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Nonlinear stability of a spatially symmetric solution of the relativistic Poisson-Vlasov equation. (English) Zbl 0649.35007

It is proven that the distribution functions f(\b{x},\b{p}), spatially homogeneous, possibly nonregular, nonincreasing in \(\| \underline p\|\), stationary solutions of the relativistic Poisson-Vlasov equation, are nonlinearly (Lyapunov) stable. No regularity assumption is done on the solutions (and on the initial data), and this might be relevant in some physical application. Similar method of proof may be employed in the relativistic Maxwell-Vlasov model.
Reviewer: E.Pagani

MSC:

35B35 Stability in context of PDEs
35Q99 Partial differential equations of mathematical physics and other areas of application
82D10 Statistical mechanics of plasmas
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[1] A.A. Arsen’ev , Global existence of a weak solution of Vlasov’s system of equations , U.R.S.S. Comput. Math. and Math. Phys. , 15 ( 1 ) ( 1975 ), pp. 131 - 143 . MR 371322
[2] A.A. Arsen’ev , Existence and uniqueness of the classical solutions of Vlasov’s system of equations , U.R.S.S. Comput. Math. and Math. Phys. , 15 ( 5 ) ( 1975 ), pp. 252 - 258 . MR 395633 | Zbl 0345.35083 · Zbl 0345.35083 · doi:10.1016/0041-5553(75)90123-8
[3] C. Bardos - P. Degond , Global existence for the Vlasov-Poisson equation in three space variables with small initial data , C. R. Acad. Sc. Paris , 297 , Sez. 1 ( 1983 ), p. 131 . MR 732496
[4] J. Batt , Global symmetric solutions of the initial-value problem in stellar dynamics , J. Diff. Eq. , 25 ( 1977 ), pp. 342 - 364 . MR 487082 | Zbl 0366.35020 · Zbl 0366.35020 · doi:10.1016/0022-0396(77)90049-3
[5] J. Batt , The nonlinear Vlasov-Poisson system of partial differential equations in stellar dynamics , Publ. C.N.E.R. Math. Pures Appl. Année 83 , vol. 5 , fasc. 2 ( 1983 ), pp. 1 - 30 .
[6] J. Cooper , Galerkin approximations for the one-dimensional Vlasov-Poisson equation , Math. Meth. in the Appl. Sci. , 5 ( 1983 ), pp. 516 - 529 . MR 723605 | Zbl 0541.65084 · Zbl 0541.65084 · doi:10.1002/mma.1670050131
[7] J. Cooper - A. Klimas , Boundary value problem for the Vlasov-Xaxwell equations in one dimension , J. Math. Anal. Appl. , 75 ( 1980 ), pp. 306 - 329 . MR 581821 | Zbl 0454.35075 · Zbl 0454.35075 · doi:10.1016/0022-247X(80)90082-7
[8] S.R. De Groot - C.G. Van Weert - W.A. Van Leeuwen , Relativistic Kinetic Theory. Principles and Application , North Holland , Amsterdam ( 1980 ). MR 635279
[9] C.S. Gardner , Bound on the energy available from a plasma , Phys. Fluids , 6 ( 1963 ), pp. 839 - 840 . MR 152330
[10] R. Glassey - J. SCHAEFFER, On symmetric solutions of the relativistic Vlasov-Poisson system , Comm. Math. Phys. , 101 ( 1985 ), pp. 459 - 473 . Article | MR 815195 | Zbl 0582.35110 · Zbl 0582.35110 · doi:10.1007/BF01210740
[11] R. Glassey - W. Strauss , Singularity formation in a collisionless plasma could occurr only at high velocities , Arch. Rat. Mech. Anal. (in print). Zbl 0595.35072 · Zbl 0595.35072 · doi:10.1007/BF00250732
[12] D.D. Holm - J.E. Marsden - T. Ratiu - A. Weinstein , Nonlinear stability of fluid and plasma equilibria , Physics Reports , 123 ( 1985 ), pp. 1 - 116 . MR 794110 | Zbl 0717.76051 · Zbl 0717.76051 · doi:10.1016/0370-1573(85)90028-6
[13] E. Horst , On the classical solutions of the initial-value problem for the unmodified nonlinear Vlasov equation I, II , Math. Meth. Appl. Sci. , 3 ( 1981 ), pp. 229 - 248 ; 4 ( 1982 ), pp. 19 - 32 . Zbl 0463.35071 · Zbl 0463.35071 · doi:10.1002/mma.1670030117
[14] E. Horst - R. Hunze , Weak solution of the initial-value problem for the unmodified nonlinear Vlasov equation , Math. Meth. Appl. Sci. , 6 ( 1984 ), pp. 262 - 279 . MR 751745 | Zbl 0556.35022 · Zbl 0556.35022 · doi:10.1002/mma.1670060118
[15] R. Illner - H. Neunzert , An existence theorem for the unmodified Vlasov equation , Math. Meth. Appl. Sci. , 1 ( 1979 ), pp. 530 - 554 . MR 548686 | Zbl 0415.35076 · Zbl 0415.35076 · doi:10.1002/mma.1670010410
[16] C. Marchioro - M. PULVIRENTI, Some considerations on the nonlinear stability of stationary planar Euler flows , Comm. Math. Phys. , 100 ( 1985 ). pp. 343 - 354 . Article | MR 802550 | Zbl 0625.76060 · Zbl 0625.76060 · doi:10.1007/BF01206135
[17] C. Marchioro - M. Pulvirenti , A note on the nonlinear stability of a spatial symmetric Vlasov-Poisson flow , Math. Meth. Appl. Sci. (in print). MR 845931 | Zbl 0609.35008 · Zbl 0609.35008 · doi:10.1002/mma.1670080119
[18] J.E. Marsden - A. Weinstein , The Hamiltonian structure of the Maxwell-Vlasov equations , Physica , 4D ( 1982 ), pp. 394 - 406 . MR 657741 · Zbl 1194.35463 · doi:10.1016/0167-2789(82)90043-4
[19] H. Neunzert , An introduction to the nonlinear Boltzmann-Vlasov equation , Lecture Notes in Mathematics , 1048 ( Springer-Verlag , Berlin , 1984 ), pp. 60 - 110 . MR 740721 | Zbl 0575.76120 · Zbl 0575.76120
[20] H. Neunzert , Approximation methods for the nonmodified Vlasov-Poisson system , Proceedings of the Workshop on Math. Aspects of Fluid and Plasma Dynamics ( Trieste , Italy, May 30 - June 2, 1984 ), edited by C. Cercignani, S. Rionero, M. Tessarotto, pp. 439 - 455 .
[21] M.N. Rosenbluth , Topics in microinstabilities , in Advanced Plasma Physics , M. Rosenbluth, ed. ( Academic Press , New York , 1964 ). MR 170672
[22] S. Ukai - T. OKABE, On classical solutions in the large in time of two dimensional Vlasov’s equation , Osaka J. Math. , 15 ( 1978 ), pp. 245 - 261 . MR 504289 | Zbl 0405.35002 · Zbl 0405.35002
[23] N.G. Van Kampen - B.V. Felderhof , Theoretical Methods in Plasma Physics , North Holland , Amsterdam ( 1967 ). Zbl 0159.29601 · Zbl 0159.29601
[24] E. Weibel , L’equation de Vlasov dans la théorie spéciale de la relativité , Plasma Phys. , 9 ( 1967 ), pp. 665 - 670 .
[25] S. Wollman , The spherically symmetric Vlasov-Poisson system , J. Diff. Equations , 35 ( 1980 ), pp. 30 - 35 . MR 556789 | Zbl 0402.76089 · Zbl 0402.76089 · doi:10.1016/0022-0396(80)90046-7
[26] S. Wollman , An existence and uniqueness theorem for the Vlasov-Maxwell system , Commun. Pure Appl. Math. , 37 ( 1984 ), pp. 457 - 462 . MR 745326 | Zbl 0592.45010 · Zbl 0592.45010 · doi:10.1002/cpa.3160370404
[27] S. Wollman , Global in time solutions of the two dimensional Vlasov-Poisson system , Commun. Pure Appl. Math. , 33 ( 1980 ), pp. 173 - 197 . MR 562549 | Zbl 0437.45023 · Zbl 0437.45023 · doi:10.1002/cpa.3160330205
[28] P. Degond , Local existence of solutions of the Maxwell-Vlasov equation and convergence to the Vlasov-Poisson equation for infinite light velocity , Int. Rep. 117 , Centre de Mathématiques Appliquées , École Polytéchnique , Paris ( 1984 ). · Zbl 0619.35088
[29] J. Schaeffer , The classical limit of the relativistic Vlasov-Maxwell system , Commun. Math. Phys. , 104 ( 1986 ), pp. 403 - 421 . Article | MR 840744 | Zbl 0597.35109 · Zbl 0597.35109 · doi:10.1007/BF01210948
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