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On critical Choquard equation with potential well
Boundedness and large time behavior in a two-dimensional Keller-Segel-Navier-Stokes system with signal-dependent diffusion and sensitivity
Department of Mathematics, South China University of Technology, Guangzhou 510640, China |
$\begin{cases}\tag{*}n_t+u·\nabla n = \nabla ·(d(c)\nabla n)-\nabla ·(χ (c) n\nabla c)+a n-bn^2, &x∈ Ω, ~~t>0, \\ c_t+u·\nabla c = Δ c+ n-c,&x∈ Ω, ~~t>0, \\ u_t+ u·\nabla u = Δ u-\nabla P+n\nabla φ,&x∈ Ω, ~~t>0, \\\nabla · u = 0& x∈ Ω, \ t>0, \end{cases}$ |
$Ω\subset \mathbb{R}^2$ |
$a≥0$ |
$b>0$ |
$d(c)$ |
$χ(c)$ |
$(d(c), χ (c))∈ [C^2([0, ∞))]^2$ |
$d(c), χ(c)>0$ |
$c≥0$ |
$d'(c)<0$ |
$\lim\limits_{c\to∞}d(c) = 0$ |
$\lim\limits_{c\to∞} \frac{χ (c)}{d(c)}$ |
$\lim\limits_{c\to∞}\frac{d'(c)}{d(c)}$ |
$\lim\limits_{c\to∞}d(c) = 0$ |
$d(c)$ |
$(n, c, u)$ |
$(\frac{a}{b}, \frac{a}{b}, 0)$ |
$b>\frac{K_0}{16}$ |
$K_0 = \max\limits_{0≤c ≤∞}\frac{|χ(c)|^2}{d(c)}$ |
References:
[1] |
N. Bellomo, A. Bellouquid, Y. S. Tao and M. Winkler,
Towards 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] |
J. P. Bourguignon and H. Brezis,
Remarks on Euler Equation, J. Functional Analysis, 15 (1974), 341-363.
doi: 10.1016/0022-1236(74)90027-5. |
[3] |
M. Chae, K. Kang and J. Lee,
Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297.
doi: 10.3934/dcds.2013.33.2271. |
[4] |
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. |
[5] |
A. Chertock, K. Fellner, A. Kurganov, A. Lorz and P. A. Markowich,
Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: A high-resolution numerical approach, J. Fluid Mech., 694 (2012), 155-190.
doi: 10.1017/jfm.2011.534. |
[6] |
S. Childress and J. K. Percus,
Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237.
doi: 10.1016/0025-5564(81)90055-9. |
[7] |
T. Ciéslak and C. Stinner,
Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.
doi: 10.1016/j.jde.2012.01.045. |
[8] |
T. Ciéslak and C. Stinner,
New critical exponents in a fully parabolic quasilinear Keller-Segel and applications to volume filling models, J. Differential Equations, 258 (2015), 2080-2113.
doi: 10.1016/j.jde.2014.12.004. |
[9] |
M. DiFrancesco, A. Lorz and P. A. Markowich,
Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior, Discrete Contin. Dyn. Syst., 28 (2010), 1437-1453.
doi: 10.3934/dcds.2010.28.1437. |
[10] |
R. J. Duan, A. Lorz and P. A. Markowich,
Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673.
doi: 10.1080/03605302.2010.497199. |
[11] |
R. J. Duan and Z. Xiang,
A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, Int. Math. Res. Not. IMRN, 7 (2014), 1833-1852.
doi: 10.1093/imrn/rns270. |
[12] |
M. Eisenbach, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion. Chemotaxis, Imperial College Press, London, 2004. |
[13] |
E. Espejo and T. Suzuki,
Reaction terms avoiding aggregation in slow fluids, Nonlinear Anal. Real World Appl., 21 (2015), 110-126.
doi: 10.1016/j.nonrwa.2014.07.001. |
[14] |
X. Fu, H. Tang, C. Liu, J. D. Huang, T. Hwa and P. Lenz,
Stripe formation in bacterial system with density-suppressed motility, Phys. Rev. Lett., 108 (2012), 198102.
doi: 10.1103/PhysRevLett.108.198102. |
[15] |
Y. Giga,
Solutions for semilinear parabolic equations in Lp and regularity of weak solution of the Navier-Stokes system, J. Differential Equations, 61 (1986), 186-212.
doi: 10.1016/0022-0396(86)90096-3. |
[16] |
Y. Giga and H. Sohr,
Abstract Lp estimate for the Cauchy problem with application to the Navier-Stokes system in exterior domains, J. Funct. Anal., 102 (1991), 72-94.
doi: 10.1016/0022-1236(91)90136-S. |
[17] |
M. Herrero and J. Velázquez,
A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683.
|
[18] |
T. Hillen and K. Painter,
A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[19] |
T. Hillen and A. Potapov,
The one-dimensional chemotaxis model: Global existence and asymptotic profile, Math. Methods Appl. Sci., 27 (2004), 1783-1801.
doi: 10.1002/mma.569. |
[20] |
D. Horstmann,
From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅰ, Jahresber. Deutsch. Math.-Verien., 105 (2003), 103-165.
|
[21] |
D. Horstmann,
From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅱ, Jahresber. Deutsch. Math.-Verien., 106 (2004), 51-69.
|
[22] |
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. |
[23] |
W. Jäger and S. Luckhaus,
On explosions of solutions to a system of partial differential equations modeling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.
doi: 10.1090/S0002-9947-1992-1046835-6. |
[24] |
H. Y. Jin, Y. J. Kim and Z. A. Wang, Boundedness, stabilization and pattern formation driven by density-suppressed motility, SIAM J. Appl. Math., 2018, to appear. |
[25] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
|
[26] |
R. Kowalczyk and Z. Szyma |
[27] |
O. Ladyzhenskaya, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, RI, 1968. |
[28] |
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. |
[29] |
A. Lorz,
Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.
doi: 10.1142/S0218202510004507. |
[30] |
A. Lorz,
A coupled Keller-Segel-Stokes model: Global existence for small initial data and blow-up delay, Commun. Math. Sci., 10 (2012), 555-574.
doi: 10.4310/CMS.2012.v10.n2.a7. |
[31] |
T. Nagai,
Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
|
[32] |
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.
|
[33] |
K. Osaki and A. Yagi,
Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
|
[34] |
K. Painter and J. A. Sherratt,
Modeling the movement of interacting cell populations, J. Theor. Biol., 225 (2003), 327-339.
doi: 10.1016/S0022-5193(03)00258-3. |
[35] |
M. 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. |
[36] |
T. Senba and T. Suzuki,
Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367.
doi: 10.4310/MAA.2001.v8.n2.a9. |
[37] |
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. |
[38] |
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. |
[39] |
Y. S. 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. |
[40] |
Y. S. 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. |
[41] |
Y. S. Tao and M. Winkler,
Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2016), 2555-2573.
doi: 10.1007/s00033-015-0541-y. |
[42] |
Y. Tao and M. Winkler,
Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system, Math. Models Meth. Appl. Sci., 27 (2017), 1645-1683.
doi: 10.1142/S0218202517500282. |
[43] |
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. Nat. Acad. Sci., USA, 102 (2005), 2277–2282
doi: 10.1073/pnas.0406724102. |
[44] |
M. Winkler,
Does a "volume-filling effect" always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.
doi: 10.1002/mma.1146. |
[45] |
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. |
[46] |
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. |
[47] |
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. |
[48] |
M. Winkler,
Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Rational Mech. Anal., 211 (2014), 455-487.
doi: 10.1007/s00205-013-0678-9. |
[49] |
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. |
[50] |
M. Winkler,
Global existence and slow grow-up in a quasilinear Keller-Segel system with exponentially decaying diffusivity, Nonlinearity, 30 (2017), 735-764.
doi: 10.1088/1361-6544/aa565b. |
[51] |
C. Yoon and Y. J. Kim,
Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion, Acta Application Mathematics, 149 (2017), 101-123.
doi: 10.1007/s10440-016-0089-7. |
show all references
References:
[1] |
N. Bellomo, A. Bellouquid, Y. S. Tao and M. Winkler,
Towards 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] |
J. P. Bourguignon and H. Brezis,
Remarks on Euler Equation, J. Functional Analysis, 15 (1974), 341-363.
doi: 10.1016/0022-1236(74)90027-5. |
[3] |
M. Chae, K. Kang and J. Lee,
Existence of smooth solutions to coupled chemotaxis-fluid equations, Discrete Contin. Dyn. Syst., 33 (2013), 2271-2297.
doi: 10.3934/dcds.2013.33.2271. |
[4] |
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. |
[5] |
A. Chertock, K. Fellner, A. Kurganov, A. Lorz and P. A. Markowich,
Sinking, merging and stationary plumes in a coupled chemotaxis-fluid model: A high-resolution numerical approach, J. Fluid Mech., 694 (2012), 155-190.
doi: 10.1017/jfm.2011.534. |
[6] |
S. Childress and J. K. Percus,
Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237.
doi: 10.1016/0025-5564(81)90055-9. |
[7] |
T. Ciéslak and C. Stinner,
Finite-time blowup and global-in-time unbounded solutions to a parabolic-parabolic quasilinear Keller-Segel system in higher dimensions, J. Differential Equations, 252 (2012), 5832-5851.
doi: 10.1016/j.jde.2012.01.045. |
[8] |
T. Ciéslak and C. Stinner,
New critical exponents in a fully parabolic quasilinear Keller-Segel and applications to volume filling models, J. Differential Equations, 258 (2015), 2080-2113.
doi: 10.1016/j.jde.2014.12.004. |
[9] |
M. DiFrancesco, A. Lorz and P. A. Markowich,
Chemotaxis-fluid coupled model for swimming bacteria with nonlinear diffusion: global existence and asymptotic behavior, Discrete Contin. Dyn. Syst., 28 (2010), 1437-1453.
doi: 10.3934/dcds.2010.28.1437. |
[10] |
R. J. Duan, A. Lorz and P. A. Markowich,
Global solutions to the coupled chemotaxis-fluid equations, Comm. Partial Differential Equations, 35 (2010), 1635-1673.
doi: 10.1080/03605302.2010.497199. |
[11] |
R. J. Duan and Z. Xiang,
A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion, Int. Math. Res. Not. IMRN, 7 (2014), 1833-1852.
doi: 10.1093/imrn/rns270. |
[12] |
M. Eisenbach, A note on global existence for the chemotaxis-Stokes model with nonlinear diffusion. Chemotaxis, Imperial College Press, London, 2004. |
[13] |
E. Espejo and T. Suzuki,
Reaction terms avoiding aggregation in slow fluids, Nonlinear Anal. Real World Appl., 21 (2015), 110-126.
doi: 10.1016/j.nonrwa.2014.07.001. |
[14] |
X. Fu, H. Tang, C. Liu, J. D. Huang, T. Hwa and P. Lenz,
Stripe formation in bacterial system with density-suppressed motility, Phys. Rev. Lett., 108 (2012), 198102.
doi: 10.1103/PhysRevLett.108.198102. |
[15] |
Y. Giga,
Solutions for semilinear parabolic equations in Lp and regularity of weak solution of the Navier-Stokes system, J. Differential Equations, 61 (1986), 186-212.
doi: 10.1016/0022-0396(86)90096-3. |
[16] |
Y. Giga and H. Sohr,
Abstract Lp estimate for the Cauchy problem with application to the Navier-Stokes system in exterior domains, J. Funct. Anal., 102 (1991), 72-94.
doi: 10.1016/0022-1236(91)90136-S. |
[17] |
M. Herrero and J. Velázquez,
A blow-up mechanism for a chemotaxis model, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 24 (1997), 633-683.
|
[18] |
T. Hillen and K. Painter,
A users guide to PDE models for chemotaxis, J. Math. Biol., 58 (2009), 183-217.
doi: 10.1007/s00285-008-0201-3. |
[19] |
T. Hillen and A. Potapov,
The one-dimensional chemotaxis model: Global existence and asymptotic profile, Math. Methods Appl. Sci., 27 (2004), 1783-1801.
doi: 10.1002/mma.569. |
[20] |
D. Horstmann,
From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅰ, Jahresber. Deutsch. Math.-Verien., 105 (2003), 103-165.
|
[21] |
D. Horstmann,
From 1970 until present: The Keller-Segel model in chemotaxis and its consequences. Ⅱ, Jahresber. Deutsch. Math.-Verien., 106 (2004), 51-69.
|
[22] |
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. |
[23] |
W. Jäger and S. Luckhaus,
On explosions of solutions to a system of partial differential equations modeling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824.
doi: 10.1090/S0002-9947-1992-1046835-6. |
[24] |
H. Y. Jin, Y. J. Kim and Z. A. Wang, Boundedness, stabilization and pattern formation driven by density-suppressed motility, SIAM J. Appl. Math., 2018, to appear. |
[25] |
E. F. Keller and L. A. Segel,
Initiation of slime mold aggregation viewed as an instability, J. Theor. Biol., 26 (1970), 399-415.
|
[26] |
R. Kowalczyk and Z. Szyma |
[27] |
O. Ladyzhenskaya, V. Solonnikov and N. Uralceva, Linear and Quasilinear Equations of Parabolic Type, AMS, Providence, RI, 1968. |
[28] |
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. |
[29] |
A. Lorz,
Coupled chemotaxis fluid model, Math. Models Methods Appl. Sci., 20 (2010), 987-1004.
doi: 10.1142/S0218202510004507. |
[30] |
A. Lorz,
A coupled Keller-Segel-Stokes model: Global existence for small initial data and blow-up delay, Commun. Math. Sci., 10 (2012), 555-574.
doi: 10.4310/CMS.2012.v10.n2.a7. |
[31] |
T. Nagai,
Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601.
|
[32] |
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.
|
[33] |
K. Osaki and A. Yagi,
Finite dimensional attractors for one-dimensional Keller-Segel equations, Funkcial. Ekvac., 44 (2001), 441-469.
|
[34] |
K. Painter and J. A. Sherratt,
Modeling the movement of interacting cell populations, J. Theor. Biol., 225 (2003), 327-339.
doi: 10.1016/S0022-5193(03)00258-3. |
[35] |
M. 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. |
[36] |
T. Senba and T. Suzuki,
Parabolic system of chemotaxis: Blowup in a finite and the infinite time, Methods Appl. Anal., 8 (2001), 349-367.
doi: 10.4310/MAA.2001.v8.n2.a9. |
[37] |
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. |
[38] |
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. |
[39] |
Y. S. 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. |
[40] |
Y. S. 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. |
[41] |
Y. S. Tao and M. Winkler,
Boundedness and decay enforced by quadratic degradation in a three-dimensional chemotaxis-fluid system, Z. Angew. Math. Phys., 66 (2016), 2555-2573.
doi: 10.1007/s00033-015-0541-y. |
[42] |
Y. Tao and M. Winkler,
Effects of signal-dependent motilities in a Keller-Segel-type reaction-diffusion system, Math. Models Meth. Appl. Sci., 27 (2017), 1645-1683.
doi: 10.1142/S0218202517500282. |
[43] |
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. Nat. Acad. Sci., USA, 102 (2005), 2277–2282
doi: 10.1073/pnas.0406724102. |
[44] |
M. Winkler,
Does a "volume-filling effect" always prevent chemotactic collapse?, Math. Methods Appl. Sci., 33 (2010), 12-24.
doi: 10.1002/mma.1146. |
[45] |
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. |
[46] |
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. |
[47] |
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. |
[48] |
M. Winkler,
Stabilization in a two-dimensional chemotaxis-Navier-Stokes system, Arch. Rational Mech. Anal., 211 (2014), 455-487.
doi: 10.1007/s00205-013-0678-9. |
[49] |
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. |
[50] |
M. Winkler,
Global existence and slow grow-up in a quasilinear Keller-Segel system with exponentially decaying diffusivity, Nonlinearity, 30 (2017), 735-764.
doi: 10.1088/1361-6544/aa565b. |
[51] |
C. Yoon and Y. J. Kim,
Global existence and aggregation in a Keller-Segel model with Fokker-Planck diffusion, Acta Application Mathematics, 149 (2017), 101-123.
doi: 10.1007/s10440-016-0089-7. |
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