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February  2014, 7(1): 161-176. doi: 10.3934/dcdss.2014.7.161

Brownian point vortices and dd-model

Takashi Suzuki 1,

1. 

Division of Mathematical Science, Department of System Innovation, Graduate School of Engineering Science, Osaka University, 1-3 Machikane-yama, Toyonaka, Osaka, 560-8531

Received  January 2012 Revised  August 2012 Published  July 2013

We study the kinetic mean field equation on two-dimensional Brownian vortices; derivation, similarity to the DD-model, and existence and non-existence of global-in-time solution.
Keywords: chemotaxis., Brownian point vortex, DD model.
Mathematics Subject Classification: Primary: 35K57, 35Q82; Secondary: 35Q9.
Citation: Takashi Suzuki. Brownian point vortices and dd-model. Discrete and Continuous Dynamical Systems - S, 2014, 7 (1) : 161-176. doi: 10.3934/dcdss.2014.7.161
References:
[1]

N. D. Alikakos, $L^p$ bounds of solutions of reaction-diffusion equations, Comm. Partial Differential Equations, 4 (1979), 827-868. doi: 10.1080/03605307908820113.

[2]

F. Bavaud, Equilibrium properties of the Vlasov functional: The generalized Poisson-Boltzmann-Emden equation, Rev. Modern Physics, 63 (1991), 129-149. doi: 10.1103/RevModPhys.63.129.

[3]

P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.

[4]

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.

[5]

E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, Comm. Math. Phys., 143 (1992), 501-525. doi: 10.1007/BF02099262.

[6]

P.-H. Chavanis, Kinetic theory of $2D$ point vortices from a BBGKY-like hiearchy, Physica A, 387 (2008), 1123-1154. doi: 10.1016/j.physa.2007.10.022.

[7]

P.-H. Chavanis, Two-dimensional Brownian vortices, Physica A, 387 (2008), 6917-6942. doi: 10.1016/j.physa.2008.09.019.

[8]

C. Conca and E. Espejo, Threshold condition for global existence and blow-up to a radially symmetric drift-diffusion system, Applied Mathematics Letters, 25 (2012), 352-356. doi: 10.1016/j.aml.2011.09.013.

[9]

C. Conca, E. Espejo and K. Vilches, Remarks on the blow-up and global existence for a two-species chemotactic Keller-Segel system in $R^2$, Euro. J. Appl. Math., 22 (2011), 553-580. doi: 10.1017/S0956792511000258.

[10]

E. E. Espejo, M. Kurokiba and T. Suzuki, Blowup threshold and collapse mass separation for a drift-diffusion system in dimension two, preprint.

[11]

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.

[12]

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.

[13]

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.

[14]

G. L. Eyink and H. Spohn, Negative-temperature states and large-scale, long-lived vortices in two-dimensional turbulence, J. Statistical Physics, 70 (1993), 833-886. doi: 10.1007/BF01053597.

[15]

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.

[16]

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. doi: 10.2307/2153966.

[17]

G. Joyce and D. Montgomery, Negative temperature states for two-dimensional guiding-centre plasma, J. Plasma Phys., 10 (1973), 107-121.

[18]

M. K. H. Kiessling, Statistical mechanics of classical particles with logarithmic interaction, Comm. Pure Appl. Math., 46 (1993), 27-56. doi: 10.1002/cpa.3160460103.

[19]

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, 16 (2003), 427-452.

[20]

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.

[21]

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

[22]

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.

[23]

K. Nagasaki and T. Suzuki, Asymptotic analysis for two-dimensional elliptic eivgnvalue problem with exponentially dominated nonlinearities, Asymptoitc Analysis, 3 (1990), 173-188.

[24]

P. K. Newton, "The $N$-Vortex Problem: Analytical Techniques," Applied Mathematical Sciences, 145, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4684-9290-3.

[25]

L. Onsager, Statistical hydrodynamics, Suppl. Nuovo Cimento, 6 (1949), 279-287. doi: 10.1007/BF02780991.

[26]

T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology, Adv. Differential Equations, 6 (2001), 21-50.

[27]

T. Senba and T. Suzuki, Parabolic system of chemotaxis; blowup in a finite and in the infinite time, Meth. Appl. Anal., 8 (2001), 349-368.

[28]

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.

[29]

T. Suzuki, Global analysis for a two-dimensional elliptic eigenvalue problem with the exponential nonlinearity, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 9 (1992), 367-397.

[30]

T. Suzuki, "Free Energy and Self-Interacting Particles,'' Progress in Nonlinear Differential Equations and their Applications, 62, Birkhäuser Boston, Inc., Boston, 2005. doi: 10.1007/0-8176-4436-9.

[31]

T. Suzuki, "Mean Field Theories and Dual Variation,'' Atlantis Studies in Mathematics for Engineering and Science, 2, Atlantis Press, Paris; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.

[32]

T. Suzuki and T. Senba, "Applied Analysis, Mathematical Methods in Natural Science,'' Second edition, Imperial College Press, London, 2011.

[33]

T. Suzuki, Exclusion of boundary blowup for $2D$ chemotaxis system provided with Dirichlet condition for the Poisson part, preprint. doi: 10.1016/j.matpur.2013.01.004.

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: 10.1080/03605307908820113.

[2]

F. Bavaud, Equilibrium properties of the Vlasov functional: The generalized Poisson-Boltzmann-Emden equation, Rev. Modern Physics, 63 (1991), 129-149. doi: 10.1103/RevModPhys.63.129.

[3]

P. Biler, Local and global solvability of some parabolic systems modelling chemotaxis, Adv. Math. Sci. Appl., 8 (1998), 715-743.

[4]

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.

[5]

E. Caglioti, P.-L. Lions, C. Marchioro and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, Comm. Math. Phys., 143 (1992), 501-525. doi: 10.1007/BF02099262.

[6]

P.-H. Chavanis, Kinetic theory of $2D$ point vortices from a BBGKY-like hiearchy, Physica A, 387 (2008), 1123-1154. doi: 10.1016/j.physa.2007.10.022.

[7]

P.-H. Chavanis, Two-dimensional Brownian vortices, Physica A, 387 (2008), 6917-6942. doi: 10.1016/j.physa.2008.09.019.

[8]

C. Conca and E. Espejo, Threshold condition for global existence and blow-up to a radially symmetric drift-diffusion system, Applied Mathematics Letters, 25 (2012), 352-356. doi: 10.1016/j.aml.2011.09.013.

[9]

C. Conca, E. Espejo and K. Vilches, Remarks on the blow-up and global existence for a two-species chemotactic Keller-Segel system in $R^2$, Euro. J. Appl. Math., 22 (2011), 553-580. doi: 10.1017/S0956792511000258.

[10]

E. E. Espejo, M. Kurokiba and T. Suzuki, Blowup threshold and collapse mass separation for a drift-diffusion system in dimension two, preprint.

[11]

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.

[12]

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.

[13]

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.

[14]

G. L. Eyink and H. Spohn, Negative-temperature states and large-scale, long-lived vortices in two-dimensional turbulence, J. Statistical Physics, 70 (1993), 833-886. doi: 10.1007/BF01053597.

[15]

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.

[16]

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. doi: 10.2307/2153966.

[17]

G. Joyce and D. Montgomery, Negative temperature states for two-dimensional guiding-centre plasma, J. Plasma Phys., 10 (1973), 107-121.

[18]

M. K. H. Kiessling, Statistical mechanics of classical particles with logarithmic interaction, Comm. Pure Appl. Math., 46 (1993), 27-56. doi: 10.1002/cpa.3160460103.

[19]

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, 16 (2003), 427-452.

[20]

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.

[21]

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

[22]

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.

[23]

K. Nagasaki and T. Suzuki, Asymptotic analysis for two-dimensional elliptic eivgnvalue problem with exponentially dominated nonlinearities, Asymptoitc Analysis, 3 (1990), 173-188.

[24]

P. K. Newton, "The $N$-Vortex Problem: Analytical Techniques," Applied Mathematical Sciences, 145, Springer-Verlag, New York, 2001. doi: 10.1007/978-1-4684-9290-3.

[25]

L. Onsager, Statistical hydrodynamics, Suppl. Nuovo Cimento, 6 (1949), 279-287. doi: 10.1007/BF02780991.

[26]

T. Senba and T. Suzuki, Chemotactic collapse in a parabolic-elliptic system of mathematical biology, Adv. Differential Equations, 6 (2001), 21-50.

[27]

T. Senba and T. Suzuki, Parabolic system of chemotaxis; blowup in a finite and in the infinite time, Meth. Appl. Anal., 8 (2001), 349-368.

[28]

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.

[29]

T. Suzuki, Global analysis for a two-dimensional elliptic eigenvalue problem with the exponential nonlinearity, Ann. Inst. Henri Poincaré Anal. Non Linéaire, 9 (1992), 367-397.

[30]

T. Suzuki, "Free Energy and Self-Interacting Particles,'' Progress in Nonlinear Differential Equations and their Applications, 62, Birkhäuser Boston, Inc., Boston, 2005. doi: 10.1007/0-8176-4436-9.

[31]

T. Suzuki, "Mean Field Theories and Dual Variation,'' Atlantis Studies in Mathematics for Engineering and Science, 2, Atlantis Press, Paris; World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2008.

[32]

T. Suzuki and T. Senba, "Applied Analysis, Mathematical Methods in Natural Science,'' Second edition, Imperial College Press, London, 2011.

[33]

T. Suzuki, Exclusion of boundary blowup for $2D$ chemotaxis system provided with Dirichlet condition for the Poisson part, preprint. doi: 10.1016/j.matpur.2013.01.004.

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