Transport equations with unbounded force fields and application to the vlasov - Poisson equation

Mathematical Models and Methods in Applied Sciences

Volumn 19, Issue 2, 2009, Pages 199-228

  Salort, Delphine ^a  

^a INSTITUT JACQUES MONOD   (France)

Author keywords

Dispersion estimate; Vlasov Poisson equation

Indexed keywords

A-PRIORI; DISPERSIVE EFFECTS; EXISTENCE AND UNIQUENESS; FORCE FIELDS; GLOBAL WEAK SOLUTIONS; KINETIC EQUATIONS; LINEAR TRANSPORT EQUATIONS; NEW APPROACHES; NONLINEAR TERMS; TRANSPORT EQUATIONS; VLASOV-POISSON EQUATION;

COSMOLOGY; EQUATIONS OF MOTION; INTEGRAL EQUATIONS; POISSON DISTRIBUTION; THREE DIMENSIONAL;

POISSON EQUATION;

EID: 65249138175     PISSN: 02182025     EISSN: None     Source Type: Journal    
DOI: 10.1142/S0218202509003401     Document Type: Article

References (22)

1. A. A Arsen'ev, Some estimates for the solution of the Vlasov equation, Zh. Vychisl. Mat. Mat. Fiz 25 (1985) 80-87.
2. A. A Arsen'ev, Global existence of a weak solution of Vlasov's system of equations, U.S.S.R. Comput. Math. Phys. 15 (1975) 131-153.
3. E. Hörst and R. Hunze, Weak solutions of the initial value problem for the unmodified nonlinear Vlasov equation, Math. Meth. Appl. Sci. 6 (1984) 262-279.
4. R. J. DiPerna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math. 6 (1989) 729-757.
5. R. J. DiPerna and P.-L. Lions, Solutions globales d'équations du type Vlasov-Poisson, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988) 655-658.
6. G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl. 86 (2006) 68-79
7. B. Perthame and P.-L. Lions, Propagation of moments and regularity for the threedimensional Vlasov-Poisson system, Invent. Math. 105 (1991) 415-430.
8. J. Batt, The present state of the existence theory of the Vlasov-Poisson and Vlasov- Maxwell system of partial differential equations in plasma physics, in Applications of Mathematics in Technology (Rome, 1984) (1984), pp. 375-385.
9. J. Batt and G. Rein, Global classical solutions of the periodic Vlasov-Poisson systemin three dimensions, MC. R. Acad. Sci. Paris Sér. I Math. 313 (1991) 411-416.
10. C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in three space variables with small initial data, Math. Ann. Inst. H. Poincaré Anal. NonLinéaire 2 (1985) 101-118.
11. E. Hörst, On the classical solutions of the initial value problem for the unmodified nonlinear Vlasov equation. I. General theory, Math. Meth. Appl. Sci. 3 (1981) 229-248.
12. E. Hörst, Global strong solutions of Vlasov's equation necessary and sufficient conditions for their existence, in Partial Differential Equations (Warsaw, 1984) 19 (1987)143-153.
13. K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Diff. Eqns. 95 (1992) 281-303.
14. J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. Partial Diff. Eqns. 16 (1991) 1313-1335.
15. B. Perthame, Mathematical tools for kinetic equations, Bull. Amer. Math. Soc. (N.S.) 41 (2004) 205-244.
16. I. Gasser, P.-E. Jabin and B. Perthame, Regularity and propagation of moments in some nonlinear Vlasov systems, Proc. Roy. Soc. Edinburgh Sect. A. 130 (2000) 1259-1273.
17. H. Bahouri and J.-Y. Chemin, équations d'ondes quasilinéaires et estimations de Strichartz, Amer. J. Math., 121 (1999) 1337-1377.
18. N. Burq, P. Gérard and N. Tzvetkov, Strichartz inequalities and the nonlinear
19. Schrödinger equation on compact manifolds, J. Amer. Math. 126 (2004) 569-605.
20. D. Salort, Dispersion and Strichartz estimates for the Liouville equation, J. Diff. Eqs. 233 (2007) 543-584.
21. F. Castella, Propagation of space moments in the Vlasov-Poisson equation and further results, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999) 503-533.
22. J.-Y. Chemin, Perfect Incompressible Fluids, Oxford Lecture Series in Mathematics and its Applications, Vol. 14 (Oxford Univ. Press, 1998).