Variational principles for parabolic equations

Ivan Hlaváček

Aplikace matematiky (1969)

  • Volume: 14, Issue: 4, page 278-297
  • ISSN: 0862-7940

Abstract

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New types of variational principles, each of them equivalent to the linear mixed problem for parabolic equation with initial and combined boundary conditions having been suggested by physicists, are discussed. Though the approach used here is purely mathematical so that it makes possible application to all mixed problems of mathematical physics with parabolic equations, only the example of heat conductions is used to show the physical interpretation. The principles under consideration are of two kinds. The first kind presents a variational characterization of the original problem, expressed in terms of a scalar function (temperature). The principles of the second kind characterize the same problem, formulated in terms of other variables, e.g. of a vector function (heat flux or entropy displacements).

How to cite

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Hlaváček, Ivan. "Variational principles for parabolic equations." Aplikace matematiky 14.4 (1969): 278-297. <http://eudml.org/doc/14603>.

@article{Hlaváček1969,
abstract = {New types of variational principles, each of them equivalent to the linear mixed problem for parabolic equation with initial and combined boundary conditions having been suggested by physicists, are discussed. Though the approach used here is purely mathematical so that it makes possible application to all mixed problems of mathematical physics with parabolic equations, only the example of heat conductions is used to show the physical interpretation. The principles under consideration are of two kinds. The first kind presents a variational characterization of the original problem, expressed in terms of a scalar function (temperature). The principles of the second kind characterize the same problem, formulated in terms of other variables, e.g. of a vector function (heat flux or entropy displacements).},
author = {Hlaváček, Ivan},
journal = {Aplikace matematiky},
keywords = {partial differential equations},
language = {eng},
number = {4},
pages = {278-297},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Variational principles for parabolic equations},
url = {http://eudml.org/doc/14603},
volume = {14},
year = {1969},
}

TY - JOUR
AU - Hlaváček, Ivan
TI - Variational principles for parabolic equations
JO - Aplikace matematiky
PY - 1969
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 14
IS - 4
SP - 278
EP - 297
AB - New types of variational principles, each of them equivalent to the linear mixed problem for parabolic equation with initial and combined boundary conditions having been suggested by physicists, are discussed. Though the approach used here is purely mathematical so that it makes possible application to all mixed problems of mathematical physics with parabolic equations, only the example of heat conductions is used to show the physical interpretation. The principles under consideration are of two kinds. The first kind presents a variational characterization of the original problem, expressed in terms of a scalar function (temperature). The principles of the second kind characterize the same problem, formulated in terms of other variables, e.g. of a vector function (heat flux or entropy displacements).
LA - eng
KW - partial differential equations
UR - http://eudml.org/doc/14603
ER -

References

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  1. G. Adler, Sulla caratterizzabilita dell'equazione del calore dal punto di vista del calcolo delle variazioni, Matematikai kutató intézenétek közlemenyei II. (1957), 153-157. (1957) MR0102656
  2. P. Rosen, On variational principles for irreversible processes, Proceedings of the Iowa Thermodynamic symposium (1953), 34-42. (1953) MR0057190
  3. M. A. Biot, 10.1063/1.1722351, J. Appl. Phys. 27 (1956), 240-253. (1956) Zbl0071.41204MR0077441DOI10.1063/1.1722351
  4. M. A. Biot, Variational principles in irreversible thermodynamics with application to viscoelasticity, Physical Review 97, (1955), 1463-1469. (1955) Zbl0065.42003MR0070514
  5. M. A. Biot, Linear thermodynamics and the mechanics of solids, Proceedings of the Third U. S. National Congress of Applied Mechanics (1958), 1-18. (1958) MR0102941
  6. R. A. Schapery, Irreversible thermodynamics and variational principles with applications to viscoelasticity, Aeronaut. Research Labs., Wright-Patterson Air Force Base, Ohio (1962). (1962) 
  7. M. E. Gurtin, 10.1090/qam/99951, Quart. Appl. Math., 22, (1964), 252-256. (1964) Zbl0173.37602DOI10.1090/qam/99951
  8. G. Herrmann, 10.1090/qam/161512, Quart. Appl. Math. 27 (1963), 2, 151-155. (1963) MR0161512DOI10.1090/qam/161512
  9. M. Ben-Amoz, 10.1115/1.3627345, Trans. ASME (1965), E 32, 4, 943-945. (1965) DOI10.1115/1.3627345
  10. M. E. Gurtin, 10.1007/BF00248489, Archive for Ratl. Mech. Anal. 16 (1964), 1, 34-50. (1964) Zbl0124.40001MR0214322DOI10.1007/BF00248489
  11. R. Courant D. Hilbert, Методы математической физики I, (Methoden der Mathematischen Physik I). Гостехиздат 1951, 231-232. (1951) 
  12. I. Hlaváček, Derivation of non-classical variational principles in the theory of elasticity, Aplikace matematiky 12 (1967), 1, 15 - 29. (1967) MR0214324
  13. В. А. Диткин А. 71. Прудников, Операционное исчисление, Издат. Высшая школа, Москва 1966. (1966) Zbl1230.03072
  14. I. Hlaváček, Variational principles in the linear theory of elasticity for general boundary conditions, Aplikace matematiky 12 (1967), 6, 425-448. (1967) MR0231575
  15. M. И. Вишик, Задача Копти для уравнений с операторноми коэффициентами, смешанная краевая задача для систем дифференциальных уравнений и приближенный метод их решения, Матем. сборник, 39 (81), (1956), 1, 51-148. (1956) Zbl0995.90522MR0080248
  16. О. А. Ладыженская, О решении нестационарных операторных уравнений, Матем. сборник, 39 (81), (1956), 4, 491-524. (1956) Zbl0995.90522MR0086987

Citations in EuDML Documents

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  1. Karel Rektorys, On application of direct variational methods to the solution of parabolic boundary value problems of arbitrary order in the space variables
  2. Ivan Hlaváček, On a conjugate semi-variational method for parabolic equations
  3. Ivan Hlaváček, Variational formulation of the Cauchy problem for equations with operator coefficients
  4. Ivan Hlaváček, On a semi-variational method for parabolic equations. I
  5. Alexander Mielke, Ulisse Stefanelli, Weighted energy-dissipation functionals for gradient flows
  6. Alexander Mielke, Ulisse Stefanelli, Weighted energy-dissipation functionals for gradient flows
  7. Joachim A. Nitsche, L ∞ -convergence of finite element Galerkin approximations for parabolic problems

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