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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 & 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.  doi: 10.1080/03605307908820113.  Google Scholar

[2]

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

[3]

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

[4]

P. Biler, W. Hebisch and T. Nadzieja, The Debye system: existence and large time behavior of solutions,, Nonlinear Analysis, 23 (1994), 1189.  doi: 10.1016/0362-546X(94)90101-5.  Google Scholar

[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.  doi: 10.1007/BF02099262.  Google Scholar

[6]

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

[7]

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

[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.  doi: 10.1016/j.aml.2011.09.013.  Google Scholar

[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.  doi: 10.1017/S0956792511000258.  Google Scholar

[10]

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

[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.   Google Scholar

[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.  doi: 10.1524/anly.2009.1029.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/BF01053597.  Google Scholar

[15]

H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis,, Math. Nachr., 195 (1998), 77.  doi: 10.1002/mana.19981950106.  Google Scholar

[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.  doi: 10.2307/2153966.  Google Scholar

[17]

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

[18]

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

[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.   Google Scholar

[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.  doi: 10.1016/j.jmaa.2007.11.017.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[23]

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

[24]

P. K. Newton, "The $N$-Vortex Problem: Analytical Techniques,", Applied Mathematical Sciences, 145 (2001).  doi: 10.1007/978-1-4684-9290-3.  Google Scholar

[25]

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

[26]

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

[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.   Google Scholar

[28]

I. Shafrir and G. Wolansky, Moser-Trudinger and logarithmic HLS inequalities for systems,, J. Euro. Math. Soc., 7 (2005), 413.  doi: 10.4171/JEMS/34.  Google Scholar

[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.   Google Scholar

[30]

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

[31]

T. Suzuki, "Mean Field Theories and Dual Variation,'', Atlantis Studies in Mathematics for Engineering and Science, 2 (2008).   Google Scholar

[32]

T. Suzuki and T. Senba, "Applied Analysis, Mathematical Methods in Natural Science,'', Second edition, (2011).   Google Scholar

[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.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

P. Biler, W. Hebisch and T. Nadzieja, The Debye system: existence and large time behavior of solutions,, Nonlinear Analysis, 23 (1994), 1189.  doi: 10.1016/0362-546X(94)90101-5.  Google Scholar

[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.  doi: 10.1007/BF02099262.  Google Scholar

[6]

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

[7]

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

[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.  doi: 10.1016/j.aml.2011.09.013.  Google Scholar

[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.  doi: 10.1017/S0956792511000258.  Google Scholar

[10]

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

[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.   Google Scholar

[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.  doi: 10.1524/anly.2009.1029.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1007/BF01053597.  Google Scholar

[15]

H. Gajewski and K. Zacharias, Global behaviour of a reaction-diffusion system modelling chemotaxis,, Math. Nachr., 195 (1998), 77.  doi: 10.1002/mana.19981950106.  Google Scholar

[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.  doi: 10.2307/2153966.  Google Scholar

[17]

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

[18]

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

[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.   Google Scholar

[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.  doi: 10.1016/j.jmaa.2007.11.017.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[23]

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

[24]

P. K. Newton, "The $N$-Vortex Problem: Analytical Techniques,", Applied Mathematical Sciences, 145 (2001).  doi: 10.1007/978-1-4684-9290-3.  Google Scholar

[25]

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

[26]

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

[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.   Google Scholar

[28]

I. Shafrir and G. Wolansky, Moser-Trudinger and logarithmic HLS inequalities for systems,, J. Euro. Math. Soc., 7 (2005), 413.  doi: 10.4171/JEMS/34.  Google Scholar

[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.   Google Scholar

[30]

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

[31]

T. Suzuki, "Mean Field Theories and Dual Variation,'', Atlantis Studies in Mathematics for Engineering and Science, 2 (2008).   Google Scholar

[32]

T. Suzuki and T. Senba, "Applied Analysis, Mathematical Methods in Natural Science,'', Second edition, (2011).   Google Scholar

[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.  Google Scholar

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