Abstract
We show that a smooth, small enough Cauchy datum launches a unique classical solution of the relativistic Vlasov–Darwin (RVD) system globally in time. A similar result is claimed in Seehafer (Commun Math Sci 6:749–769, 2008) following the work in Pallard (Int Mat Res Not 57191:1–31, 2006). Our proof does not require estimates derived from the conservation of the total energy, nor those previously given on the transversal component of the electric field. These estimates are crucial in the references cited above. Instead, we exploit the formulation of the RVD system in terms of the generalized space and momentum variables. By doing so, we produce a simple a priori estimate on the transversal component of the electric field. We widen the functional space required for the Cauchy datum to extend the solution globally in time, and we improve decay estimates given in Seehafer (2008) on the electromagnetic field and its space derivatives. Our method extends the constructive proof presented in Rein (Handbook of differential equations: evolutionary equations, vol 3. Elsevier, Amsterdam, 2007) to solve the Cauchy problem for the Vlasov–Poisson system with a small initial datum.
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References
Batt J.: Global symmetric solutions of the initial value problem of stellar dynamics. J. Differ. Equ. 25, 342–364 (1977)
Benachour S., Filbet F., Laurencot P., Sonnendrücker E.: Global existence for the Vlasov-Darwin system in \({\mathbb{R}^3}\) for small initial data. Math. Methods Appl. Sci. 26, 297–319 (2003)
Bouchut F., Golse F., Pallard C.: Nonreasonant smoothing for coupled wave + transport equations and the Vlasov-Maxwell system. Revista Mat. Iberoamericana 20, 865–892 (2004)
Bouchut F., Golse F., Pallard C.: On classical solutions to the 3D relativistic Vlasov-Maxwell system: Glassey-Strauss’ theorem revisited. Arch. Rational Mech. Anal. 170, 1–15 (2003)
Gilbarg D., Trudinger N.: Elliptic Partial Differential Equations of Second Order. Classic in Mathematics Series. Springer, Berlin (2001)
Glassey R.: The Cauchy Problem in Kinetic Theory. SIAM, Philadelphia (1996)
Glassey R., Strauss W.: Singularity formation in a collisionless plasma could ocurr only at high velocities. Arch. Rational Mech. Anal. 92, 56–90 (1986)
Hartman P.: Ordinary Differential Equations. Wiley, New York (1964)
Jackson J.: Classical Electrodynamics. Wiley, New York (1999)
Krause T.B., Apte A., Morrison P.J.: A unified approach to the Darwin approximation. Phys. Plasmas 14, 102–112 (2002)
Lieb E., Loss M.: Analysis. Graduate Studies in Mathematics, Vol. 14. AMS, Providence (1997)
McOwen R.: Partial Differential Equations: Methods and Applications. Pearson Education, New Jersey (2003)
Pallard C.: The initial value problem for the relativistic Vlasov-Darwin system. Int. Mat. Res. Not. 57191, 1–31 (2006)
Rein G.: Collisionless kinetic equations from astrophysics: the Vlasov-Poisson system. Handbook of Differential Equations: Evolutionary Equations, Vol. 3. Elsevier, Amsterdam (2007)
Seehafer M.: Global classical solutions of the Vlasov-Darwin system for small initial data. Commun. Math. Sci. 6, 749–769 (2008)
Slezák, B.: On the inverse function theorem and implicit function theorem in Banach spaces. Function Spaces, Poznań, 1986, 186–190 (Ed. Musielak, J.). Teubner, Leipzig, 1988
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Communicated by D. Kinderlehrer
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Sospedra-Alfonso, R., Agueh, M. & Illner, R. Global Classical Solutions of the Relativistic Vlasov–Darwin System with Small Cauchy Data: The Generalized Variables Approach. Arch Rational Mech Anal 205, 827–869 (2012). https://doi.org/10.1007/s00205-012-0518-3
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DOI: https://doi.org/10.1007/s00205-012-0518-3