The mathematics of the Thomson effect
Author:
Giovanni Cimatti
Journal:
Quart. Appl. Math. 67 (2009), 617-626
MSC (2000):
Primary 35J60, 35K50, 80A22
DOI:
https://doi.org/10.1090/S0033-569X-09-01140-6
Published electronically:
July 29, 2009
MathSciNet review:
2588227
Full-text PDF Free Access
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Additional Information
Abstract: The paper deals with the elliptic system of the thermistor problem in 3 dimensions taking into account the Thomson effect. Existence and uniqueness results are presented. The proofs are based on a reduction to a two-point problem for an ordinary differential equation.
References
- Walter Allegretto, Yanping Lin, and Shuqing Ma, On the time periodic thermistor problem, European J. Appl. Math. 15 (2004), no. 1, 55–77. MR 2056329, DOI https://doi.org/10.1017/S0956792503005308
- S. N. Antontsev and M. Chipot, The thermistor problem: existence, smoothness uniqueness, blowup, SIAM J. Math. Anal. 25 (1994), no. 4, 1128–1156. MR 1278895, DOI https://doi.org/10.1137/S0036141092233482
- Giovanni Cimatti, Existence and uniqueness for the equations of the Joule-Thomson effect, Appl. Anal. 41 (1991), no. 1-4, 131–144. MR 1103851, DOI https://doi.org/10.1080/00036819108840019
- Giovanni Cimatti, Stability and multiplicity of solutions for the thermistor problem, Ann. Mat. Pura Appl. (4) 181 (2002), no. 2, 181–212. MR 1911935, DOI https://doi.org/10.1007/s102310100034
- H. Diesselhorst, Über das Probleme eines elektrisch erwämtern Leiters, Ann. Phys., 1 (1900), 312-325.
- S. D. Howison, J. F. Rodrigues, and M. Shillor, Stationary solutions to the thermistor problem, J. Math. Anal. Appl. 174 (1993), no. 2, 573–588. MR 1215637, DOI https://doi.org/10.1006/jmaa.1993.1142
- L. Landau and E. Lifschitz, Électrodynamique des Milieux Continus, Editiones Mir, (1969), Moscow.
- W. Voigt, Über ein von Herrn Fr. Kohlrausch aufgestelltes Problem der Wärmelehre, Gött. Nachr. 231 (1899), 228-239.
References
- W. Allegretto, Yanping Lin and Shuqing Ma, On the time periodic thermistor problem, European J. Appl. Math. 15 (2004), 123-134. MR 2056329 (2005e:35220)
- S.N. Antonsev and M. Chipot, The thermistor problem: Existence, smoothness, uniqueness, blow up, SIAM J. Math. Anal. 25 (1994), 67-83. MR 1278895 (95f:35013)
- G. Cimatti, Existence and Uniqueness for the Equations of the Joule-Thomson Effect, Appl. Anal., 41 (1991), 131-144. MR 1103851 (92f:35135)
- G. Cimatti, Stability and Multiplicity of Solutions for the Thermistor Problem, Ann. Mat. Pura Appl., 181 (2002), 181-212. MR 1911935 (2003g:35056)
- H. Diesselhorst, Über das Probleme eines elektrisch erwämtern Leiters, Ann. Phys., 1 (1900), 312-325.
- S.D. Howison, J.F. Rodrigues and M. Shilor, Stationary solutions for the thermistor problem, J. Math. Appl., 174 (1993), 46-58. MR 1215637 (94h:80005)
- L. Landau and E. Lifschitz, Électrodynamique des Milieux Continus, Editiones Mir, (1969), Moscow.
- W. Voigt, Über ein von Herrn Fr. Kohlrausch aufgestelltes Problem der Wärmelehre, Gött. Nachr. 231 (1899), 228-239.
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Additional Information
Giovanni Cimatti
Affiliation:
Dipartimento di Matematica, Largo Bruno Pontecorvo 5, Pisa, Italy
Email:
cimatti@dm.unipi.it
Keywords:
Thermistor problem,
Thomson effect,
existence and uniqueness results,
multiple solutions
Received by editor(s):
April 3, 2008
Published electronically:
July 29, 2009
Article copyright:
© Copyright 2009
Brown University
The copyright for this article reverts to public domain 28 years after publication.