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NON-RESONANT COUPLING OF WAVE-TRANSPORT SYSTEMS IN DISTORTED GEOMETRY

CHRISTOPHE PALLARD

CHRISTOPHE PALLARD

Département de mathématiques et applications, Ecole normale supérieure, 45 rue d'Ulm, F75230 Paris cedex 05, France

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https://doi.org/10.1142/S0219891605000403Cited by:1 (Source: Crossref)

Abstract

We consider coupled systems of a second order hyperbolic equation and a first order kinetic equation

The unknowns are u ≡ u(t,x,ξ) and f ≡ f(t,x,ξ) with phase space

, while the coefficients K_α are essentially linear combinations of ξ-moments of u and their first derivatives. Under suitable assumptions on the vector field p, we show that moments of u as well as their first and second derivatives remain bounded as long as the support of f in the ξ variable remains compact. Examples of application are the relativistic Vlasov–Maxwell, Vlasov–Nordström and relativistic Vlasov–Klein–Gordon systems.

Keywords:

References

H. Andréasson, S. Calogero and G. Rein, Global classical solutions to the spherically symmetric Nordström–Vlasov system, to appear in Math. Proc. Camb. Phil. Soc . Google Scholar
F. Bouchut, F. Golse and C. Pallard, Arch. Rational Mech. Anal. 170, 1 (2003). Crossref, Web of Science, Google Scholar
F. Bouchut, F. Golse and C. Pallard, Revista Mat. Iberoamericana 20(3), 865 (2004). Crossref, Web of Science, Google Scholar
F. Bouchut , F. Golse and M. Pulvirenti , Series in Applied Mathematics 4 , eds. L. Desvillettes and B. Perthame ( Gauthier-Villars, Editions Scientifiques et Médicales Elsevier, North-Holland , Paris, Amsterdam , 2000 ) . Google Scholar
S. Calogero, Class. Quantum Grav. 20, 1729 (2003). Crossref, Web of Science, Google Scholar
S. Calogero and H. Lee, Comm. Math. Sci. 2, 19 (2004). Crossref, Google Scholar
S. Calogero and G. Rein, Comm. Partial Diff. Eqns. 28, 1863 (2003). Crossref, Web of Science, Google Scholar
S. Calogero and G. Rein, J. Differential Equations 204, 323 (2004). Crossref, Web of Science, Google Scholar
S. Friedrich, Global small solutions of the Vlasov–Nordström system, arXiv:math-ph/0407023 . Google Scholar
I. Gelfand and G. Shilov , Generalized functions I ( Academic press , New York, London , 1964 ) . Google Scholar
R. Glassey , The Cauchy Problem in Kinetic Theory ( Society for Industrial and Applied Mathematics (SIAM) , Philadelphia, PA , 1996 ) . Crossref, Google Scholar
R. Glassey and J. Schaeffer, Arch. Rat. Mech. Anal. 141, 331 (1998). Crossref, Web of Science, Google Scholar
R. Glassey and J. Schaeffer, Arch. Rat. Mech. Anal. 141, 355 (1998). Crossref, Web of Science, Google Scholar
R. Glassey and W. Strauss, Arch. Rational Mech. Anal. 92(1), 59 (1986). Crossref, Web of Science, Google Scholar
L. Hörmander , The analysis of partial differential operators. I , Grundlehren der Mathematischen Wissenschaften 256 ( Springer–Verlag , Berlin , 1983 ) . Crossref, Google Scholar
L. Hörmander , The analysis of partial differential operators. II , Grundlehren der Mathematischen Wissenschaften 257 ( Springer–Verlag , Berlin , 1983 ) . Crossref, Google Scholar
L. Hörmander , The analysis of partial differential operators. III , Grundlehren der Mathematischen Wissenschaften 274 ( Springer–Verlag , Berlin , 1983 ) . Crossref, Google Scholar
M. Kunzingeret al., Comm. Math. Phys. 238(1–2), 367 (2003). Crossref, Web of Science, Google Scholar
M. Kunzinger, G. Rein, R. Steinbauer and G. Teschl, On classical solutions of the relativistic Vlasov–Klein–Gordon system, arXiv:math.AP/0407028 . Google Scholar
H. Lee, Global existence of solutions to the Nordström–Vlasov system in two space dimensions, arXiv:math-ph/0312014 . Google Scholar
C. Pallard, Bull. Sci. Math. 127, 705 (2003). Crossref, Web of Science, Google Scholar
C. Pallard, On global smooth solutions to the 3D Vlasov–Nordström system, to appear in Ann. Inst. H. Poincaré Anal. Non Linéaire . Google Scholar