- Previous Article
- DCDS Home
- This Issue
-
Next Article
Concentrated solutions for a critical elliptic equation
Boundedness and stabilization of a three-dimensional parabolic-elliptic Keller-Segel-Stokes system
School of Mathematics and Information Sciences, Yantai University, Yantai 264005, China |
$\begin{align} \left\{ \begin{array}{l} n_t+u\cdot\nabla n = \Delta n-\nabla\cdot(nS(n)\nabla c),\quad x\in \Omega, t>0,\\ u\cdot\nabla c = \Delta c-c+n,\quad x\in \Omega, t>0,\\ u_t+\nabla P = \Delta u+n\nabla \phi,\quad x\in \Omega, t>0,\\ \nabla\cdot u = 0,\quad x\in \Omega, t>0\\ \end{array}\right. \end{align} \;\;\;\;\;\;\;\;\;\;\;\;(KSF)$ |
$ n $ |
$ c $ |
$ u $ |
$ \Omega \subseteq \mathbb{R}^3 $ |
$ S\in C^2(\bar{\Omega}) $ |
$ |S(n)|\leq C_S(1 + n)^{-\alpha} \; \; \; \; \text{for all}\; \; n\geq0 $ |
$ C_S > 0 $ |
$ \alpha> 0 $ |
$ \alpha> \frac{1}{2} $ |
$ C_S $ |
$ (\bar{n}_0, \bar{n}_0, 0) $ |
$ \bar{n}_0 = \frac{1}{|\Omega|}\int_{\Omega}n_0 $ |
$ n_0\not \equiv0 $ |
$ \Omega $ |
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] |
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. |
[3] |
Y. Ke and J. Zheng, An optimal result for global existence in a three-dimensional Keller-Segel-Navier-Stokes system involving tensor-valued sensitivity with saturation, Calc. Var. Partial Differential Equations, 58 (2019), 27pp.
doi: 10.1007/s00526-019-1568-2. |
[4] |
K. J. Painter and T. Hillen,
Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.
|
[5] |
Y. Peng and Z. Xiang, Global existence and boundedness in a 3D Keller–Segel–Stokes system with nonlinear diffusion and rotational flux, Z. Angew. Math. Phys., 68 (2017), 26pp.
doi: 10.1007/s00033-017-0816-6. |
[6] |
H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2001.
doi: 10.1007/978-3-0348-8255-2. |
[7] |
Y. Tao and M. Winkler,
Locally bounded global solutions in a three-dimensional chemotaxis–Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178.
doi: 10.1016/j.anihpc.2012.07.002. |
[8] |
Y. Wang, M. Winkler and Z. Xiang,
Global classical solutions in a two-dimensional chemotaxis–Navier–Stokes system with subcritical sensitivity, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 18 (2018), 421-466.
doi: 10.2422/2036-2145.201603_004. |
[9] |
Y. Wang and Z. Xiang,
Global existence and boundedness in a Keller–Segel–Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations, 259 (2015), 7578-7609.
doi: 10.1016/j.jde.2015.08.027. |
[10] |
Y. Wang and Z. Xiang,
Global existence and boundedness in a Keller–Segel–Stokes system involving a tensor-valued sensitivity with saturation: The 3D case, J. Differential Equations, 261 (2016), 4944-4973.
doi: 10.1016/j.jde.2016.07.010. |
[11] |
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. |
[12] |
M. Winkler,
Boundedness and large time behavior in a three-dimensional chemotaxis–Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differential Equations, 54 (2015), 3789-3828.
doi: 10.1007/s00526-015-0922-2. |
[13] |
M. Winkler,
Does fluid interaction affect regularity in the three-dimensional Keller–Segel System with saturated sensitivity?, J. Math. Fluid Mech., 20 (2018), 1889-1909.
doi: 10.1007/s00021-018-0395-0. |
[14] |
M. Winkler,
Global existence and stabilization in a degenerate chemotaxis–Stokes system with mildly strong diffusion enhancement, J. Differential Equations, 264 (2018), 6109-6151.
doi: 10.1016/j.jde.2018.01.027. |
[15] |
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. |
[16] |
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. |
[17] |
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. |
[18] |
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. |
[19] |
Q. Zhang and 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. |
[20] |
J. Zheng, Eventual smoothness and stabilization in a three-dimensional Keller-Segel-Navier-Stokes system with rotational flux, Calc. Var. Partial Differential Equations, 61 (2022), 34pp.
doi: 10.1007/s00526-021-02164-6. |
[21] |
J. Zheng, Global classical solutions and stabilization in a two-dimensional parabolic–elliptic Keller–Segel–Stokes system, J. Math. Fluid Mech., 23 (2021), 25pp.
doi: 10.1007/s00021-021-00600-3. |
[22] |
J. Zheng,
Global existence and boundedness in a three-dimensional chemotaxis–Stokes system with nonlinear diffusion and general sensitivity, Ann. Mat. Pura Appl. (4), 201 (2022), 243-288.
doi: 10.1007/s10231-021-01115-4. |
[23] |
J. Zheng,
A new result for the global existence (and boundedness) and regularity of a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization, J. Differential Equations, 272 (2021), 164-202.
doi: 10.1016/j.jde.2020.09.029. |
[24] |
J. Zheng,
An optimal result for global existence and boundedness in a three-dimensional Keller–Segel–Stokes system with nonlinear diffusion, J. Differential Equations, 267 (2019), 2385-2415.
doi: 10.1016/j.jde.2019.03.013. |
[25] |
J. Zheng and Y. Ke,
Blow-up prevention by nonlinear diffusion in a 2D Keller–Segel–Navier–Stokes system with rotational flux, J. Differential Equations, 268 (2020), 7092-7120.
doi: 10.1016/j.jde.2019.11.071. |
[26] |
J. Zheng and Y. Ke,
Global bounded weak solutions for a chemotaxis–Stokes system with nonlinear diffusion and rotation, J. Differential Equations, 289 (2021), 182-235.
doi: 10.1016/j.jde.2021.04.020. |
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] |
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. |
[3] |
Y. Ke and J. Zheng, An optimal result for global existence in a three-dimensional Keller-Segel-Navier-Stokes system involving tensor-valued sensitivity with saturation, Calc. Var. Partial Differential Equations, 58 (2019), 27pp.
doi: 10.1007/s00526-019-1568-2. |
[4] |
K. J. Painter and T. Hillen,
Volume-filling and quorum-sensing in models for chemosensitive movement, Can. Appl. Math. Q., 10 (2002), 501-543.
|
[5] |
Y. Peng and Z. Xiang, Global existence and boundedness in a 3D Keller–Segel–Stokes system with nonlinear diffusion and rotational flux, Z. Angew. Math. Phys., 68 (2017), 26pp.
doi: 10.1007/s00033-017-0816-6. |
[6] |
H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach, Birkhäuser Advanced Texts: Basler Lehrbücher, Birkhäuser Verlag, Basel, 2001.
doi: 10.1007/978-3-0348-8255-2. |
[7] |
Y. Tao and M. Winkler,
Locally bounded global solutions in a three-dimensional chemotaxis–Stokes system with nonlinear diffusion, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 157-178.
doi: 10.1016/j.anihpc.2012.07.002. |
[8] |
Y. Wang, M. Winkler and Z. Xiang,
Global classical solutions in a two-dimensional chemotaxis–Navier–Stokes system with subcritical sensitivity, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5), 18 (2018), 421-466.
doi: 10.2422/2036-2145.201603_004. |
[9] |
Y. Wang and Z. Xiang,
Global existence and boundedness in a Keller–Segel–Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations, 259 (2015), 7578-7609.
doi: 10.1016/j.jde.2015.08.027. |
[10] |
Y. Wang and Z. Xiang,
Global existence and boundedness in a Keller–Segel–Stokes system involving a tensor-valued sensitivity with saturation: The 3D case, J. Differential Equations, 261 (2016), 4944-4973.
doi: 10.1016/j.jde.2016.07.010. |
[11] |
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. |
[12] |
M. Winkler,
Boundedness and large time behavior in a three-dimensional chemotaxis–Stokes system with nonlinear diffusion and general sensitivity, Calc. Var. Partial Differential Equations, 54 (2015), 3789-3828.
doi: 10.1007/s00526-015-0922-2. |
[13] |
M. Winkler,
Does fluid interaction affect regularity in the three-dimensional Keller–Segel System with saturated sensitivity?, J. Math. Fluid Mech., 20 (2018), 1889-1909.
doi: 10.1007/s00021-018-0395-0. |
[14] |
M. Winkler,
Global existence and stabilization in a degenerate chemotaxis–Stokes system with mildly strong diffusion enhancement, J. Differential Equations, 264 (2018), 6109-6151.
doi: 10.1016/j.jde.2018.01.027. |
[15] |
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. |
[16] |
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. |
[17] |
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. |
[18] |
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. |
[19] |
Q. Zhang and 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. |
[20] |
J. Zheng, Eventual smoothness and stabilization in a three-dimensional Keller-Segel-Navier-Stokes system with rotational flux, Calc. Var. Partial Differential Equations, 61 (2022), 34pp.
doi: 10.1007/s00526-021-02164-6. |
[21] |
J. Zheng, Global classical solutions and stabilization in a two-dimensional parabolic–elliptic Keller–Segel–Stokes system, J. Math. Fluid Mech., 23 (2021), 25pp.
doi: 10.1007/s00021-021-00600-3. |
[22] |
J. Zheng,
Global existence and boundedness in a three-dimensional chemotaxis–Stokes system with nonlinear diffusion and general sensitivity, Ann. Mat. Pura Appl. (4), 201 (2022), 243-288.
doi: 10.1007/s10231-021-01115-4. |
[23] |
J. Zheng,
A new result for the global existence (and boundedness) and regularity of a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization, J. Differential Equations, 272 (2021), 164-202.
doi: 10.1016/j.jde.2020.09.029. |
[24] |
J. Zheng,
An optimal result for global existence and boundedness in a three-dimensional Keller–Segel–Stokes system with nonlinear diffusion, J. Differential Equations, 267 (2019), 2385-2415.
doi: 10.1016/j.jde.2019.03.013. |
[25] |
J. Zheng and Y. Ke,
Blow-up prevention by nonlinear diffusion in a 2D Keller–Segel–Navier–Stokes system with rotational flux, J. Differential Equations, 268 (2020), 7092-7120.
doi: 10.1016/j.jde.2019.11.071. |
[26] |
J. Zheng and Y. Ke,
Global bounded weak solutions for a chemotaxis–Stokes system with nonlinear diffusion and rotation, J. Differential Equations, 289 (2021), 182-235.
doi: 10.1016/j.jde.2021.04.020. |
[1] |
Kentarou Fujie, Takasi Senba. Global existence and boundedness in a parabolic-elliptic Keller-Segel system with general sensitivity. Discrete and Continuous Dynamical Systems - B, 2016, 21 (1) : 81-102. doi: 10.3934/dcdsb.2016.21.81 |
[2] |
Tobias Black. Global generalized solutions to a parabolic-elliptic Keller-Segel system with singular sensitivity. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 119-137. doi: 10.3934/dcdss.2020007 |
[3] |
Johannes Lankeit. Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 233-255. doi: 10.3934/dcdss.2020013 |
[4] |
Mengyao Ding, Sining Zheng. $ L^γ$-measure criteria for boundedness in a quasilinear parabolic-elliptic Keller-Segel system with supercritical sensitivity. Discrete and Continuous Dynamical Systems - B, 2019, 24 (7) : 2971-2988. doi: 10.3934/dcdsb.2018295 |
[5] |
Sachiko Ishida. $L^\infty$-decay property for quasilinear degenerate parabolic-elliptic Keller-Segel systems. Conference Publications, 2013, 2013 (special) : 335-344. doi: 10.3934/proc.2013.2013.335 |
[6] |
Ansgar Jüngel, Oliver Leingang. Blow-up of solutions to semi-discrete parabolic-elliptic Keller-Segel models. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4755-4782. doi: 10.3934/dcdsb.2019029 |
[7] |
Jinhuan Wang, Li Chen, Liang Hong. Parabolic elliptic type Keller-Segel system on the whole space case. Discrete and Continuous Dynamical Systems, 2016, 36 (2) : 1061-1084. doi: 10.3934/dcds.2016.36.1061 |
[8] |
Hi Jun Choe, Bataa Lkhagvasuren, Minsuk Yang. Wellposedness of the Keller-Segel Navier-Stokes equations in the critical Besov spaces. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2453-2464. doi: 10.3934/cpaa.2015.14.2453 |
[9] |
Piotr Biler, Ignacio Guerra, Grzegorz Karch. Large global-in-time solutions of the parabolic-parabolic Keller-Segel system on the plane. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2117-2126. doi: 10.3934/cpaa.2015.14.2117 |
[10] |
Chao Deng, Tong Li. Global existence and large time behavior of a 2D Keller-Segel system in logarithmic Lebesgue spaces. Discrete and Continuous Dynamical Systems - B, 2019, 24 (1) : 183-195. doi: 10.3934/dcdsb.2018093 |
[11] |
Hao Yu, Wei Wang, Sining Zheng. Boundedness of solutions to a fully parabolic Keller-Segel system with nonlinear sensitivity. Discrete and Continuous Dynamical Systems - B, 2017, 22 (4) : 1635-1644. doi: 10.3934/dcdsb.2017078 |
[12] |
Luca Battaglia. A general existence result for stationary solutions to the Keller-Segel system. Discrete and Continuous Dynamical Systems, 2019, 39 (2) : 905-926. doi: 10.3934/dcds.2019038 |
[13] |
Hai-Yang Jin, Zhi-An Wang. Global stabilization of the full attraction-repulsion Keller-Segel system. Discrete and Continuous Dynamical Systems, 2020, 40 (6) : 3509-3527. doi: 10.3934/dcds.2020027 |
[14] |
Hao Yu, Wei Wang, Sining Zheng. Global boundedness of solutions to a Keller-Segel system with nonlinear sensitivity. Discrete and Continuous Dynamical Systems - B, 2016, 21 (4) : 1317-1327. doi: 10.3934/dcdsb.2016.21.1317 |
[15] |
Ling Liu, Jiashan Zheng, Gui Bao. Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system modeling coral fertilization. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3437-3460. doi: 10.3934/dcdsb.2020068 |
[16] |
Miaoqing Tian, Sining Zheng. Global boundedness versus finite-time blow-up of solutions to a quasilinear fully parabolic Keller-Segel system of two species. Communications on Pure and Applied Analysis, 2016, 15 (1) : 243-260. doi: 10.3934/cpaa.2016.15.243 |
[17] |
Youshan Tao, Michael Winkler. Global existence and boundedness in a Keller-Segel-Stokes model with arbitrary porous medium diffusion. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1901-1914. doi: 10.3934/dcds.2012.32.1901 |
[18] |
Tomasz Cieślak, Kentarou Fujie. Global existence in the 1D quasilinear parabolic-elliptic chemotaxis system with critical nonlinearity. Discrete and Continuous Dynamical Systems - S, 2020, 13 (2) : 165-176. doi: 10.3934/dcdss.2020009 |
[19] |
Monica Marras, Stella Vernier Piro, Giuseppe Viglialoro. Lower bounds for blow-up in a parabolic-parabolic Keller-Segel system. Conference Publications, 2015, 2015 (special) : 809-816. doi: 10.3934/proc.2015.0809 |
[20] |
Kentarou Fujie, Chihiro Nishiyama, Tomomi Yokota. Boundedness in a quasilinear parabolic-parabolic Keller-Segel system with the sensitivity $v^{-1}S(u)$. Conference Publications, 2015, 2015 (special) : 464-472. doi: 10.3934/proc.2015.0464 |
2021 Impact Factor: 1.588
Tools
Metrics
Other articles
by authors
[Back to Top]