slow chemical diffusion, SIAM J. Math. Anal., 46 (2014), 3761-3781.
doi: 10.1137/140971853. |
[21] |
M. Porzio and V. Vespri,
Hölder estimates for local solutions of some doubly nonlinear degenerate parabolic equations, J. Differential Equations, 103 (1993), 146-178.
doi: 10.1006/jdeq.1993.1045. |
[22] |
C. Stinner, J. I. Tello and M. Winkler,
Competitive exclusion in a two-species chemotaxis model, J. Math. Biol., 68 (2014), 1607-1626.
doi: 10.1007/s00285-013-0681-7. |
[23] |
Y. Tao,
Boundedness in a chemotaxis model with oxygen consumption by bacteria, J. Math. Anal. Appl., 381 (2011), 521-529.
doi: 10.1016/j.jmaa.2011.02.041. |
[24] |
Y. Tao and M. Winkler,
Eventual smoothness and stabilization of large-data solutions in a three-dimensional chemotaxis system with consumption of chemoattractant, J. Differential Equations, 252 (2012), 2520-2543.
doi: 10.1016/j.jde.2011.07.010. |
[25] |
Y. Tao and M. Winkler,
Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2015), 2555-2573.
doi: 10.1007/s00033-015-0541-y. |
[26] |
Y. Tao and M. Winkler,
Large time behavior in a multidimensional chemotaxis-haptotaxis model with slow signal diffusion, SIAM J. Math. Anal., 47 (2015), 4229-4250.
doi: 10.1137/15M1014115. |
[27] |
Y. Tao and M. Winkler, Blow-up prevention by quadratic degradation in a two-dimensional Keller-Segel-Navier-Stokes system, Z. Angew. Math. Phys., 67 (2016), Art. 138, 23 pp.
doi: 10.1007/s00033-016-0732-1. |
[28] |
I. Tuval, L. Cisneros, C. Dombrowski, C. W. Wolgemuth, J. O. Kessler and R. E. Goldstein,
Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. U.S.A., 102 (2005), 2277-2282.
doi: 10.1073/pnas.0406724102. |
[29] |
M. Winkler,
Aggregation vs. global diffusive behavior in the higher-dimensional Keller-Segel model, J. Differential Equations, 248 (2010), 2889-2905.
doi: 10.1016/j.jde.2010.02.008. |
[30] |
M. Winkler,
Boundedness in the higher-dimensional parabolic-parabolic chemotaxis system with logistic source, Comm. Partial Differential Equations, 35 (2010), 1516-1537.
doi: 10.1080/03605300903473426. |
[31] |
M. Winkler,
Global large-data solutions in a chemotaxis-(Navier-)Stokes system modeling cellular swimming in fluid drops, Comm. Partial Differential Equations, 37 (2012), 319-351.
doi: 10.1080/03605302.2011.591865. |
[32] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pures Appl., 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[33] |
M. Winkler,
Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Ration. Mech. Anal., 211 (2014), 455-487.
doi: 10.1007/s00205-013-0678-9. |
[34] |
M. Winkler,
Global asymptotic stability of constant equilibria in a fully parabolic chemotaxis system with strong logistic dampening, J. Differential Equations, 257 (2014), 1056-1077.
doi: 10.1016/j.jde.2014.04.023. |
[35] |
M. Winkler,
How far do chemotaxis-driven forces influence regularity in the Navier-Stokes system?, Trans. Amer. Math. Soc., 369 (2017), 3067-3125.
doi: 10.1090/tran/6733. |
[36] |
T. Xiang,
How strong a logistic damping can prevent blow-up for the minimal Keller-Segel chemotaxis system?, J. Math. Anal. Appl., 459 (2018), 1172-1200.
doi: 10.1016/j.jmaa.2017.11.022. |
[37] |
Q. Zhang and Y. Li,
Convergence rates of solutions for a two-dimensional chemotaxis-Navier-Stokes system, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 2751-2759.
doi: 10.3934/dcdsb.2015.20.2751. |
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