Abstract.
We study an optimal inequality which relates potential and kinetic energies in an appropriate framework for bounded solutions of the Vlasov-Poisson (VP) system. Optimal distribution functions, which are completely characterized, minimize the total energy. From this variational approach, we deduce bounds for the kinetic and potential energies in terms of conserved quantities (mass and total energy) of the solutions of the VP system and a nonlinear stability result. Then we apply our estimates to the study of the large-time asymptotics and observe two different regimes.
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Bardos, C., Degond, P.: Global existence for the Vlasov-Poisson equation in~3 space variables with small initial data. Ann. Inst. H. Poincaré, Analyse Non Linéaire 2, 101–118 (1985)
Batt, J.: Asymptotic properties of spherically symmetric self–gravitating mass systems for t → ∞. Transport Theory and Statist. Phys. 16, 763–778 (1987)
Batt, J., Faltenbacher, W., Horst, E.: Stationary spherically symmetric models in stellar dynamics. Arch. Ration. Mech. Anal. 93, 159–183 (1986)
Batt, J., Pfaffelmoser, K.: On the radius continuity of the models of polytropic gas spheres which correspond to the positive solutions of the generalized Emdem-Fowler equation. Math. Meth. Appl. Sci. 10, 499–516 (1988)
Castella, F.: Propagation of space moments in the Vlasov-Poisson equation and further results. Ann. Inst. H. Poincaré Anal. Non Linéaire 16, 503–533 (1999)
Guo, Y.: Variational method for stable polytropic galaxies. Arch. Ration. Mech. Anal. 150, 209–224 (1999)
Guo, Y., Rein, G.: Stable steady states in stellar Dynamics. Arch. Ration. Mech. Anal. 147, 225–243 (1999)
Horst, E.: On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation: I, General Theory. Math. Methods Appl. Sci. 4, 229–248 (1982)
Horst, E.: On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation: II, Special Cases. Math. Methods Appl. Sci. 3, 19–32 (1982)
Illner, R., Rein, G.: Time decay of the solutions of the Vlasov-Poisson system in the plasma physical case. Math. Methods Appl. Sci. 19, 1409–1413 (1996)
Kurth, R.: A global particular solution to the initial value problem of stellar dynamics. Quart. Appl. Math. 36, 325–329 (1978)
Lieb, E., Loss, M.: Analysis. American Mathematical Society, 1997
Lieb, E.H.: Existence an uniqueness of the mininimizing solution of Choquard’s Nonlinear Equation. Studies in Appl. Math. 57, 93–105 (1977)
Lieb, E.H.: Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities. Ann. Math. 118, 349–374 (1983)
Lions, P.-L.: The concentration-compactness principle in the calculus of variations, Part I: The locally compact case. Ann. Inst. H. Poincaré 1, 109–145 (1984)
Perthame, B.: Time decay, propagation of low moments and dispersive effects for kinetic equations. Comm. Part. Diff. Eqs. 21, 659–686 (1996)
Rein, G.: Reduction and a concentration-compactness principle for Energy-Casimir functionals. Siam J. Math. Anal. 33, 896–912 (2001)
Ruiz Arriola, E., Soler, J.: A variational approach to the Schrödinger–Poisson system: asymptotic behaviour, breathers and stability. J. Stat. Phys. 103, 1069–1106 (2001)
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Dolbeault, J., Sánchez, Ó. & Soler, J. Asymptotic Behaviour for the Vlasov-Poisson System in the Stellar-Dynamics Case. Arch. Rational Mech. Anal. 171, 301–327 (2004). https://doi.org/10.1007/s00205-003-0283-4
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DOI: https://doi.org/10.1007/s00205-003-0283-4