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A stability result for the Stokes-Boussinesq equations in infinite 3d channels
Blowup threshold and collapse mass separation for a drift-diffusion system in space-dimension two
1. | Departamento de Matemáticas, Universidad de los Andes, Bogotá |
2. | Muroran Institute of Technology, 27-1 Mizumoto, Muroran, 050-8585, Japan |
3. | Division of Mathematical Science, Department of System Innovation, Graduate School of Engineering Science, Osaka University, 1-3 Machikane-yama, Toyonaka, Osaka, 560-8531 |
References:
[1] |
N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.
doi: 0.1080/03605307908820113. |
[2] |
P. Biler, Local and global solvability of some systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743. |
[3] |
P. Biler, W. Hebisch and T. Nadzieja, The Debye system: existence and large time behavior of solutions, Nonlinear Analysis, 23 (1994), 1189-1209.
doi: 10.1016/0362-546X(94)90101-5. |
[4] |
P. Biler, D. Hilhorst and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitating interaction of particles, II, Colloq. Math., 67 (1994), 297-308. |
[5] |
C. Conca and E. E. Espejo, Threshold condition for global existence and blow-up to a radially symmetric drift-diffusion system, Applied Math Letters, (2012), 352-356.
doi: 10.1016/j.aml.2011.09.013. |
[6] |
J. Dolbeault and C. Schmeiser, The two-dimensional Keller-Segel model after blow-up, Discrete and Continuous Dynamical Systems B, 25 (2009), 109-121.
doi: 10.3934/dcds.2009.25.109. |
[7] |
E. E. Espejo, A. Stevens and T. Suzuki, Simultaneous blowup and mass separation during collapse in an interacting system of chemotactic species, Differential and Integral Equations, 25 (2012), 251-288. |
[8] |
E. E. Espejo, A. Stevens and J. J. L. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis, 29 (2009), 317-338.
doi: 10.1524/anly.2009.1029. |
[9] |
E. E. Espejo, A. Stevens and J. J. L. Velázquez, A note on non-simultaneous blow-up for a drift-diffusion model, Differential and Integral Equations, 23 (2010), 451-462. |
[10] |
H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.
doi: 10.1002/mana.19981950106. |
[11] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1983. |
[12] |
T. Iwaniec and A. Verde, On the operator $L(f) = f \log |f|$, J. Funct. Anal., 169 (1999), 391-420.
doi: 10.1006/jfan.1999.3443. |
[13] |
W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. |
[14] |
M. Kurokiba and T. Ogawa, Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type, Differential and Integral Equations, 4 (2003), 427-452. |
[15] |
M. Kurokiba and T. Ogawa, Wellposedness of the drit-diffusion system in $L^p$ arising from the semiconductor device simulation, J. Math. Anal. Appl., 342 (2008), 1052-1067.
doi: 10.1016/j.jmaa.2007.11.017. |
[16] |
M. Kurokiba, T. Nagai and T. Ogawa, The uniform boundedness and threshold for the global existence of the radial solution to a drift-diffusion system, Comm. Pure Appl. Anal., 5 (2006), 97-106.
doi: 10.3934/cpaa.2006.5.97. |
[17] |
T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. |
[18] |
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. |
[19] |
T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. |
[20] |
F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Mech. Appl. Anal., 9 (2002), 533-562. |
[21] |
M. M. Rao and Z. D. Ren, "Theory of Orlicz Spaces," Marcel Dekker, New York, 1991. |
[22] |
T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology, Adv. Differential Equations, 6 (2001), 21-50. |
[23] |
T. Senba and T. Suzuki, Parabolic system of chemotaxis: blowup in a finite and the infinite time, Meth. Appl. Anal., 8 (2001), 349-368. |
[24] |
T. Senba and T. Suzuki, Weak solutions to a parabolic-elliptic system of chemotaxis, J. Funct. Anal., 191 (2002), 17-51.
doi: 10.1006/jfan.2001.3802. |
[25] |
I. Shafrir and G. Wolansky, Moser-Trudinger and logarithmic HLS inequalities for systems, J. Euro. Math. Soc., 7 (2005), 413-448.
doi: 10.4171/JEMS/34. |
[26] |
T. Suzuki, "Free Energy and Self-Interacting Particles," Birkhäuser, Boston, 2005 |
[27] |
T. Suzuki, "Mean Field Theories and Dual Variation," Atlantis Press, Amsterdam-Paris, 2008. |
[28] |
T. Suzuki, 2D Brownian point vortices and the drift-diffusion model, Discrete and Continuous Dynamical Systems Ser. S., accepted for publication. |
[29] |
T. Suzuki, Exclusion of boundary blowup for 2D chemotaxis system provided with Dirichlet boundary condition for the Poisson part, J. Math. Pure Appl., accepted for publication.
doi: 10.1016/j.matpur.2013.01.004. |
[30] |
T. Suzuki and T. Senba, "Applied Analysis - Mathematical Methods in Natural Science," 2nd edition, Imperial College Press, 2011. |
[31] |
G. Wolansky, Multi-components chemotactic system in the absence of conflicts, Euro. J. Appl. Math., 3 (2002), 641-661.
doi: 10.1017/S0956792501004843. |
show all references
References:
[1] |
N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868.
doi: 0.1080/03605307908820113. |
[2] |
P. Biler, Local and global solvability of some systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743. |
[3] |
P. Biler, W. Hebisch and T. Nadzieja, The Debye system: existence and large time behavior of solutions, Nonlinear Analysis, 23 (1994), 1189-1209.
doi: 10.1016/0362-546X(94)90101-5. |
[4] |
P. Biler, D. Hilhorst and T. Nadzieja, Existence and nonexistence of solutions for a model of gravitating interaction of particles, II, Colloq. Math., 67 (1994), 297-308. |
[5] |
C. Conca and E. E. Espejo, Threshold condition for global existence and blow-up to a radially symmetric drift-diffusion system, Applied Math Letters, (2012), 352-356.
doi: 10.1016/j.aml.2011.09.013. |
[6] |
J. Dolbeault and C. Schmeiser, The two-dimensional Keller-Segel model after blow-up, Discrete and Continuous Dynamical Systems B, 25 (2009), 109-121.
doi: 10.3934/dcds.2009.25.109. |
[7] |
E. E. Espejo, A. Stevens and T. Suzuki, Simultaneous blowup and mass separation during collapse in an interacting system of chemotactic species, Differential and Integral Equations, 25 (2012), 251-288. |
[8] |
E. E. Espejo, A. Stevens and J. J. L. Velázquez, Simultaneous finite time blow-up in a two-species model for chemotaxis, Analysis, 29 (2009), 317-338.
doi: 10.1524/anly.2009.1029. |
[9] |
E. E. Espejo, A. Stevens and J. J. L. Velázquez, A note on non-simultaneous blow-up for a drift-diffusion model, Differential and Integral Equations, 23 (2010), 451-462. |
[10] |
H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis, Math. Nachr., 195 (1998), 77-114.
doi: 10.1002/mana.19981950106. |
[11] |
D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations of Second Order," Springer-Verlag, Berlin, 1983. |
[12] |
T. Iwaniec and A. Verde, On the operator $L(f) = f \log |f|$, J. Funct. Anal., 169 (1999), 391-420.
doi: 10.1006/jfan.1999.3443. |
[13] |
W. Jäger and S. Luckhaus, On explosions of solutions to a system of partial differential equations modelling chemotaxis, Trans. Amer. Math. Soc., 329 (1992), 819-824. |
[14] |
M. Kurokiba and T. Ogawa, Finite time blow-up of the solution for a nonlinear parabolic equation of drift-diffusion type, Differential and Integral Equations, 4 (2003), 427-452. |
[15] |
M. Kurokiba and T. Ogawa, Wellposedness of the drit-diffusion system in $L^p$ arising from the semiconductor device simulation, J. Math. Anal. Appl., 342 (2008), 1052-1067.
doi: 10.1016/j.jmaa.2007.11.017. |
[16] |
M. Kurokiba, T. Nagai and T. Ogawa, The uniform boundedness and threshold for the global existence of the radial solution to a drift-diffusion system, Comm. Pure Appl. Anal., 5 (2006), 97-106.
doi: 10.3934/cpaa.2006.5.97. |
[17] |
T. Nagai, Blow-up of radially symmetric solutions to a chemotaxis system, Adv. Math. Sci. Appl., 5 (1995), 581-601. |
[18] |
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. |
[19] |
T. Nagai, Blowup of nonradial solutions to parabolic-elliptic systems modeling chemotaxis in two-dimensional domains, J. Inequal. Appl., 6 (2001), 37-55. |
[20] |
F. Poupaud, Diagonal defect measures, adhesion dynamics and Euler equation, Mech. Appl. Anal., 9 (2002), 533-562. |
[21] |
M. M. Rao and Z. D. Ren, "Theory of Orlicz Spaces," Marcel Dekker, New York, 1991. |
[22] |
T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology, Adv. Differential Equations, 6 (2001), 21-50. |
[23] |
T. Senba and T. Suzuki, Parabolic system of chemotaxis: blowup in a finite and the infinite time, Meth. Appl. Anal., 8 (2001), 349-368. |
[24] |
T. Senba and T. Suzuki, Weak solutions to a parabolic-elliptic system of chemotaxis, J. Funct. Anal., 191 (2002), 17-51.
doi: 10.1006/jfan.2001.3802. |
[25] |
I. Shafrir and G. Wolansky, Moser-Trudinger and logarithmic HLS inequalities for systems, J. Euro. Math. Soc., 7 (2005), 413-448.
doi: 10.4171/JEMS/34. |
[26] |
T. Suzuki, "Free Energy and Self-Interacting Particles," Birkhäuser, Boston, 2005 |
[27] |
T. Suzuki, "Mean Field Theories and Dual Variation," Atlantis Press, Amsterdam-Paris, 2008. |
[28] |
T. Suzuki, 2D Brownian point vortices and the drift-diffusion model, Discrete and Continuous Dynamical Systems Ser. S., accepted for publication. |
[29] |
T. Suzuki, Exclusion of boundary blowup for 2D chemotaxis system provided with Dirichlet boundary condition for the Poisson part, J. Math. Pure Appl., accepted for publication.
doi: 10.1016/j.matpur.2013.01.004. |
[30] |
T. Suzuki and T. Senba, "Applied Analysis - Mathematical Methods in Natural Science," 2nd edition, Imperial College Press, 2011. |
[31] |
G. Wolansky, Multi-components chemotactic system in the absence of conflicts, Euro. J. Appl. Math., 3 (2002), 641-661.
doi: 10.1017/S0956792501004843. |
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