On classical solutions and the classical limit of the Vlasov-Darwin system
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In this paper we study the initial value problem of the non-relativistic Vlasov-Darwin system with generalized variables (VDG). We first prove local existence and uniqueness of a nonnegative classical solution to VDG in three space variables, and establish the blow-up criterion. Then we show that it converges to the well-known Vlasov-Poisson system when the light velocity $c$ tends to infinity in a pointwise sense.
Abstract
In this paper we study the initial value problem of the non-relativistic Vlasov-Darwin system with generalized variables (VDG). We first prove local existence and uniqueness of a nonnegative classical solution to VDG in three space variables, and establish the blow-up criterion. Then we show that it converges to the well-known Vlasov-Poisson system when the light velocity $c$ tends to infinity in a pointwise sense.
참고문헌 (33)
- K. Asano, "On local solutions of the initial value problem for the Vlasov-Maxwell equation", Comm. Math. Phys., 106(4), pp.551~568 (1986)[DOI]
- C. Bardos, "Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data", Ann. Inst. H. Poincare Anal. Non Lineaire, 2(2), pp.101~118 (1985)[DOI]
- J. Batt, "Global symmetric solutions of the initial value problem of stellar dynamics", J. Differential Equations, 25(3), pp.342~364 (1977)[DOI]
- S. Bauer, "The Darwin approximation of the relativistic Vlasov-Maxwell system", Ann. Henri Poincare, 6(2), pp.283~308 (2005)[DOI]
- S. Benachour, "Global existence for the Vlasov-Darwin system in <TEX>${\mathbb{R}}^3$<", TEX> for small initial data, 2003, pp.26(4), 297~319
- P. Degond, "An analysis of the Darwin model of approximation to Maxwell's equations", Forum Math., 4(1), pp.13~44 (1992)
- R. J. DiPerna, "Global weak solutions of Vlasov-Maxwell systems", Comm. Pure Appl. Math., 42(6), pp.729~757 (1989)[DOI]
- D. Gilbarg, "Elliptic Partial Differential Equations of Second Order, reprint of the 1998 edition, Classics in Mathematics" (2001), Springer-Verlag
- R. T. Glassey, "The Cauchy Problem in Kinetic Theory" (1996)
- R. T. Glassey, "Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data", Comm. Math. Phys., 119(3), pp.353~384 (1988)[DOI]
- R. T. Glassey, "On the "one and one-half dimensional" relativistic Vlasov-Maxwell system", Math. Methods Appl. Sci., 13(2), pp.169~179 (1990)[DOI]
- R. T. Glassey, "The "two and one-half-dimensional" relativistic Vlasov Maxwell system", Comm. Math. Phys., 185(2), pp.257~284 (1997)[DOI]
- R. T. Glassey, "The relativistic Vlasov-Maxwell system in two space dimensions. I", Arch. Rational Mech. Anal., 141(4), pp.331~354 (1998)[DOI]
- R. T. Glassey, "The relativistic Vlasov-Maxwell system in two space dimensions. II", Arch. Rational Mech. Anal., 141(4), pp.355~374 (1998)[DOI]
- R. T. Glassey, "Singularity formation in a collisionless plasma could occur only at high velocities", Arch. Rational Mech. Anal., 92(1), pp.59~90 (1986)[DOI]
- R. T. Glassey, "Absence of shocks in an initially dilute collisionless plasma", Comm. Math. Phys., 113(2), pp.191~208 (1987)[DOI]
- F. Golse, "Mean field kinetic equations" (2013)
- E. Horst, "On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation. I. General theory", Math. Methods Appl. Sci, 3(2), pp.229~248 (1981)[DOI]
- E. Horst, "On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation. II. Special cases", Math. Methods Appl. Sci., 4(1), pp.19~32 (1982)[DOI]
- E. Horst, "Weak solutions of the initial value problem for the unmodified nonlinear Vlasov equation", Math. Methods Appl. Sci., 6(2), pp.262~279 (1984)[DOI]
- E. H. Lieb, "Analysis, Graduate Studies in Mathematics, 14" (1997)
- P.-L. Lions, "Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system", Invent. Math., 105(2), pp.415~430 (1991)[DOI]
- G. Loeper, "Uniqueness of the solution to the Vlasov-Poisson system with bounded density", J. Math. Pures Appl., 86(1), pp.68~79 (2006)[DOI]
- C. Pallard, "The initial value problem for the relativistic Vlasov-Darwin system", Int. Math. Res. Not., 2006, pp.31~ (2006)
- K. Pfaffelmoser, "Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data", J. Differential Equations, 95(2), pp.281~303 (1992)[DOI]
- G. Rein, "Collisionless kinetic equations from astrophysics-the Vlasov-Poisson system;Handbook of differential equations: evolutionary equations. Vol. III, Handb. Differ. Equ", pp.2007, Elsevier
- J. Schaeffer, "Asymptotic growth bounds for the Vlasov-Poisson system", Math. Methods Appl. Sci., 34(3), pp.262~277 (2011)[DOI]
- J. Schaeffer, "Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions", Comm. Partial Differential Equations, 16(8-9), pp.1313~1335 (1991)[DOI]
- J. Schaeffer, "The classical limit of the relativistic Vlasov-Maxwell system", Comm. Math. Phys., 104(3), pp.403~421 (1986)[DOI]
- M. Seehafer, "Global classical solutions of the Vlasov-Darwin system for small initial data", Commun. Math. Sci., 6(3), pp.749~764 (2008)[DOI]
- R. Sospedra-Alfonso, "Uniqueness of the compactly supported weak solutions of the relativistic Vlasov-Darwin system", Acta Appl. Math., 124, pp.207~227 (2013)[DOI]
- R. Sospedra-Alfonso, "Global classical solutions of the relativistic Vlasov-Darwin system with small Cauchy data: the generalized variables approach", Arch. Ration. Mech. Anal., 205(3), pp.827~869 (2012)[DOI]
- S. Wollman, "An existence and uniqueness theorem for the Vlasov-Maxwell system", Comm. Pure Appl. Math., 37(4), pp.457~462 (1984)[DOI]
