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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
| 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 |
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 and Related Models,
2019, 12
(6)
: 1297-1312.
doi: 10.3934/krm.2019050
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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. |
| [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. |
| [3] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, 2011. |
| [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.
|
| [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. |
| [6] |
M. G. Crandall and P. H. Rabinowitz,
Nonlinear Sturm-Liouville eigenvalue problems and topological degree, J. Math. Mech., 19 (1969/1970), 1083-1102.
|
| [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. |
| [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. |
| [9] |
S. K. Dhali and P. F. Williams,
Twodimensional studies of streamers in gases, J. Appl. Phys., 62 (1987), 4694-4707.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [15] |
R. Morrow,
Theory of negative corona in oxygen, Phys. Rev. A, 32 (1985), 1799-1809.
doi: 10.1103/PhysRevA.32.1799. |
| [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. |
| [17] |
Y. P. Raizer, Gas Discharge Physics, Springer, 2001. |
| [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. |
| [19] |
M. Suzuki and A. Tani, Bifurcation analysis of the Degond-Lucquin-Desreux-Morrow model for gas discharge, submitted.
doi: 10.1137/17M111852X. |
| [20] |
J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.
doi: 10.1017/CBO9781139171755.
|
| [21] |
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. |
| [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. |
| [3] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, 2011. |
| [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.
|
| [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. |
| [6] |
M. G. Crandall and P. H. Rabinowitz,
Nonlinear Sturm-Liouville eigenvalue problems and topological degree, J. Math. Mech., 19 (1969/1970), 1083-1102.
|
| [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. |
| [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. |
| [9] |
S. K. Dhali and P. F. Williams,
Twodimensional studies of streamers in gases, J. Appl. Phys., 62 (1987), 4694-4707.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [15] |
R. Morrow,
Theory of negative corona in oxygen, Phys. Rev. A, 32 (1985), 1799-1809.
doi: 10.1103/PhysRevA.32.1799. |
| [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. |
| [17] |
Y. P. Raizer, Gas Discharge Physics, Springer, 2001. |
| [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. |
| [19] |
M. Suzuki and A. Tani, Bifurcation analysis of the Degond-Lucquin-Desreux-Morrow model for gas discharge, submitted.
doi: 10.1137/17M111852X. |
| [20] |
J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.
doi: 10.1017/CBO9781139171755.
|
| [21] |
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