American Institute of Mathematical Sciences

Advanced
December  2019, 12(6): 1297-1312. doi: 10.3934/krm.2019050

Large amplitude stationary solutions of the Morrow model of gas ionization

Walter A. Strauss 1, and  Masahiro Suzuki 2,,

1. 

Department of Mathematics and Lefschetz Center for Dynamical Systems, Brown University, Providence, RI 02912, USA
2. 

Department of Computer Science and Engineering, Nagoya Institute of Technology, Nagoya, 466-8555, Japan

* Corresponding author: Masahiro Suzuki

Received  January 2019 Revised  May 2019 Published  September 2019

Fund Project: M. Suzuki is supported by JSPS KAKENHI Grant Numbers 26800067 and 18K03364.

We consider the steady states of a gas between two parallel plates that is ionized by a strong electric field so as to create a plasma. We use global bifurcation theory to prove that there is a curve $ \mathcal{K} $ of such states with the following property. The curve begins at the sparking voltage and either the particle density becomes unbounded or the curve ends at the anti-sparking voltage.

Keywords: Hyperbolic-parabolic-elliptic coupled system, boundary value problem, global bifurcation, gas discharge, sparking voltage.
Mathematics Subject Classification: Primary: 35A01, 35M12, 70K50; Secondary: 76X05, 82D10.
Citation: Walter A. Strauss, Masahiro Suzuki. Large amplitude stationary solutions of the Morrow model of gas ionization. Kinetic & Related Models, 2019, 12 (6) : 1297-1312. doi: 10.3934/krm.2019050
References:
[1]

I. Abbas and P. Bayle, A critical analysis of ionising wave propagation mechanisms in breakdown, J. Phys. D: Appl. Phys., 13 (1980), 1055-1068.  doi: 10.1088/0022-3727/13/6/015.  Google Scholar

[2]

Y. A. Abramovich and C. D. Aliprantis, An Invitation to Operator Theory, Graduate Studies in Mathematics, 50. American Mathematical Society, Providence, 2002. doi: 10.1090/gsm/050.  Google Scholar

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, 2011.  Google Scholar

[4] B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation: An Introduction, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2003.  doi: 10.1515/9781400884339.  Google Scholar
[5]

A. Constantin, W. Strauss and E. Vǎrvǎrucǎ, Global bifurcation of steady gravity water waves with critical layers, Acta. Math., 217 (2016), 195-262.  doi: 10.1007/s11511-017-0144-x.  Google Scholar

[6]

M. G. Crandall and P. H. Rabinowitz, Nonlinear Sturm-Liouville eigenvalue problems and topological degree, J. Math. Mech., 19 (1969/1970), 1083-1102.   Google Scholar

[7]

E. N. Dancer, Bifurcation theory for analytic operators, Proc. London Math. Soc., 26 (1973), 359-384.  doi: 10.1112/plms/s3-26.2.359.  Google Scholar

[8]

P. Degond and B. Lucquin-Desreux, Mathematical models of electrical discharges in air at atmospheric pressure: A derivation from asymptotic analysis, Int. J. Compu. Sci. Math., 1 (2007), 58-97.  doi: 10.1504/IJCSM.2007.013764.  Google Scholar

[9]

S. K. Dhali and P. F. Williams, Twodimensional studies of streamers in gases, J. Appl. Phys., 62 (1987), 4694-4707.   Google Scholar

[10]

P. A. Durbin and L. Turyn, Analysis of the positive DC corona between coaxial cylinders, J. Phys. D: Appl. Phys., 20 (1987), 1490-1496.  doi: 10.1088/0022-3727/20/11/020.  Google Scholar

[11]

H. Kielhöfer, Bifurcation Theory, An Introduction with Applications to Partial Differential Equations, Second edition, Applied Mathematical Sciences, 156. Springer, New York, 2012. doi: 10.1007/978-1-4614-0502-3.  Google Scholar

[12]

A. A. Kulikovsky, Positive streamer between parallel plate electrodes in atmospheric pressure air, IEEE Trans. Plasma Sci., 30 (1997), 441-450.  doi: 10.1088/0022-3727/30/3/017.  Google Scholar

[13]

A. A. Kulikovsky, The role of photoionization in positive streamer dynamics, J. Phys. D: Appl. Phys., 33 (2000), 1514-1524.  doi: 10.1088/0022-3727/33/12/314.  Google Scholar

[14]

A. Luque, V. Ratushnaya and U. Ebert, Positive and negative streamers in ambient air: Modeling evolution and velocities, J. Phys. D: Appl. Phys., 41 (2008), 234005. doi: 10.1088/0022-3727/41/23/234005.  Google Scholar

[15]

R. Morrow, Theory of negative corona in oxygen, Phys. Rev. A, 32 (1985), 1799-1809.  doi: 10.1103/PhysRevA.32.1799.  Google Scholar

[16]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[17]

Y. P. Raizer, Gas Discharge Physics, Springer, 2001. Google Scholar

[18]

M. Suzuki and A. Tani, Time-local solvability of the Degond-Lucquin-Desreux-Morrow model for gas discharge, SIAM Math. Anal., 50 (2018), 5096-5118.  doi: 10.1137/17M111852X.  Google Scholar

[19]

M. Suzuki and A. Tani, Bifurcation analysis of the Degond-Lucquin-Desreux-Morrow model for gas discharge, submitted. doi: 10.1137/17M111852X.  Google Scholar

[20] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9781139171755.  Google Scholar
[21]

http://www.physics.csbsju.edu/370/jcalvert/dischg.htm.html. Google Scholar

show all references

References:
[1]

I. Abbas and P. Bayle, A critical analysis of ionising wave propagation mechanisms in breakdown, J. Phys. D: Appl. Phys., 13 (1980), 1055-1068.  doi: 10.1088/0022-3727/13/6/015.  Google Scholar

[2]

Y. A. Abramovich and C. D. Aliprantis, An Invitation to Operator Theory, Graduate Studies in Mathematics, 50. American Mathematical Society, Providence, 2002. doi: 10.1090/gsm/050.  Google Scholar

[3]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, 2011.  Google Scholar

[4] B. Buffoni and J. Toland, Analytic Theory of Global Bifurcation: An Introduction, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2003.  doi: 10.1515/9781400884339.  Google Scholar
[5]

A. Constantin, W. Strauss and E. Vǎrvǎrucǎ, Global bifurcation of steady gravity water waves with critical layers, Acta. Math., 217 (2016), 195-262.  doi: 10.1007/s11511-017-0144-x.  Google Scholar

[6]

M. G. Crandall and P. H. Rabinowitz, Nonlinear Sturm-Liouville eigenvalue problems and topological degree, J. Math. Mech., 19 (1969/1970), 1083-1102.   Google Scholar

[7]

E. N. Dancer, Bifurcation theory for analytic operators, Proc. London Math. Soc., 26 (1973), 359-384.  doi: 10.1112/plms/s3-26.2.359.  Google Scholar

[8]

P. Degond and B. Lucquin-Desreux, Mathematical models of electrical discharges in air at atmospheric pressure: A derivation from asymptotic analysis, Int. J. Compu. Sci. Math., 1 (2007), 58-97.  doi: 10.1504/IJCSM.2007.013764.  Google Scholar

[9]

S. K. Dhali and P. F. Williams, Twodimensional studies of streamers in gases, J. Appl. Phys., 62 (1987), 4694-4707.   Google Scholar

[10]

P. A. Durbin and L. Turyn, Analysis of the positive DC corona between coaxial cylinders, J. Phys. D: Appl. Phys., 20 (1987), 1490-1496.  doi: 10.1088/0022-3727/20/11/020.  Google Scholar

[11]

H. Kielhöfer, Bifurcation Theory, An Introduction with Applications to Partial Differential Equations, Second edition, Applied Mathematical Sciences, 156. Springer, New York, 2012. doi: 10.1007/978-1-4614-0502-3.  Google Scholar

[12]

A. A. Kulikovsky, Positive streamer between parallel plate electrodes in atmospheric pressure air, IEEE Trans. Plasma Sci., 30 (1997), 441-450.  doi: 10.1088/0022-3727/30/3/017.  Google Scholar

[13]

A. A. Kulikovsky, The role of photoionization in positive streamer dynamics, J. Phys. D: Appl. Phys., 33 (2000), 1514-1524.  doi: 10.1088/0022-3727/33/12/314.  Google Scholar

[14]

A. Luque, V. Ratushnaya and U. Ebert, Positive and negative streamers in ambient air: Modeling evolution and velocities, J. Phys. D: Appl. Phys., 41 (2008), 234005. doi: 10.1088/0022-3727/41/23/234005.  Google Scholar

[15]

R. Morrow, Theory of negative corona in oxygen, Phys. Rev. A, 32 (1985), 1799-1809.  doi: 10.1103/PhysRevA.32.1799.  Google Scholar

[16]

P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal., 7 (1971), 487-513.  doi: 10.1016/0022-1236(71)90030-9.  Google Scholar

[17]

Y. P. Raizer, Gas Discharge Physics, Springer, 2001. Google Scholar

[18]

M. Suzuki and A. Tani, Time-local solvability of the Degond-Lucquin-Desreux-Morrow model for gas discharge, SIAM Math. Anal., 50 (2018), 5096-5118.  doi: 10.1137/17M111852X.  Google Scholar

[19]

M. Suzuki and A. Tani, Bifurcation analysis of the Degond-Lucquin-Desreux-Morrow model for gas discharge, submitted. doi: 10.1137/17M111852X.  Google Scholar

[20] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9781139171755.  Google Scholar
[21]

http://www.physics.csbsju.edu/370/jcalvert/dischg.htm.html. Google Scholar

Figure 4.  voltage–current curve
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 1.  case 1
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  case 2
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  alternative (ⅱ)
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Tatsien Li, Libin Wang. Global classical solutions to a kind of mixed initial-boundary value problem for quasilinear hyperbolic systems. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 59-78. doi: 10.3934/dcds.2005.12.59
[2]

Shitao Liu, Roberto Triggiani. Determining damping and potential coefficients of an inverse problem for a system of two coupled hyperbolic equations. Part I: Global uniqueness. Conference Publications, 2011, 2011 (Special) : 1001-1014. doi: 10.3934/proc.2011.2011.1001
[3]

Francesca R. Guarguaglini. Global solutions for a chemotaxis hyperbolic-parabolic system on networks with nonhomogeneous boundary conditions. Communications on Pure & Applied Analysis, 2020, 19 (2) : 1057-1087. doi: 10.3934/cpaa.2020049
[4]

G. Métivier, K. Zumbrun. Symmetrizers and continuity of stable subspaces for parabolic-hyperbolic boundary value problems. Discrete & Continuous Dynamical Systems - A, 2004, 11 (1) : 205-220. doi: 10.3934/dcds.2004.11.205
[5]

Matthias Eller, Daniel Toundykov. Carleman estimates for elliptic boundary value problems with applications to the stablization of hyperbolic systems. Evolution Equations & Control Theory, 2012, 1 (2) : 271-296. doi: 10.3934/eect.2012.1.271
[6]

Chan-Gyun Kim, Yong-Hoon Lee. A bifurcation result for two point boundary value problem with a strong singularity. Conference Publications, 2011, 2011 (Special) : 834-843. doi: 10.3934/proc.2011.2011.834
[7]

Mark I. Vishik, Sergey Zelik. Attractors for the nonlinear elliptic boundary value problems and their parabolic singular limit. Communications on Pure & Applied Analysis, 2014, 13 (5) : 2059-2093. doi: 10.3934/cpaa.2014.13.2059
[8]

Tong Yang, Fahuai Yi. Global existence and uniqueness for a hyperbolic system with free boundary. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 763-780. doi: 10.3934/dcds.2001.7.763
[9]

Francisco Ortegón Gallego, María Teresa González Montesinos. Existence of a capacity solution to a coupled nonlinear parabolic--elliptic system. Communications on Pure & Applied Analysis, 2007, 6 (1) : 23-42. doi: 10.3934/cpaa.2007.6.23
[10]

Jun Zhou. Initial boundary value problem for a inhomogeneous pseudo-parabolic equation. Electronic Research Archive, 2020, 28 (1) : 67-90. doi: 10.3934/era.2020005
[11]

Bastian Gebauer, Nuutti Hyvönen. Factorization method and inclusions of mixed type in an inverse elliptic boundary value problem. Inverse Problems & Imaging, 2008, 2 (3) : 355-372. doi: 10.3934/ipi.2008.2.355
[12]

Angelo Favini, Rabah Labbas, Stéphane Maingot, Maëlis Meisner. Boundary value problem for elliptic differential equations in non-commutative cases. Discrete & Continuous Dynamical Systems - A, 2013, 33 (11&12) : 4967-4990. doi: 10.3934/dcds.2013.33.4967
[13]

Shaoyong Lai, Yong Hong Wu, Xu Yang. The global solution of an initial boundary value problem for the damped Boussinesq equation. Communications on Pure & Applied Analysis, 2004, 3 (2) : 319-328. doi: 10.3934/cpaa.2004.3.319
[14]

Takashi Narazaki. Global solutions to the Cauchy problem for the weakly coupled system of damped wave equations. Conference Publications, 2009, 2009 (Special) : 592-601. doi: 10.3934/proc.2009.2009.592
[15]

Julián López-Gómez, Marcela Molina-Meyer, Paul H. Rabinowitz. Global bifurcation diagrams of one node solutions in a class of degenerate boundary value problems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (3) : 923-946. doi: 10.3934/dcdsb.2017047
[16]

Hua Chen, Wenbin Lv, Shaohua Wu. A free boundary problem for a class of parabolic-elliptic type chemotaxis model. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2577-2592. doi: 10.3934/cpaa.2018122
[17]

Shao-Yuan Huang, Shin-Hwa Wang. On S-shaped bifurcation curves for a two-point boundary value problem arising in a theory of thermal explosion. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4839-4858. doi: 10.3934/dcds.2015.35.4839
[18]

Louis Tebou. Stabilization of some coupled hyperbolic/parabolic equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1601-1620. doi: 10.3934/dcdsb.2010.14.1601
[19]

Inmaculada Antón, Julián López-Gómez. Global bifurcation diagrams of steady-states for a parabolic model related to a nuclear engineering problem. Conference Publications, 2013, 2013 (special) : 21-30. doi: 10.3934/proc.2013.2013.21
[20]

Leo G. Rebholz, Dehua Wang, Zhian Wang, Camille Zerfas, Kun Zhao. Initial boundary value problems for a system of parabolic conservation laws arising from chemotaxis in multi-dimensions. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 3789-3838. doi: 10.3934/dcds.2019154

2019 Impact Factor: 1.311

Tools

Metrics

  • PDF downloads (77)
  • HTML views (125)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences