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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
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Figure 1.  case 1
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Figure 2.  case 2
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Figure 3.  alternative (ⅱ)
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