×

Stationary solutions of the flat Vlasov-Poisson system. (English) Zbl 1447.35326

Summary: The stationary solutions are triples \((f,\rho,U)\) of three functions: the distribution function \(f=f(x,v)\), the potential \(U=U(x)\) and the local density \(\rho=\rho(x)\), \(x,v\in\mathbb{R}^2\), which are linked by the Vlasov-Poisson system. We prove the existence of wide classes of spherically symmetric stationary solutions with the property that \(\rho\) depends on \(|x|=r\) and \(f\) on the energy \(E:=U(x)+\frac{v^2}2\). First we answer the question of which given functions \(\rho\) are the local density of a stationary solution (inverse problem). Our result is (up to technicalities) that every \(\rho\geq0\) which is strictly decreasing on an interval \([0,R)\) and zero on its complement \((R\leqq\infty)\) belongs to this class. Second, we ask: which given functions \(q\) induce distribution functions \(f\) of the form \(f=q(-E_0 -E)\) \((E_0\geqq0)\) of a stationary solution? (direct problem). This question is answered for many \(q\) which are positive for positive and vanish for negative arguments in an approximative and constructive way which is based on numerical methods.

MSC:

35Q83 Vlasov equations
35Q60 PDEs in connection with optics and electromagnetic theory
85A05 Galactic and stellar dynamics
35R30 Inverse problems for PDEs
35C15 Integral representations of solutions to PDEs
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Andréasson, H.; Rein, G., On the rotation curves for axially symmetric disk solutions of the Vlasov-Poisson system, Mon. Not. R. Astron. Soc., 446, 3932-3942, (2015) · doi:10.1093/mnras/stu2346
[2] Batt J., Faltenbacher W., Horst E.: Stationary spherically symmetric models in stellar dynamics. Arch. Ration. Mech. Anal. 93, 159-83 (1986) · Zbl 0605.70008
[3] Batt, J.; Pfaffelmoser, K., On the radius continuity of the models of polytropic gas spheres which correspond to the positive solutions of the generalized Emden-Fowler equation, Math. Methodes Appl. Sci., 10, 499-516, (1988) · Zbl 0676.34017 · doi:10.1002/mma.1670100502
[4] Batt, J.; Li, Y., The positive solutions of the Matukuma equation and the problem of finite radius and finite mass, Arch. Ration. Mech. Anal., 198, 613-675, (2010) · Zbl 1229.34013 · doi:10.1007/s00205-010-0315-9
[5] Binney J., Tremaine S.: Galactic Dynamics. Princeton University Press, Princeton (1987) · Zbl 1130.85301
[6] Dietz S.: Flache Lösungen des Vlasov-Poisson-Systems. PhD thesis, Ludwig-Maximilians-Universität Munich 2002
[7] Dubinski, J.; Kuijken, K., Nearly self-consistent disk-bulge-halo models for galaxies, Mon. Not. R. Astron. Soc., 277, 1341-1353, (1995) · doi:10.1093/mnras/277.4.1341
[8] Dubinski, J.; Widrow, L., Equilibrium disk-bulge-halo models for the Milky Way and Andromeda galaxies, Astrophys. J., 631, 838-855, (2005) · doi:10.1086/432710
[9] Fiřt, R.; Rein, G., Stability of disk-like galaxies—Part I: stability via reduction, Analysis, 26, 507-525, (2006) · Zbl 1135.35084
[10] Fiřt, R., Stability of disk-like galaxies—Part II: the Kuzmin disk, Analysis, 27, 405-424, (2007) · Zbl 1144.35340
[11] Fiřt, R.; Rein, G.; Seehafer, M., Flat galaxies with dark matter halos—existence and stability, Commun. Math. Phys., 291, 225-255, (2009) · Zbl 1179.85012 · doi:10.1007/s00220-009-0872-7
[12] Geigant E.: Inversionsmethoden zur Konstruktion von stationären Lösungen des selbst-konsistenten Problems des Stellardynamik. Diplomarbeit, Ludwig-Maximilians-Universität Munich 1993
[13] Gradshteyn I.S., Ryzhik I.M.: Table of Integrals, Series, and Products. Academic Press, Boston (2000) · Zbl 0981.65001
[14] Guo, Y., Variational method in polytropic galaxies, Arch. Ration. Mech. Anal., 130, 163-182, (1995) · Zbl 0828.76093 · doi:10.1007/BF00375154
[15] Guo, Y.; Rein, G., Stable steady states in stellar dynamics, Arch. Ration. Mech. Anal., 147, 225-243, (1999) · Zbl 0935.70011 · doi:10.1007/s002050050150
[16] Guo, Y.; Rein, G., Existence and stability of Camm type steady states in galactic dynamics, Indiana Univ. Math. J., 48, 1237-1255, (1999) · Zbl 0945.35003 · doi:10.1512/iumj.1999.48.1819
[17] Guo, Y.; Rein, G., Isotropic steady states in galactic dynamics, Commun. Math. Phys., 219, 607-629, (2001) · Zbl 0974.35093 · doi:10.1007/s002200100434
[18] Korn G.A., Korn T.M.: Mathematical Handbook for Scientists and Engineers. McGraw-Hill, New York (1961) · Zbl 0121.00103
[19] Lieb, E., Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities, Ann. Math., 118, 349-374, (1983) · Zbl 0527.42011 · doi:10.2307/2007032
[20] Natanson I.P.: Theorie der Funktionen einer reellen Veränderlichen. Akademie-Verlag, Berlin (1961) · Zbl 0097.26601
[21] Rein G.: Collisionless kinetic equations from astrophysics—the Vlasov-Poisson system. In: Dafermos, C.M., Feireisl, E. (eds.) Handbook of Differential Equations: Evolutionary Equations, Vol. 3, 383-476. Elsevier, Amsterdam 2007 · Zbl 1193.35230
[22] Rein, G., Flat steady states in stellar dynamics—existence and stability, Commun. Math. Phys., 205, 229-247, (1999) · Zbl 0937.85003 · doi:10.1007/s002200050674
[23] Stoer J.: Einführung in die Numerische Mathematik I. Springer, Berlin (1972) · Zbl 0245.65001 · doi:10.1007/978-3-662-06865-6
[24] Willers F.A.: Methoden der Praktischen Analysis. Walter de Gruyter, Berlin (1957) · Zbl 0077.11006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.