Abstract
Transport equations in configuration space (linear and non-linear heat equations) and in phase space (Vlasov-Poisson systems for plasmas, beams and gravitating gases) are considered in the frame of transformation group techniques. Both self-similar and more qeneral groups are introduced to find specially interesting solutions. Two kinds of results are obtained time evolution of given initial situations and systematic derivation of possible scaling laws for a given mathematical model. These last results are specially interesting for extrapolating performances of Fusion Machines.
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Bibliography
G. Kalman; Annals of Physics 10, 1, (1960)
R. Balescu: “Statistical mechanics of charged particles”. Wiley and Sons (London 1963)
D. Anderson, A. Bondeson, M. Lisak, R. Nakach and H. Wilhelmsson: EUR-CEA. FC 925 (October 1977). To be published in Physica Scripta.
F. Grant and M.R. Feix: Physics of Fluids 10, 696 (1967)
H.L. Berk and K.V. Roberts: Physics of Fluids 10, 1595 (1967)
S. Lie: “Theorie der transformations gruppen” I, II, III. Chelsea Publ. Company. (New-York 1970)
L.V. Ovsjannikov: “Group properties of differential equations” Translated by G. Bluman (1967), Cal. Inst. of Tech.
W.F. Ames: “Non-linear partial differential equations in engineering”, Vol II, Academic Press (1972)
H.S. Woodard: “Similarity solutions for partial differential equations generated by finite and infinitesimal group”. PHD Thesis University of Iowa (1971)
G.W. Bluman and J.D. Cole: “Similarity methods for differential equations”. Appl. Math. Sci. 13 Springer Verlag, N.Y. Heidelberg Berlin
J.R. Burgan, J. Gutiérrez, E. Fijalkow, M. Navet and M.R. Feix Journal de Physique Lettres 38 161 (1977)
W.F. Ames: “Non-linear partial differential equations in engineering”. Vol I page 153 Academic Press (1965). The solutions obtained in this reference are found with particular auxilary conditions
N.N. Bogoliubov: “Studies in Statistical Mechanics” (Translated by E.K. Gora), Part A, Vol I. North Holland Publishing Co. (Amsterdam 1962)
N. Rostoker and M.N. Rosenbluth: Physics of Fluids 3, 1 (1960)
J.R. Burgan, J. Gutierrez, E. Fijalkow, M. Navet and M.R. Feix: J. Plasma Physics 19, 135 (1978)
A. Munier, M.R. Feix, E. Fijalkow, J.R. Burgan, J. Gutierrez “Time dependent solutions for a self gravitating star system”. Presented to publication.
J.W. Connor and J.B. Taylor: Nuclear Fusion, 17, 1047 (1977)
J.W. Connor and J.B. Taylor: Private communication
E.D. Courant and H.S. Snyder: Annals of Physics 3, 1 (1958)
H.R. Lewis and W.B. Riesenfeld: Journal of Mathematical Physics 10, 1458 (1968)
J.R. Burgan, J. Gutierrez, A. Munier, E.. Fijalkow and M.R. Feix: “Group transformations for phase space fluids”. Note technique CRPE/47, page 9 (C.R.P.E./P.C.E. Orléans 1977)
J.R. Burgan: “Sur des groupes de transformation en physique mathematique” P.H.D. Thesis. Université d'Orléans (1978)
J.R. Burgan, M.R. Feix, E. Fijalkow, J. Gutiérrez, A. Munier “Utilisation des groupes de transformation pour la resolution des équations aux derives partielles”. Proceedings de la première rencontre interdisciplinaire sur les problèmes inverses. Springer Verlag (1978).
G.L. Camm: Self gravitating star systems. Mont. Not. Roy. Soc., 110, 306 (1949); 112, 155, (1951).
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Gutierrez, J., Munier, A., Burgan, J.R., Feix, M.R. (1979). Non-linear transport equations : Properties deduced through transformation groups. In: Rañada, A.F. (eds) Nonlinear Problems in Theoretical Physics. Lecture Notes in Physics, vol 98. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-09246-3_8
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DOI: https://doi.org/10.1007/3-540-09246-3_8
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