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Qualitative analysis of a Lotka-Volterra competition-diffusion-advection system
Global classical solutions to two-dimensional chemotaxis-shallow water system
College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China |
We consider the Cauchy problem of two-dimensional chemotaxis-shallow water system in the present paper. For regular initial data with small energy but possibly large oscillations, we prove the global well-posedness of classical solution. Then, we show the large-time behavior of the solution using the time-independent lower-order estimates as well.
References:
[1] |
N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler,
Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[2] |
M. Chae, K. Kang and J. Lee,
On existence of the smooth solutions to the coupled chemotaxis-fluid equations, Discrete Contin. Dyn. Syst. A, 33 (2013), 2271-2297.
|
[3] |
M. Chae, K. Kang and J. Lee,
Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Partial Differential Equations, 39 (2014), 1205-1235.
doi: 10.1080/03605302.2013.852224. |
[4] |
J. H. Che, L. Chen, B. Duan and Z. Luo,
On the existence of local strong solutions to chemotaxis-shallow water system with large data and vacuum, J. Differential Equations, 261 (2016), 6758-6789.
doi: 10.1016/j.jde.2016.09.005. |
[5] |
R. J. Duan, X. Li and Z. Y. Xiang,
Global existence and large time behavior for a two-dimensional chemotaxis-Navier-Stokes system, J. Differential Equations, 263 (2017), 6284-6316.
doi: 10.1016/j.jde.2017.07.015. |
[6] |
R. Duan, A. Lorz and P. Markowich,
Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673.
doi: 10.1080/03605302.2010.497199. |
[7] |
R. J. Duan and Z. Y. Xiang,
A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, Int. Math. Res. Not. IMRN, 2014 (2014), 1833-1852.
doi: 10.1093/imrn/rns270. |
[8] |
E. Espejo and M. Winkler,
Global classical solvability and stabilization in a two-dimensional chemotaxis-Navier-Stokes system modeling coral fertilization, Nonlinearity, 31 (2018), 1227-1259.
doi: 10.1088/1361-6544/aa9d5f. |
[9] |
T. Hillen and K. J. Painter,
A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[10] |
D. Horstmann,
From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.
|
[11] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[12] |
X. D. Huang and J. Li,
Global classical and weak solutions to the three-dimensional full compressible Navier-Stokes system with vacuum and large oscillations, Arch. Ration. Mech. Anal., 227 (2018), 995-1059.
doi: 10.1007/s00205-017-1188-y. |
[13] |
X. D. Huang, J. Li and Z. P. Xin,
Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585.
doi: 10.1002/cpa.21382. |
[14] |
E. F. Keller and L. A. Segel,
Initiation of slide mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[15] |
E. F. Keller and L. A. Segel,
Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[16] |
J.-G. Liu and A. Lorz,
A coupled chemotaxis-fluid model: Global existence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 643-652.
doi: 10.1016/j.anihpc.2011.04.005. |
[17] |
A. Lorz,
Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.
doi: 10.1142/S0218202510004507. |
[18] |
A. Matsumura and T. Nishida,
The initial value problems for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.
doi: 10.1215/kjm/1250522322. |
[19] |
L. Nirenberg,
On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 13 (1959), 115-162.
|
[20] |
Z. Tan and X. Zhang,
Decay estimates of the coupled chemotaxis-fluid equations in $\mathbb{R}^3$, J. Math. Anal. Appl., 410 (2014), 27-38.
doi: 10.1016/j.jmaa.2013.08.008. |
[21] |
Y. S. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[22] |
Q. Tao and Z.-A. Yao,
Global existence and large time behavior for a two-dimensional chemotaxis-shallow water system, J. Differential Equations, 265 (2018), 3092-3129.
doi: 10.1016/j.jde.2018.05.002. |
[23] |
I. Tuval, L. Cisneros, C. Dombrowski, C. Wolgemuth, J. Kessler and R. Goldstein,
Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277-2282.
doi: 10.1073/pnas.0406724102. |
[24] |
W. K. Wang and Y. C. Wang,
The $L^p$ decay estimates for the chemotaxis-shallow water system, J. Math. Anal. Appl., 474 (2019), 640-665.
doi: 10.1016/j.jmaa.2019.01.066. |
[25] |
Y. L. Wang, M. Winkler and Z. Y. Xiang,
Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (2018), 421-466.
doi: 10.1109/tps.2017.2783887. |
[26] |
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. |
[27] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pure. Appl., 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[28] |
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. |
[29] |
M. Winkler,
Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352.
doi: 10.1016/j.anihpc.2015.05.002. |
[30] |
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. |
[31] |
M. Winkler,
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: 10.1016/j.jfa.2018.12.009. |
[32] |
Q. Zhang and X. X. Zheng,
Global well-posedness for the two-dimensional incompressible chemotaxis-Navier-Stokes equations, SIAM J. Math. Anal., 46 (2014), 3078-3105.
doi: 10.1137/130936920. |
show all references
References:
[1] |
N. Bellomo, A. Bellouquid, Y. Tao and M. Winkler,
Toward a mathematical theory of Keller-Segel models of pattern formation in biological tissues, Math. Models Methods Appl. Sci., 25 (2015), 1663-1763.
doi: 10.1142/S021820251550044X. |
[2] |
M. Chae, K. Kang and J. Lee,
On existence of the smooth solutions to the coupled chemotaxis-fluid equations, Discrete Contin. Dyn. Syst. A, 33 (2013), 2271-2297.
|
[3] |
M. Chae, K. Kang and J. Lee,
Global existence and temporal decay in Keller-Segel models coupled to fluid equations, Comm. Partial Differential Equations, 39 (2014), 1205-1235.
doi: 10.1080/03605302.2013.852224. |
[4] |
J. H. Che, L. Chen, B. Duan and Z. Luo,
On the existence of local strong solutions to chemotaxis-shallow water system with large data and vacuum, J. Differential Equations, 261 (2016), 6758-6789.
doi: 10.1016/j.jde.2016.09.005. |
[5] |
R. J. Duan, X. Li and Z. Y. Xiang,
Global existence and large time behavior for a two-dimensional chemotaxis-Navier-Stokes system, J. Differential Equations, 263 (2017), 6284-6316.
doi: 10.1016/j.jde.2017.07.015. |
[6] |
R. Duan, A. Lorz and P. Markowich,
Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673.
doi: 10.1080/03605302.2010.497199. |
[7] |
R. J. Duan and Z. Y. Xiang,
A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, Int. Math. Res. Not. IMRN, 2014 (2014), 1833-1852.
doi: 10.1093/imrn/rns270. |
[8] |
E. Espejo and M. Winkler,
Global classical solvability and stabilization in a two-dimensional chemotaxis-Navier-Stokes system modeling coral fertilization, Nonlinearity, 31 (2018), 1227-1259.
doi: 10.1088/1361-6544/aa9d5f. |
[9] |
T. Hillen and K. J. Painter,
A user's guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[10] |
D. Horstmann,
From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. I, Jahresber. Deutsch. Math.-Verein., 105 (2003), 103-165.
|
[11] |
D. Horstmann and M. Winkler,
Boundedness vs. blow-up in a chemotaxis system, J. Differential Equations, 215 (2005), 52-107.
doi: 10.1016/j.jde.2004.10.022. |
[12] |
X. D. Huang and J. Li,
Global classical and weak solutions to the three-dimensional full compressible Navier-Stokes system with vacuum and large oscillations, Arch. Ration. Mech. Anal., 227 (2018), 995-1059.
doi: 10.1007/s00205-017-1188-y. |
[13] |
X. D. Huang, J. Li and Z. P. Xin,
Global well-posedness of classical solutions with large oscillations and vacuum to the three-dimensional isentropic compressible Navier-Stokes equations, Comm. Pure Appl. Math., 65 (2012), 549-585.
doi: 10.1002/cpa.21382. |
[14] |
E. F. Keller and L. A. Segel,
Initiation of slide mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[15] |
E. F. Keller and L. A. Segel,
Model for chemotaxis, J. Theor. Biol., 30 (1971), 225-234.
doi: 10.1016/0022-5193(71)90050-6. |
[16] |
J.-G. Liu and A. Lorz,
A coupled chemotaxis-fluid model: Global existence, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28 (2011), 643-652.
doi: 10.1016/j.anihpc.2011.04.005. |
[17] |
A. Lorz,
Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.
doi: 10.1142/S0218202510004507. |
[18] |
A. Matsumura and T. Nishida,
The initial value problems for the equations of motion of viscous and heat-conductive gases, J. Math. Kyoto Univ., 20 (1980), 67-104.
doi: 10.1215/kjm/1250522322. |
[19] |
L. Nirenberg,
On elliptic partial differential equations, Ann. Sc. Norm. Super. Pisa Cl. Sci., 13 (1959), 115-162.
|
[20] |
Z. Tan and X. Zhang,
Decay estimates of the coupled chemotaxis-fluid equations in $\mathbb{R}^3$, J. Math. Anal. Appl., 410 (2014), 27-38.
doi: 10.1016/j.jmaa.2013.08.008. |
[21] |
Y. S. Tao and M. Winkler,
Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with subcritical sensitivity, J. Differential Equations, 252 (2012), 692-715.
doi: 10.1016/j.jde.2011.08.019. |
[22] |
Q. Tao and Z.-A. Yao,
Global existence and large time behavior for a two-dimensional chemotaxis-shallow water system, J. Differential Equations, 265 (2018), 3092-3129.
doi: 10.1016/j.jde.2018.05.002. |
[23] |
I. Tuval, L. Cisneros, C. Dombrowski, C. Wolgemuth, J. Kessler and R. Goldstein,
Bacterial swimming and oxygen transport near contact lines, Proc. Natl. Acad. Sci. USA, 102 (2005), 2277-2282.
doi: 10.1073/pnas.0406724102. |
[24] |
W. K. Wang and Y. C. Wang,
The $L^p$ decay estimates for the chemotaxis-shallow water system, J. Math. Anal. Appl., 474 (2019), 640-665.
doi: 10.1016/j.jmaa.2019.01.066. |
[25] |
Y. L. Wang, M. Winkler and Z. Y. Xiang,
Global classical solutions in a two-dimensional chemotaxis-Navier-Stokes system with subcritical sensitivity, Ann. Sc. Norm. Super. Pisa Cl. Sci., 18 (2018), 421-466.
doi: 10.1109/tps.2017.2783887. |
[26] |
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. |
[27] |
M. Winkler,
Finite-time blow-up in the higher-dimensional parabolic-parabolic Keller-Segel system, J. Math. Pure. Appl., 100 (2013), 748-767.
doi: 10.1016/j.matpur.2013.01.020. |
[28] |
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. |
[29] |
M. Winkler,
Global weak solutions in a three-dimensional chemotaxis-Navier-Stokes system, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1329-1352.
doi: 10.1016/j.anihpc.2015.05.002. |
[30] |
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. |
[31] |
M. Winkler,
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: 10.1016/j.jfa.2018.12.009. |
[32] |
Q. Zhang and X. X. Zheng,
Global well-posedness for the two-dimensional incompressible chemotaxis-Navier-Stokes equations, SIAM J. Math. Anal., 46 (2014), 3078-3105.
doi: 10.1137/130936920. |
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