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The Euler Equations of Compressible Fluid Flow
Author:
Demetrios Christodoulou
Journal:
Bull. Amer. Math. Soc. 44 (2007), 581-602
MSC (2000):
Primary 76L05, 76-03; Secondary 01A50, 01A55, 01A60, 35L65, 35L67, 76N15
Posted:
June 18, 2007
MathSciNet review:
2338367
Full-text PDF
References |
Similar Articles |
Additional Information
- [Be]
Bernoulli, D. Hydrodynamica, Argentorati (1738).
- [Bo]
Boltzmann, L. ``Über die Bezeihung zwischen dem zweiten Hauptsatzes der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respective den Sätzen über das Wärmegleichgewicht'', Wien Ber. 76, 73 (1877).
- [Ch1]
Demetrios
Christodoulou, The formation of shocks in 3-dimensional
fluids, EMS Monographs in Mathematics, European Mathematical Society
(EMS), Zürich, 2007. MR 2284927
(2008e:76104)
- [Ch2]
Demetrios
Christodoulou, The action principle and partial differential
equations, Annals of Mathematics Studies, vol. 146, Princeton
University Press, Princeton, NJ, 2000. MR 1739321
(2003a:58001)
- [C-K]
Demetrios
Christodoulou and Sergiu
Klainerman, The global nonlinear stability of the Minkowski
space, Princeton Mathematical Series, vol. 41, Princeton
University Press, Princeton, NJ, 1993. MR 1316662
(95k:83006)
- [Cl1]
Clausius, R. ``Über die bewegende Kraft der Wärme'', Annalen der Physik und Chemie 79, 368-397, 500-524 (1850).
- [Cl2]
Clausius, R. ``Über verschiedene fü die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie'', Annalen der Physik und Chemie 125, 353-400 (1865).
- [DA]
D'Alembert, J.-B. le R. ``Recherches sur la courbe que forme une corde tenduë mise en vibration'', Mém. Acad. Sci. Berlin 2, 214-219 (1849).
- [Ei]
Einstein, A. ``Zur Electrodynamic bewegter Körper'', Annalem der Physik 17, 891-921 (1905).
- [Eu1]
Euler, L. ``Principes generaux de l'etat d'equilibre des fluides'', Mémoires de l'Academie des Sciences de Berlin 11, 217-273 (1757).
- [Eu2]
Euler, L. ``Principes generaux du mouvement des fluides'', Mémoires de l'Academie des Sciences de Berlin 11, 274-315 (1757).
- [Eu3]
Euler, L. ``Continuation des recherrches sur la theorie du mouvement des fluides'', Mémoires de l'Academie des Sciences de Berlin 11, 316-361 (1757).
- [Eu4]
Euler, L. ``Principia motus fluidorum'', Novi Commentarii Academiae Scientiarum Petropolitanae 6, 271-311 (1761).
- [Eu5]
Euler, L. ``De motu fluidorum a diverso caloris gradu oriundo'', Novi Commentarii Academiae Scientiarum Petropolitanae 11, 232-267 (1767).
- [Eu6]
Euler, L. ``Sectio secunda de principiis motus fluidorum'', Novi Commentarii Academiae Scientiarum Petropolitanae 14, 270-386 (1770).
- [Eu7]
Euler, L. ``Sectio tertia de motu fluidorum lineari potissimum aquae'', Novi Commentarii Academiae Scientiarum Petropolitanae 15, 219-360 (1771).
- [Eu8]
Euler, L. ``Sectio quarta de motu aeris in tubis'', Novi Commentarii Academiae Scientiarum Petropolitanae 16, 281-425 (1772).
- [F]
K.
O. Friedrichs, Symmetric hyperbolic linear differential
equations, Comm. Pure Appl. Math. 7 (1954),
345–392. MR 0062932
(16,44c)
- [F-L]
K.
O. Friedrichs and P.
D. Lax, Systems of conservation equations with a convex
extension, Proc. Nat. Acad. Sci. U.S.A. 68 (1971),
1686–1688. MR 0285799
(44 #3016)
- [Gl]
James
Glimm, Solutions in the large for nonlinear hyperbolic systems of
equations, Comm. Pure Appl. Math. 18 (1965),
697–715. MR 0194770
(33 #2976)
- [He]
Helmholtz, H. v. Über die Erhaltung der Kraft, G. Reimer, Berlin, 1847.
- [Hu]
Hugoniot, H. ``Sur la propagation du mouvement dans les corps et spécialement dans les gaz parfaits'', Journal de l'école polytechnique 58, 1-125 (1889).
- [J]
Fritz
John, Formation of singularities in one-dimensional nonlinear wave
propagation, Comm. Pure Appl. Math. 27 (1974),
377–405. MR 0369934
(51 #6163)
- [Kr]
Kruzhkov, S. N. ``First order quasilinear equations in several independent variables'', Math. USSR Sbornik 10, No. 2 (1970).
- [K-T]
Joseph
B. Keller and Lu
Ting, Periodic vibrations of systems governed by nonlinear partial
differential equations, Comm. Pure Appl. Math. 19
(1966), 371–420. MR 0205520
(34 #5347)
- [La]
Laplace, P. S. ``Sur la vitesse du son dans l'air et dans l'eau'', Ann. de Chim. et de Phys. iii, 238 (1816).
- [Ln]
L.
D. Landau and E.
M. Lifshitz, Course of theoretical physics. Vol. 6, 2nd ed.,
Pergamon Press, Oxford, 1987. Fluid mechanics; Translated from the third
Russian edition by J. B. Sykes and W. H. Reid. MR 961259
(89i:00006)
- [Lx1]
Peter
Lax, Shock waves and entropy, Contributions to nonlinear
functional analysis (Proc. Sympos., Math. Res. Center, Univ. Wisconsin,
Madison, Wis., 1971), Academic Press, New York, 1971,
pp. 603–634. MR 0393870
(52 #14677)
- [Lx2]
Peter
D. Lax, Development of singularities of solutions of nonlinear
hyperbolic partial differential equations, J. Mathematical Phys.
5 (1964), 611–613. MR 0165243
(29 #2532)
- [Lx3]
Peter
D. Lax, Hyperbolic systems of conservation laws and the
mathematical theory of shock waves, Society for Industrial and Applied
Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical
Sciences Regional Conference Series in Applied Mathematics, No. 11. MR 0350216
(50 #2709)
- [Ma1]
A.
Majda, Compressible fluid flow and systems of conservation laws in
several space variables, Applied Mathematical Sciences, vol. 53,
Springer-Verlag, New York, 1984. MR 748308
(85e:35077)
- [Ma2]
Majda, A. The Stability of Multi-Dimensional Shock Fronts - A New Problem for Linear Hyperbolic Equations, Mem. Amer. Math. Society 275, 1983.
- [Ma3]
Andrew
Majda, The existence of multidimensional shock fronts, Mem.
Amer. Math. Soc. 43 (1983), no. 281, v+93. MR 699241
(85f:35139)
- [M-B]
Andrew
J. Majda and Andrea
L. Bertozzi, Vorticity and incompressible flow, Cambridge
Texts in Applied Mathematics, vol. 27, Cambridge University Press,
Cambridge, 2002. MR 1867882
(2003a:76002)
- [Mi]
Minkowski, H. ``Raum und Zeit'', Address at the 80th Assembly of German Natural Scientists and Physicians, Cologne (1908).
- [Mo]
Cathleen
S. Morawetz, The decay of solutions of the exterior
initial-boundary value problem for the wave equation, Comm. Pure Appl.
Math. 14 (1961), 561–568. MR 0132908
(24 #A2744)
- [N]
Noether, E. ``Invariante Variationsprobleme'', Nach. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1918, 235-257 (1918).
- [Ra]
Rankine, W.J.M. ``On the thermodynamic theory of waves of finite longitudinal disturbance'', Philosophical Transactions of the Royal Society of London 160, 277-288 (1870).
- [Ri]
Riemann, B. ``Über die Fortpfanzung ebener Luftwellen von endlicher Schwingungswete'', Abhandlungen der Gesellshaft der Wissenshaften zu Göttingen, Mathematisch-physikalishe Klasse 8, 43 (1858-59).
- [S]
Thomas
C. Sideris, Formation of singularities in three-dimensional
compressible fluids, Comm. Math. Phys. 101 (1985),
no. 4, 475–485. MR 815196
(87d:35127)
- [Z-R]
Zel'dovich, Y.B., and Raizer, Y.P. Physics of Shock Waves and High-Temperature Hydrodynamic Phaenomena, New York, 1966, 1967, Chapter I, Section 19, and Chapter XI, Section 20.
- [Be]
- Bernoulli, D. Hydrodynamica, Argentorati (1738).
- [Bo]
- Boltzmann, L. ``Über die Bezeihung zwischen dem zweiten Hauptsatzes der mechanischen Wärmetheorie und der Wahrscheinlichkeitsrechnung respective den Sätzen über das Wärmegleichgewicht'', Wien Ber. 76, 73 (1877).
- [Ch1]
- Christodoulou, D. The Formation of Shocks in 3-Dimensional Fluids, EMS Monographs in Mathematics, EMS Publishing House, 2007. MR 2284927
- [Ch2]
- Christodoulou, D. The Action Principle and Partial Differential Equations, Ann. Math. Stud. 146, Princeton University Press, 2000. MR 1739321 (2003a:58001)
- [C-K]
- Christodoulou, D., and Klainerman, S. The Global Nonlinear Stability of the Minkowski Space, Princeton Mathematical Series 41, Princeton University Press, 1993. MR 1316662 (95k:83006)
- [Cl1]
- Clausius, R. ``Über die bewegende Kraft der Wärme'', Annalen der Physik und Chemie 79, 368-397, 500-524 (1850).
- [Cl2]
- Clausius, R. ``Über verschiedene fü die Anwendung bequeme Formen der Hauptgleichungen der mechanischen Wärmetheorie'', Annalen der Physik und Chemie 125, 353-400 (1865).
- [DA]
- D'Alembert, J.-B. le R. ``Recherches sur la courbe que forme une corde tenduë mise en vibration'', Mém. Acad. Sci. Berlin 2, 214-219 (1849).
- [Ei]
- Einstein, A. ``Zur Electrodynamic bewegter Körper'', Annalem der Physik 17, 891-921 (1905).
- [Eu1]
- Euler, L. ``Principes generaux de l'etat d'equilibre des fluides'', Mémoires de l'Academie des Sciences de Berlin 11, 217-273 (1757).
- [Eu2]
- Euler, L. ``Principes generaux du mouvement des fluides'', Mémoires de l'Academie des Sciences de Berlin 11, 274-315 (1757).
- [Eu3]
- Euler, L. ``Continuation des recherrches sur la theorie du mouvement des fluides'', Mémoires de l'Academie des Sciences de Berlin 11, 316-361 (1757).
- [Eu4]
- Euler, L. ``Principia motus fluidorum'', Novi Commentarii Academiae Scientiarum Petropolitanae 6, 271-311 (1761).
- [Eu5]
- Euler, L. ``De motu fluidorum a diverso caloris gradu oriundo'', Novi Commentarii Academiae Scientiarum Petropolitanae 11, 232-267 (1767).
- [Eu6]
- Euler, L. ``Sectio secunda de principiis motus fluidorum'', Novi Commentarii Academiae Scientiarum Petropolitanae 14, 270-386 (1770).
- [Eu7]
- Euler, L. ``Sectio tertia de motu fluidorum lineari potissimum aquae'', Novi Commentarii Academiae Scientiarum Petropolitanae 15, 219-360 (1771).
- [Eu8]
- Euler, L. ``Sectio quarta de motu aeris in tubis'', Novi Commentarii Academiae Scientiarum Petropolitanae 16, 281-425 (1772).
- [F]
- Friedrichs, K.O. ``Symmetric hyperbolic linear differential equations'', Comm. Pure & Appl. Math. 7, 345-392 (1954). MR 0062932 (16:44c)
- [F-L]
- Friedrichs, K.O., and Lax, P.D. ``Systems of Conservation Equations with a Convex Extension'', Proc. Nat. Acad. Sci. USA 68, 1686-1688 (1971). MR 0285799 (44:3016)
- [Gl]
- Glimm, J. ``Solutions in the large for nonlinear hyperbolic systems of equations'', Comm. Pure & Appl. Math. 18, 697-715 (1965). MR 0194770 (33:2976)
- [He]
- Helmholtz, H. v. Über die Erhaltung der Kraft, G. Reimer, Berlin, 1847.
- [Hu]
- Hugoniot, H. ``Sur la propagation du mouvement dans les corps et spécialement dans les gaz parfaits'', Journal de l'école polytechnique 58, 1-125 (1889).
- [J]
- John, F. ``Formation of singularities in one-dimensional non-linear wave propagation'', Comm. Pure & Appl. Math. 27, 377-405 (1974). MR 0369934 (51:6163)
- [Kr]
- Kruzhkov, S. N. ``First order quasilinear equations in several independent variables'', Math. USSR Sbornik 10, No. 2 (1970).
- [K-T]
- Keller, J.B. and Ting, L. ``Periodic vibrations of systems governed by nonlinear partial differential equations'', Comm. Pure & Appl. Math. 169, 371-420 (1966). MR 0205520 (34:5347)
- [La]
- Laplace, P. S. ``Sur la vitesse du son dans l'air et dans l'eau'', Ann. de Chim. et de Phys. iii, 238 (1816).
- [Ln]
- Landau, L.D. (1944). See Landau, L.D., and Lifschitz, E.M. Fluid Mechanics, 2nd edition, Oxford, Pergamon Press, 1987, pages 332-333. MR 961259 (89i:00006)
- [Lx1]
- Lax, P.D. ``Shock waves and entropy'', pp. 603-634 in Contributions to Nonlinear Functional Analysis, edited by E. Zarantonello, Academic Press, 1971. MR 0393870 (52:14677)
- [Lx2]
- Lax, P.D. ``Development of singularities of solutions of non-linear hyperbolic partial differential equations'', J. Math. Phys. 5, 611-613 (1964). MR 0165243 (29:2532)
- [Lx3]
- Lax, P.D. Hyperbolic Systems of Conservation Laws and the Mathematical Theory of Shock Waves, Regional Conf. Series in Appl. Math. 13, SIAM, 1973. MR 0350216 (50:2709)
- [Ma1]
- Majda, A. Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Appl. Math. Sci. 53, Springer-Verlag, 1984. MR 748308 (85e:35077)
- [Ma2]
- Majda, A. The Stability of Multi-Dimensional Shock Fronts - A New Problem for Linear Hyperbolic Equations, Mem. Amer. Math. Society 275, 1983.
- [Ma3]
- Majda, A. The Existence of Multi-Dimensional Shock Fronts, Mem. Amer. Math. Soc. 285, 1983. MR 699241 (85f:35139)
- [M-B]
- Majda, A., and Bertozzi, A. Vorticity and Incompressible Flow, Cambridge Texts in Applied Mathematics, Cambridge University Press, 2002. MR 1867882 (2003a:76002)
- [Mi]
- Minkowski, H. ``Raum und Zeit'', Address at the 80th Assembly of German Natural Scientists and Physicians, Cologne (1908).
- [Mo]
- Morawetz, C. ``The decay of solutions of the exterior initial-boundary value problem for the wave equation'', Comm. Pure & Appl. Math. 14, 561-568 (1961). MR 0132908 (24:A2744)
- [N]
- Noether, E. ``Invariante Variationsprobleme'', Nach. Ges. Wiss. Göttingen, Math.-Phys. Kl. 1918, 235-257 (1918).
- [Ra]
- Rankine, W.J.M. ``On the thermodynamic theory of waves of finite longitudinal disturbance'', Philosophical Transactions of the Royal Society of London 160, 277-288 (1870).
- [Ri]
- Riemann, B. ``Über die Fortpfanzung ebener Luftwellen von endlicher Schwingungswete'', Abhandlungen der Gesellshaft der Wissenshaften zu Göttingen, Mathematisch-physikalishe Klasse 8, 43 (1858-59).
- [S]
- Sideris, T. ``Formation of singularities in three-dimensional compressible fluids'', Commun. Math. Phys. 101, 475-85 (1985). MR 815196 (87d:35127)
- [Z-R]
- Zel'dovich, Y.B., and Raizer, Y.P. Physics of Shock Waves and High-Temperature Hydrodynamic Phaenomena, New York, 1966, 1967, Chapter I, Section 19, and Chapter XI, Section 20.
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Additional Information
Demetrios Christodoulou
Affiliation:
Departments of Mathematics and Physics, ETH-Zürich, ETH-Zentrum, 8092 Zürich, Switzerland
Email:
demetri@math.ethz.ch
DOI:
http://dx.doi.org/10.1090/S0273-0979-07-01181-0
PII:
S 0273-0979(07)01181-0
Received by editor(s):
May 15, 2007
Posted:
June 18, 2007
Article copyright:
© Copyright 2007 American Mathematical Society
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