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A backscattering model based on corrector theory of homogenization for the random Helmholtz equation
Global existence of weak solution in a chemotaxis-fluid system with nonlinear diffusion and rotational flux
1. | School of Mathematics, Southeast University, Nanjing 210096, China |
2. | Institute for Applied Mathematics, School of Mathematics, Southeast University, Nanjing 211189, China |
$\begin{eqnarray*} \left\{\begin{array}{lll}n_{t}+u\cdot\nabla n=\Delta n^m-\nabla\cdot(uS(x,n,c)\cdot\nabla c),&x\in\Omega,\ \ t>0,\\[1mm]c_t+u\cdot\nabla c=\Delta c-c+n,&x\in\Omega,\ \ t>0,\\[1mm]u_t+k(u\cdot\nabla)u=\Delta u+\nabla P+n\nabla\phi,&x\in\Omega,\ \ t>0\\[1mm]\nabla\cdot u=0,&x\in\Omega,\ \ t>0 \end{array}\right.\end{eqnarray*}$ |
$\Omega\subset\mathbb{R}^3$ |
$k\in\mathbb{R}$ |
$\phi\in W^{2,\infty}(\Omega)$ |
$S$ |
$\overline\Omega\times[0,\infty)^2\rightarrow\mathbb{R}^{3\times 3}$ |
$|S(x,n,c)|\leq S_0(n+1)^{-\alpha}\ \ {\rm for\ all}\ x\in\mathbb{R}^3,\ n\geq0,\ c\geq0.$ |
$m+\alpha>\frac{4}{3}$ |
$m>\frac{1}{3}$ |
References:
[1] |
V. Calvez and J. A. Carrillo,
Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl. (9), 86 (2006), 155-175.
doi: 10.1016/j.matpur.2006.04.002. |
[2] |
X. Cao and S. Ishida,
Global-in-time bounded weak solutions to a degenerate quasilinear Keller-Segel system with rotation, Nonlinearity, 27 (2014), 1899-1913.
doi: 10.1088/0951-7715/27/8/1899. |
[3] |
R. Duan, X. Li and Z. 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. |
[4] |
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. |
[5] |
H. He and Q. Zhang,
Global existence of weak solutions for the 3D chemotaxis-Navier-Stokes equations, Nonlinear Anal. Real World Appl., 35 (2017), 336-349.
doi: 10.1016/j.nonrwa.2016.11.006. |
[6] |
T. Hillen and K. Painter,
Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math., 26 (2001), 280-301.
doi: 10.1006/aama.2001.0721. |
[7] |
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. |
[8] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[9] |
A. Kiselev and L. Ryzhik,
Biomixing by chemotaxis and enhancement of biological reactions, Comm. Partial Differential Equations, 37 (2012), 298-318.
doi: 10.1080/03605302.2011.589879. |
[10] |
R. Kowalczyk,
Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), 566-588.
doi: 10.1016/j.jmaa.2004.12.009. |
[11] |
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. |
[12] |
Y. Li and Y. Li,
Finite-time blow-up in higher dimensional fully-parabolic chemotaxis system for two species, Nonlinear Anal., 109 (2014), 72-84.
doi: 10.1016/j.na.2014.05.021. |
[13] |
Y. Li and Y. Li,
Global boundedness of solutions for the chemotaxis-Navier-Stokes system in $\Bbb{R}^2$, J. Differential Equations, 261 (2016), 6570-6613.
doi: 10.1016/j.jde.2016.08.045. |
[14] |
J. Liu and Y. Wang,
Global existence and boundedness in a Keller-Segel-(Navier-)Stokes system with signal-dependent sensitivity, J. Math. Anal. Appl., 447 (2017), 499-528.
doi: 10.1016/j.jmaa.2016.10.028. |
[15] |
J. Liu and Y. Wang,
Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations, 262 (2017), 5271-5305.
doi: 10.1016/j.jde.2017.01.024. |
[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] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.
|
[18] |
K. Osaki and A. Yagi,
Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
|
[19] |
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), Art. 68, 26pp.
doi: 10.1007/s00033-017-0816-6. |
[20] |
Z. Tan and X. Zhang,
Decay estimates of the coupled chemotaxis-fluid equations in R3, J. Math. Anal. Appl., 410 (2014), 27-38.
doi: 10.1016/j.jmaa.2013.08.008. |
[21] |
Y. 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] |
Y. Tao and M. Winkler,
Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3165-3183.
doi: 10.3934/dcdsb.2015.20.3165. |
[23] |
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. |
[24] |
Y. Wang,
Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with subcritical sensitivity, Math. Models Methods Appl. Sci., 27 (2017), 2745-2780.
doi: 10.1142/S0218202517500579. |
[25] |
Y. Wang and X. Cao,
Global classical solutions of a 3D chemotaxis-Stokes system with rotation, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3235-3254.
doi: 10.3934/dcdsb.2015.20.3235. |
[26] |
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.
|
[27] |
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. |
[28] |
M. Winkler,
Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.
doi: 10.1002/mma.1146. |
[29] |
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. |
[30] |
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. |
[31] |
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. |
[32] |
M. Winkler,
Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal., 47 (2015), 3092-3115.
doi: 10.1137/140979708. |
[33] |
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. |
[34] |
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. |
[35] |
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. |
[36] |
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. |
[37] |
D. Wrzosek,
Long-time behaviour of solutions to a chemotaxis model with volume-filling effect, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 431-444.
doi: 10.1017/S0308210500004649. |
[38] |
M. Yang,
Global solutions to Keller-Segel-Navier-Stokes equations with a class of large initial data in critical Besov spaces, Math. Methods Appl. Sci., 40 (2017), 7425-7437.
doi: 10.1002/mma.4538. |
[39] |
H. Yu, W. Wang and S. Zheng,
Global classical solutions to the Keller-Segel-Navier-Stokes system with matrix-valued sensitivity, J. Math. Anal. Appl., 461 (2018), 1748-1770.
doi: 10.1016/j.jmaa.2017.12.048. |
[40] |
Q. Zhang and Y. Li,
Global existence and asymptotic properties of the solution to a two-species chemotaxis system, J. Math. Anal. Appl., 418 (2014), 47-63.
doi: 10.1016/j.jmaa.2014.03.084. |
[41] |
Q. Zhang and Y. Li,
Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484.
doi: 10.1007/s00033-015-0532-z. |
[42] |
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), 2527-2551.
doi: 10.3934/dcdsb.2015.20.2751. |
[43] |
Q. Zhang and Y. Li,
Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93.
doi: 10.1007/s00033-013-0383-4. |
[44] |
Q. Zhang and Y. Li,
Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 259 (2015), 3730-3754.
doi: 10.1016/j.jde.2015.05.012. |
[45] |
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. |
[46] |
J. Zheng,
Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 263 (2017), 2606-2629.
doi: 10.1016/j.jde.2017.04.005. |
show all references
References:
[1] |
V. Calvez and J. A. Carrillo,
Volume effects in the Keller-Segel model: Energy estimates preventing blow-up, J. Math. Pures Appl. (9), 86 (2006), 155-175.
doi: 10.1016/j.matpur.2006.04.002. |
[2] |
X. Cao and S. Ishida,
Global-in-time bounded weak solutions to a degenerate quasilinear Keller-Segel system with rotation, Nonlinearity, 27 (2014), 1899-1913.
doi: 10.1088/0951-7715/27/8/1899. |
[3] |
R. Duan, X. Li and Z. 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. |
[4] |
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. |
[5] |
H. He and Q. Zhang,
Global existence of weak solutions for the 3D chemotaxis-Navier-Stokes equations, Nonlinear Anal. Real World Appl., 35 (2017), 336-349.
doi: 10.1016/j.nonrwa.2016.11.006. |
[6] |
T. Hillen and K. Painter,
Global existence for a parabolic chemotaxis model with prevention of overcrowding, Adv. in Appl. Math., 26 (2001), 280-301.
doi: 10.1006/aama.2001.0721. |
[7] |
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. |
[8] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, Journal of Theoretical Biology, 26 (1970), 399-415.
doi: 10.1016/0022-5193(70)90092-5. |
[9] |
A. Kiselev and L. Ryzhik,
Biomixing by chemotaxis and enhancement of biological reactions, Comm. Partial Differential Equations, 37 (2012), 298-318.
doi: 10.1080/03605302.2011.589879. |
[10] |
R. Kowalczyk,
Preventing blow-up in a chemotaxis model, J. Math. Anal. Appl., 305 (2005), 566-588.
doi: 10.1016/j.jmaa.2004.12.009. |
[11] |
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. |
[12] |
Y. Li and Y. Li,
Finite-time blow-up in higher dimensional fully-parabolic chemotaxis system for two species, Nonlinear Anal., 109 (2014), 72-84.
doi: 10.1016/j.na.2014.05.021. |
[13] |
Y. Li and Y. Li,
Global boundedness of solutions for the chemotaxis-Navier-Stokes system in $\Bbb{R}^2$, J. Differential Equations, 261 (2016), 6570-6613.
doi: 10.1016/j.jde.2016.08.045. |
[14] |
J. Liu and Y. Wang,
Global existence and boundedness in a Keller-Segel-(Navier-)Stokes system with signal-dependent sensitivity, J. Math. Anal. Appl., 447 (2017), 499-528.
doi: 10.1016/j.jmaa.2016.10.028. |
[15] |
J. Liu and Y. Wang,
Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system involving a tensor-valued sensitivity with saturation, J. Differential Equations, 262 (2017), 5271-5305.
doi: 10.1016/j.jde.2017.01.024. |
[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] |
T. Nagai, T. Senba and K. Yoshida,
Application of the Trudinger-Moser inequality to a parabolic system of chemotaxis, Funkcial. Ekvac., 40 (1997), 411-433.
|
[18] |
K. Osaki and A. Yagi,
Finite dimensional attractor for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
|
[19] |
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), Art. 68, 26pp.
doi: 10.1007/s00033-017-0816-6. |
[20] |
Z. Tan and X. Zhang,
Decay estimates of the coupled chemotaxis-fluid equations in R3, J. Math. Anal. Appl., 410 (2014), 27-38.
doi: 10.1016/j.jmaa.2013.08.008. |
[21] |
Y. 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] |
Y. Tao and M. Winkler,
Boundedness vs. blow-up in a two-species chemotaxis system with two chemicals, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3165-3183.
doi: 10.3934/dcdsb.2015.20.3165. |
[23] |
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. |
[24] |
Y. Wang,
Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with subcritical sensitivity, Math. Models Methods Appl. Sci., 27 (2017), 2745-2780.
doi: 10.1142/S0218202517500579. |
[25] |
Y. Wang and X. Cao,
Global classical solutions of a 3D chemotaxis-Stokes system with rotation, Discrete Contin. Dyn. Syst. Ser. B, 20 (2015), 3235-3254.
doi: 10.3934/dcdsb.2015.20.3235. |
[26] |
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.
|
[27] |
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. |
[28] |
M. Winkler,
Does a 'volume-filling effect' always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.
doi: 10.1002/mma.1146. |
[29] |
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. |
[30] |
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. |
[31] |
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. |
[32] |
M. Winkler,
Large-data global generalized solutions in a chemotaxis system with tensor-valued sensitivities, SIAM J. Math. Anal., 47 (2015), 3092-3115.
doi: 10.1137/140979708. |
[33] |
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. |
[34] |
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. |
[35] |
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. |
[36] |
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. |
[37] |
D. Wrzosek,
Long-time behaviour of solutions to a chemotaxis model with volume-filling effect, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 431-444.
doi: 10.1017/S0308210500004649. |
[38] |
M. Yang,
Global solutions to Keller-Segel-Navier-Stokes equations with a class of large initial data in critical Besov spaces, Math. Methods Appl. Sci., 40 (2017), 7425-7437.
doi: 10.1002/mma.4538. |
[39] |
H. Yu, W. Wang and S. Zheng,
Global classical solutions to the Keller-Segel-Navier-Stokes system with matrix-valued sensitivity, J. Math. Anal. Appl., 461 (2018), 1748-1770.
doi: 10.1016/j.jmaa.2017.12.048. |
[40] |
Q. Zhang and Y. Li,
Global existence and asymptotic properties of the solution to a two-species chemotaxis system, J. Math. Anal. Appl., 418 (2014), 47-63.
doi: 10.1016/j.jmaa.2014.03.084. |
[41] |
Q. Zhang and Y. Li,
Boundedness in a quasilinear fully parabolic Keller-Segel system with logistic source, Z. Angew. Math. Phys., 66 (2015), 2473-2484.
doi: 10.1007/s00033-015-0532-z. |
[42] |
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), 2527-2551.
doi: 10.3934/dcdsb.2015.20.2751. |
[43] |
Q. Zhang and Y. Li,
Global boundedness of solutions to a two-species chemotaxis system, Z. Angew. Math. Phys., 66 (2015), 83-93.
doi: 10.1007/s00033-013-0383-4. |
[44] |
Q. Zhang and Y. Li,
Global weak solutions for the three-dimensional chemotaxis-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 259 (2015), 3730-3754.
doi: 10.1016/j.jde.2015.05.012. |
[45] |
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. |
[46] |
J. Zheng,
Global weak solutions in a three-dimensional Keller-Segel-Navier-Stokes system with nonlinear diffusion, J. Differential Equations, 263 (2017), 2606-2629.
doi: 10.1016/j.jde.2017.04.005. |
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