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A second order energy stable scheme for the Cahn-Hilliard-Hele-Shaw equations
Global existence and large time behavior of a 2D Keller-Segel system in logarithmic Lebesgue spaces
1. | School of Mathematics and Statistics, Jiangsu Normal University, Xuzhou 221116, China |
2. | Department of Mathematics, The University of Iowa, Iowa City, IA 52242, USA |
This paper is devoted to the global analysis for the two-dimensional parabolic-parabolic Keller-Segel system in the whole space. By well balanced arguments of the $L^1$ and $L^∞$ spaces, we first prove global well-posedness of the system in $L^1× L^∞$ which partially answers the question posted by Kozono et al in [
References:
[1] |
A. Blanchet, J. Dolbeault and B. Perthame,
Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Diff. Eqns., 44 (2006), 1-32.
|
[2] |
V. Calvez and L. Corrias,
The parabolic-parabolic Keller-Segel model in $R^2$, Commun. Math. Sci., 6 (2008), 417-447.
doi: 10.4310/CMS.2008.v6.n2.a8. |
[3] |
S. Childress and J. K. Percus,
Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237.
doi: 10.1016/0025-5564(81)90055-9. |
[4] |
S. Childress and J. K. Percus, Chemotactic collapse in two dimensions, Lecture Notes in Biomathematics, 55, Springer, Berlin-Heidelberg-New York, 61-66,1984. |
[5] |
L. Corrias, B. Perthame and H. Zaag,
A chemotaxis model motivated by angiogenesis, C. R. Acad. Sci. Paris, Ser. Ⅰ., 336 (2003), 141-146.
doi: 10.1016/S1631-073X(02)00008-0. |
[6] |
L. Corrias, B. Perthame and H. Zaag,
Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28.
doi: 10.1007/s00032-003-0026-x. |
[7] |
J. I. Diaz, T. Nagai and J. M. Rakotoson,
Symmetrization techniques on unbounded domains: Application to a chemotaxis system on $\mathbb{R}^n$, J. Differential Equations, 145 (1998), 156-183.
doi: 10.1006/jdeq.1997.3389. |
[8] |
C. Deng and T. Li,
Well-posedness of the 3D Parabolic-hyperbolic Keller-Segel System in the Sobolev space framework, J. Differential Equations, 257 (2014), 1311-1332.
doi: 10.1016/j.jde.2014.05.014. |
[9] |
M. Eisenbach, Chemotaxis, Imperial College Press, London, 2004. |
[10] |
Y. Guo and H. J. Hwang,
Pattern formation (Ⅰ): The Keller-Segel model, J. Differential Equations, 249 (2010), 1519-1530.
doi: 10.1016/j.jde.2010.07.025. |
[11] |
D. Horstmann,
From 1970 until present: The Keller-Segel model in chemotaxis and its consequences Ⅰ., Jahresber. Dutsch. Math. Ver., 105 (2003), 103-165.
|
[12] |
Y. Kagei and Y. Maekawa,
On asymptotic behaviors of solutions to parabolic systems modelling chemotaxis, J. Differential Equations, 253 (2012), 2951-2992.
doi: 10.1016/j.jde.2012.08.028. |
[13] |
T. Kato,
Strong ${L}^p$-solutions of the Navier-Stokes equation in $\mathbb{R}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[14] |
E. F. Keller and L. A. Segel,
Initiation of slime 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] |
E. F. Keller and L. A. Segel,
Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[17] |
H. Kozono and Y. Sugiyama,
Global strong solution to the semi-linear Keller-Segel system of parabolic-parabolic type with small data in scale invariant spaces, J. Differential Equations, 247 (2009), 1-32.
doi: 10.1016/j.jde.2009.03.027. |
[18] |
H. Kozono and Y. Sugiyama,
Keller-Segel system of parabolic-parabolic type with initial data in weak $L^{n/2}$ and its application to self-similar solutions, Indiana Univ. Math. J., 57 (2008), 1467-1500.
doi: 10.1512/iumj.2008.57.3316. |
[19] |
H. Kozono, Y. Sugiyama and T. Wachi,
Existence and uniqueness theorem on mild solutions to the Keller-Segel system in the scaling invariant space, J. Differential Equations, 252 (2012), 1213-1228.
doi: 10.1016/j.jde.2011.08.025. |
[20] |
P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Research Notes in Mathematics, Chapman & Hall/CRC, 2002. |
[21] |
H. A. Levine and B. D. Sleeman,
A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683-730.
doi: 10.1137/S0036139995291106. |
[22] |
D. Li, T. Li and K. Zhao,
On a hyperbolic-parabolic system modeling chemotaxis, Math. Model. Meth. Appl. Sci., 21 (2011), 1631-1650.
doi: 10.1142/S0218202511005519. |
[23] |
T. Li, R. H. Pan and K. Zhao,
Global dynamics of a chemotaxis model on bounded domains with large data, SIAM J. Appl. Math., 72 (2012), 417-443.
doi: 10.1137/110829453. |
[24] |
T. Li and Z. A. Wang,
Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009), 1522-1541.
|
[25] |
T. Li and Z. A. Wang,
Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differential Equations, 250 (2011), 1310-1333.
doi: 10.1016/j.jde.2010.09.020. |
[26] |
C. S. Lin, W. M. Ni and I. Takagi,
Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1998), 1-27.
doi: 10.1016/0022-0396(88)90147-7. |
[27] |
T. Nagai and T. Ikeda,
Traveling waves in a chemotaxis model, J. Math. Biol., 30 (1991), 169-184.
doi: 10.1007/BF00160334. |
[28] |
H. Othmer and A. Stevens,
Aggregation, blowup and collapse: The ABCs of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081.
doi: 10.1137/S0036139995288976. |
[29] |
C. S. Patlak,
Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[30] |
B. D. Sleeman, M. Ward and J. Wei,
The existence and stability of spike patterns in a chemotaxis model, SIAM J. Appl. Math., 65 (2005), 790-817.
doi: 10.1137/S0036139902415117. |
[31] |
Z. A. Wang and T. Hillen,
Shock formation in a chemotaxis model, Math. Meth. Appl. Sci., 31 (2008), 45-70.
doi: 10.1002/mma.898. |
[32] |
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. |
[33] |
Y. Yang, H. Chen, W. Liu and B. D. Sleeman,
The solvability of some chemotaxis systems, J. Differential Equations, 212 (2005), 432-451.
doi: 10.1016/j.jde.2005.01.002. |
show all references
References:
[1] |
A. Blanchet, J. Dolbeault and B. Perthame,
Two-dimensional Keller-Segel model: Optimal critical mass and qualitative properties of the solutions, Electron. J. Diff. Eqns., 44 (2006), 1-32.
|
[2] |
V. Calvez and L. Corrias,
The parabolic-parabolic Keller-Segel model in $R^2$, Commun. Math. Sci., 6 (2008), 417-447.
doi: 10.4310/CMS.2008.v6.n2.a8. |
[3] |
S. Childress and J. K. Percus,
Nonlinear aspects of chemotaxis, Math. Biosci., 56 (1981), 217-237.
doi: 10.1016/0025-5564(81)90055-9. |
[4] |
S. Childress and J. K. Percus, Chemotactic collapse in two dimensions, Lecture Notes in Biomathematics, 55, Springer, Berlin-Heidelberg-New York, 61-66,1984. |
[5] |
L. Corrias, B. Perthame and H. Zaag,
A chemotaxis model motivated by angiogenesis, C. R. Acad. Sci. Paris, Ser. Ⅰ., 336 (2003), 141-146.
doi: 10.1016/S1631-073X(02)00008-0. |
[6] |
L. Corrias, B. Perthame and H. Zaag,
Global solutions of some chemotaxis and angiogenesis systems in high space dimensions, Milan J. Math., 72 (2004), 1-28.
doi: 10.1007/s00032-003-0026-x. |
[7] |
J. I. Diaz, T. Nagai and J. M. Rakotoson,
Symmetrization techniques on unbounded domains: Application to a chemotaxis system on $\mathbb{R}^n$, J. Differential Equations, 145 (1998), 156-183.
doi: 10.1006/jdeq.1997.3389. |
[8] |
C. Deng and T. Li,
Well-posedness of the 3D Parabolic-hyperbolic Keller-Segel System in the Sobolev space framework, J. Differential Equations, 257 (2014), 1311-1332.
doi: 10.1016/j.jde.2014.05.014. |
[9] |
M. Eisenbach, Chemotaxis, Imperial College Press, London, 2004. |
[10] |
Y. Guo and H. J. Hwang,
Pattern formation (Ⅰ): The Keller-Segel model, J. Differential Equations, 249 (2010), 1519-1530.
doi: 10.1016/j.jde.2010.07.025. |
[11] |
D. Horstmann,
From 1970 until present: The Keller-Segel model in chemotaxis and its consequences Ⅰ., Jahresber. Dutsch. Math. Ver., 105 (2003), 103-165.
|
[12] |
Y. Kagei and Y. Maekawa,
On asymptotic behaviors of solutions to parabolic systems modelling chemotaxis, J. Differential Equations, 253 (2012), 2951-2992.
doi: 10.1016/j.jde.2012.08.028. |
[13] |
T. Kato,
Strong ${L}^p$-solutions of the Navier-Stokes equation in $\mathbb{R}^m$, with applications to weak solutions, Math. Z., 187 (1984), 471-480.
doi: 10.1007/BF01174182. |
[14] |
E. F. Keller and L. A. Segel,
Initiation of slime 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] |
E. F. Keller and L. A. Segel,
Traveling bands of chemotactic bacteria: A theoretical analysis, J. Theor. Biol., 30 (1971), 235-248.
doi: 10.1016/0022-5193(71)90051-8. |
[17] |
H. Kozono and Y. Sugiyama,
Global strong solution to the semi-linear Keller-Segel system of parabolic-parabolic type with small data in scale invariant spaces, J. Differential Equations, 247 (2009), 1-32.
doi: 10.1016/j.jde.2009.03.027. |
[18] |
H. Kozono and Y. Sugiyama,
Keller-Segel system of parabolic-parabolic type with initial data in weak $L^{n/2}$ and its application to self-similar solutions, Indiana Univ. Math. J., 57 (2008), 1467-1500.
doi: 10.1512/iumj.2008.57.3316. |
[19] |
H. Kozono, Y. Sugiyama and T. Wachi,
Existence and uniqueness theorem on mild solutions to the Keller-Segel system in the scaling invariant space, J. Differential Equations, 252 (2012), 1213-1228.
doi: 10.1016/j.jde.2011.08.025. |
[20] |
P. G. Lemarié-Rieusset, Recent Developments in the Navier-Stokes Problem, Research Notes in Mathematics, Chapman & Hall/CRC, 2002. |
[21] |
H. A. Levine and B. D. Sleeman,
A system of reaction diffusion equations arising in the theory of reinforced random walks, SIAM J. Appl. Math., 57 (1997), 683-730.
doi: 10.1137/S0036139995291106. |
[22] |
D. Li, T. Li and K. Zhao,
On a hyperbolic-parabolic system modeling chemotaxis, Math. Model. Meth. Appl. Sci., 21 (2011), 1631-1650.
doi: 10.1142/S0218202511005519. |
[23] |
T. Li, R. H. Pan and K. Zhao,
Global dynamics of a chemotaxis model on bounded domains with large data, SIAM J. Appl. Math., 72 (2012), 417-443.
doi: 10.1137/110829453. |
[24] |
T. Li and Z. A. Wang,
Nonlinear stability of traveling waves to a hyperbolic-parabolic system modeling chemotaxis, SIAM J. Appl. Math., 70 (2009), 1522-1541.
|
[25] |
T. Li and Z. A. Wang,
Asymptotic nonlinear stability of traveling waves to conservation laws arising from chemotaxis, J. Differential Equations, 250 (2011), 1310-1333.
doi: 10.1016/j.jde.2010.09.020. |
[26] |
C. S. Lin, W. M. Ni and I. Takagi,
Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations, 72 (1998), 1-27.
doi: 10.1016/0022-0396(88)90147-7. |
[27] |
T. Nagai and T. Ikeda,
Traveling waves in a chemotaxis model, J. Math. Biol., 30 (1991), 169-184.
doi: 10.1007/BF00160334. |
[28] |
H. Othmer and A. Stevens,
Aggregation, blowup and collapse: The ABCs of taxis in reinforced random walks, SIAM J. Appl. Math., 57 (1997), 1044-1081.
doi: 10.1137/S0036139995288976. |
[29] |
C. S. Patlak,
Random walk with persistence and external bias, Bull. Math. Biophys., 15 (1953), 311-338.
doi: 10.1007/BF02476407. |
[30] |
B. D. Sleeman, M. Ward and J. Wei,
The existence and stability of spike patterns in a chemotaxis model, SIAM J. Appl. Math., 65 (2005), 790-817.
doi: 10.1137/S0036139902415117. |
[31] |
Z. A. Wang and T. Hillen,
Shock formation in a chemotaxis model, Math. Meth. Appl. Sci., 31 (2008), 45-70.
doi: 10.1002/mma.898. |
[32] |
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. |
[33] |
Y. Yang, H. Chen, W. Liu and B. D. Sleeman,
The solvability of some chemotaxis systems, J. Differential Equations, 212 (2005), 432-451.
doi: 10.1016/j.jde.2005.01.002. |
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