Neiss, Robert Axel Symmetry reduction and periodic solutions in Hamiltonian Vlasov systems. (English) Zbl 1437.35662 SIAM J. Math. Anal. 52, No. 2, 1844-1863 (2020). Summary: In this paper, we discuss a general approach to finding periodic solutions bifurcating from equilibrium points of classical Vlasov systems. The main access to the problem is chosen through the Hamiltonian representation of any Vlasov system, first put forward in [J. Fröhlich et al., Commun. Math. Phys. 288, No. 3, 1023–1059 (2009; Zbl 1177.82016)] and generalized in [R. A. Neiss and P. Pickl, J. Stat. Phys. 178, No. 2, 472–498 (2020; Zbl 1442.82016); R. A. Neiss, Arch. Ration. Mech. Anal. 231, No. 1, 115–151 (2019; Zbl 1406.35405)]. The method transforms the problem into a setup of complex valued \(\mathcal{L}^2\) functions with phase equivariant Hamiltonian. Through Marsden-Weinstein symmetry reduction [J. Marsden and A. Weinstein, Rep. Math. Phys. 5, 121–130 (1974; Zbl 0327.58005)], the problem is mapped on a Hamiltonian system on the quotient manifold \(\mathbb{S}^{\mathcal{L}^2}/\mathbb{S}^1\), which actually proves to be necessary to close many trajectories of the dynamics. As a toy model to apply the method we use the harmonic Vlasov system, a nonrelativistic Vlasov equation with attractive harmonic two-body interaction potential. The simple structure of this model allows us to compute all of its solutions directly and therefore test the benefits of the Hamiltonian formalism and symmetry reduction in Vlasov systems. Cited in 2 Documents MSC: 35Q83 Vlasov equations 37J06 General theory of finite-dimensional Hamiltonian and Lagrangian systems, Hamiltonian and Lagrangian structures, symmetries, invariants 35B10 Periodic solutions to PDEs 35B32 Bifurcations in context of PDEs Keywords:Vlasov; Hamiltonian PDE; periodic bifurcation; symmetry reduction Citations:Zbl 1177.82016; Zbl 1442.82016; Zbl 1406.35405; Zbl 0327.58005 PDFBibTeX XMLCite \textit{R. A. Neiss}, SIAM J. Math. Anal. 52, No. 2, 1844--1863 (2020; Zbl 1437.35662) Full Text: DOI arXiv References: [1] A. Ambrosetti and G. Prodi, A Primer of Nonlinear Analysis, Cambridge Stud. Adv. Math. 34, Cambridge University Press, Cambridge, 1993. · Zbl 0781.47046 [2] J. Batt and G. Rein, Global classical solutions of the periodic Vlasov-Poisson system in three dimensions, C. R. Acad. Sci. Paris Sér. I Math., 313 (1991), pp. 411-416. · Zbl 0741.35058 [3] M. Bostan, Boundary value problem for the \(N\)-dimensional time periodic Vlasov-Poisson system, Math. Methods Appl. Sci., 29 (2006), pp. 1801-1848. · Zbl 1108.35023 [4] J. Fröhlich, A. Knowles, and S. Schwarz, On the mean-field limit of bosons with Coulomb two-body interaction, Comm. Math. Phys., 288 (2009), pp. 1023-1059. · Zbl 1177.82016 [5] J. Marsden and A. Weinstein, Reduction of symplectic manifolds with symmetry, Rep. Math. Phys., 5 (1974), pp. 121-130. · Zbl 0327.58005 [6] C. Mouhot, Stabilité orbitale pour le système de Vlasov-Poisson gravitationnel, d’après Lemou-Méhats-Raphaël, Guo, Lin, Rein et al., in Proceedings of Séminaire Bourbaki, Exposés, 2011/2012, pp. 1043-1058. [7] R. A. Neiss and P. Pickl, A mean field limit for the Hamiltonian Vlasov system, J. Stat. Phys., 178 (2020), pp. 472-498. · Zbl 1442.82016 [8] R. A. Neiss, Generalized symplectization of Vlasov dynamics and application to the Vlasov-Poisson system, Arch. Ration. Mech. Anal., 231 (2019), pp. 115-151. · Zbl 1406.35405 [9] G. Rein, Collisionless Kinetic Equations from Astrophysics-The Vlasov-Poisson System, Handb. Differ. Equ. 3, Elsevier/North-Holland, Amsterdam, 2007, pp. 383-476. · Zbl 1193.35230 [10] F. Schwabl, Quantenmechanik (QM I): Eine Einführung, Springer, Berlin, 2007. · Zbl 1166.81003 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.