On a Vlasov-Fokker-Planck equation for stored electron beams. (English) Zbl 1542.35383
Summary: In this paper we study a self-consistent Vlasov-Fokker-Planck equations which describes the longitudinal dynamics of an electron bunch in the storage ring of a synchrotron particle accelerator. We show existence and uniqueness of global classical solutions under physical hypotheses on the initial data. The proof relies on a mild formulation of the equation and hypoelliptic regularization estimates. We also address the problem of the long-time behavior of solutions. We prove the existence of steady states, called Haissinski solutions, given implicitly by a nonlinear integral equation. When the beam current (i.e. the nonlinearity) is small enough, we show uniqueness of steady state and local asymptotic nonlinear stability of solutions in appropriate weighted Lebesgue spaces. The proof is based on hypocoercivity estimates. Finally, we discuss the physical derivation of the equation and its particular asymmetric interaction potential.
MSC:
| 35Q83 | Vlasov equations |
| 35Q84 | Fokker-Planck equations |
| 78A40 | Waves and radiation in optics and electromagnetic theory |
| 78A35 | Motion of charged particles |
| 78A55 | Technical applications of optics and electromagnetic theory |
| 35B35 | Stability in context of PDEs |
| 82D75 | Nuclear reactor theory; neutron transport |
| 83A05 | Special relativity |
| 35A09 | Classical solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35H10 | Hypoelliptic equations |
Keywords:
Vlasov-Fokker-Planck; hypocoercivity; hypoellipticity; synchrotron radiation; Wakefield; Haissinski solutionsSoftware:
VadorReferences:
| [1] | Addala, Lanoir; Dolbeault, Jean; Li, Xingyu; Tayeb, M. Lazhar, \( \operatorname{L}^2\)-hypocoercivity and large time asymptotics of the linearized Vlasov-Poisson-Fokker-Planck system, J. Stat. Phys., 184, 1, Article 4 pp., 2021 · Zbl 1486.82034 |
| [2] | Bakry, Dominique; Gentil, Ivan; Ledoux, Michel, Analysis and Geometry of Markov Diffusion Operators, Grundlehren Math. Wiss., vol. 348, 2014, Springer: Springer Cham · Zbl 1376.60002 |
| [3] | Bolley, François; Guillin, Arnaud; Malrieu, Florent, Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation, ESAIM: Math. Model. Numer. Anal., 44, 5, 867-884, 2010 · Zbl 1201.82029 |
| [4] | Bouchut, François, Existence and uniqueness of a global smooth solution for the Vlasov- Poisson-Fokker-Planck system in three dimensions, J. Funct. Anal., 111, 1, 239-258, 1993 · Zbl 0777.35059 |
| [5] | Bouchut, François, Smoothing effect for the nonlinear Vlasov-Poisson-Fokker-Planck system, J. Differ. Equ., 122, 2, 225-238, 1995 · Zbl 0840.35053 |
| [6] | Bouin, Emeric; Dolbeault, Jean; Mischler, Stéphane; Mouhot, Clément; Schmeiser, Christian, Hypocoercivity without confinement, Pure Appl. Anal., 2, 2, 203-232, 2020 · Zbl 1448.82035 |
| [7] | Cai, Yunhai, Linear theory of microwave instability in electron storage rings, Phys. Rev. Spec. Top., Accel. Beams, 14, 6, Article 061002 pp., 2011 |
| [8] | Cai, Yunhai, Coherent synchrotron radiation by electrons moving on circular orbits, Phys. Rev. Accel. Beams, 20, 6, Article 064402 pp., 2017 |
| [9] | Chandrasekhar, S., Stochastic problems in physics and astronomy, Rev. Mod. Phys., 15, 1-89, 1943 · Zbl 0061.46403 |
| [10] | Degond, P.; Raviart, P. A., On the paraxial approximation of the stationary Vlasov-Maxwell system, Math. Models Methods Appl. Sci., 3, 4, 513-562, 1993 · Zbl 0787.35110 |
| [11] | Degond, Pierre, Global existence of smooth solutions for the Vlasov-Fokker-Planck equation in 1 and 2 space dimensions, Ann. Sci. Éc. Norm. Supér. (4), 19, 4, 519-542, 1986 · Zbl 0619.35087 |
| [12] | Dolbeault, J., Stationary states in plasma physics: Maxwellian solutions of the Vlasov- Poisson system, Math. Models Methods Appl. Sci., 1, 2, 183-208, 1991 · Zbl 0758.35062 |
| [13] | Dolbeault, Jean; Mouhot, Clément; Schmeiser, Christian, Hypocoercivity for linear kinetic equations conserving mass, Trans. Am. Math. Soc., 367, 6, 3807-3828, 2015 · Zbl 1342.82115 |
| [14] | Evain, C.; Szwaj, C.; Roussel, E.; Rodriguez, J.; Le Parquier, M.; Tordeux, M-A.; Ribeiro, F.; Labat, M.; Hubert, N.; Brubach, J-B., Stable coherent terahertz synchrotron radiation from controlled relativistic electron bunches, Nat. Phys., 15, 7, 635-639, 2019 |
| [15] | Evain, Clément; Roussel, Eleonore; Le Parquier, Marc; Szwaj, Christophe; Tordeux, M-A.; Brubach, J-B.; Manceron, Laurent; Roy, Pascale; Bielawski, Serge, Direct observation of spatiotemporal dynamics of short electron bunches in storage rings, Phys. Rev. Lett., 118, 5, Article 054801 pp., 2017 |
| [16] | Favre, Gianluca; Pirner, Marlies; Schmeiser, Christian, Hypocoercivity and reaction-diffusion limit for a nonlinear generation-recombination model, Arch. Ration. Mech. Anal., 247, 4, Article 72 pp., 2023 · Zbl 07727496 |
| [17] | Filbet, Francis; Sonnendrücker, Eric, Modeling and numerical simulation of space charge dominated beams in the paraxial approximation, Math. Models Methods Appl. Sci., 16, 05, 763-791, 2006 · Zbl 1109.78013 |
| [18] | Gualdani, Maria Pia; Mischler, Stéphane; Mouhot, Clément; Mischler, Stéphane, Factorization of Non-symmetric Operators and Exponential H-Theorem, 2017, Société Mathématique de France · Zbl 1470.47066 |
| [19] | Guillin, Arnaud; Le Bris, Pierre; Monmarché, Pierre, Convergence rates for the Vlasov-Fokker-Planck equation and uniform in time propagation of chaos in non convex cases, Electron. J. Probab., 27, Article 124 pp., 2022 · Zbl 1504.60137 |
| [20] | Haissinski, J., Exact longitudinal equilibrium distribution of stored electrons in the presence of self-fields, Il Nuovo Cimento B (1971-1996), 18, 1, 72-82, 1973 |
| [21] | Helffer, Bernard; Nier, Francis, Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians, Lect. Notes Math., vol. 1862, 2005, Springer: Springer Berlin · Zbl 1072.35006 |
| [22] | Hérau, Frédéric; Nier, Francis, Isotropic hypoelliptic and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Arch. Ration. Mech. Anal., 171, 2, 151-218, 2004 · Zbl 1139.82323 |
| [23] | Hérau, Frédéric; Thomann, Laurent, On global existence and trend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system with exterior confining potential, J. Funct. Anal., 271, 5, 1301-1340, 2016 · Zbl 1347.35221 |
| [24] | Herda, Maxime; Rodrigues, L. Miguel, Large-time behavior of solutions to Vlasov-Poisson-Fokker-Planck equations: from evanescent collisions to diffusive limit, J. Stat. Phys., 170, 5, 895-931, 2018 · Zbl 1392.35311 |
| [25] | Hörmander, Lars, Hypoelliptic second order differential equations, Acta Math., 119, 147-171, 1967 · Zbl 0156.10701 |
| [26] | Lions, P.-L.; Masmoudi, N., Uniqueness of mild solutions of the Navier-Stokes system in \(L^N\), Commun. Partial Differ. Equ., 26, 11-12, 2211-2226, 2001 · Zbl 1086.35077 |
| [27] | Mischler, S.; Mouhot, C., Exponential stability of slowly decaying solutions to the kinetic-Fokker-Planck equation, Arch. Ration. Mech. Anal., 221, 2, 677-723, 2016 · Zbl 1338.35430 |
| [28] | Monmarché, Pierre, A note on a Vlasov-Fokker-Planck equation with non-symmetric interaction, 2023, arXiv preprint · Zbl 1522.60069 |
| [29] | Murphy, J. B.; Gluckstern, R. L.; Krinsky, S., Longitudinal wake field for an electron moving on a circular orbit, Part. Accel., 57, BNL-63090, 9-64, 1996 |
| [30] | Neunzert, H.; Pulvirenti, M.; Triolo, L., On the Vlasov-Fokker-Planck equation, Math. Methods Appl. Sci., 6, 527-538, 1984 · Zbl 0561.35070 |
| [31] | Roussel, Eléonore, Spatio-temporal dynamics of relativistic electron bunches during the microbunching instability: study of the Synchrotron SOLEIL and UVSOR storage rings, 2014, Université Lille1-Sciences et Technologies, PhD thesis |
| [32] | Stupakov, G., Lecture Notes on Classical Mechanics and Electromagnetism in Accelerator Physics, 2007, The US Particle Accelerator School: The US Particle Accelerator School Lansing, Michigan |
| [33] | Venturini, Marco; Warnock, Robert; Ruth, Ronald; Ellison, James A., Coherent synchrotron radiation and bunch stability in a compact storage ring, Phys. Rev. Spec. Top., Accel. Beams, 8, 1, Article 014202 pp., 2005 |
| [34] | Dean, Harold; Victory, jun, On the existence of global weak solutions for Vlasov-Poisson-Fokker- Planck systems, J. Math. Anal. Appl., 160, 2, 525-555, 1991 · Zbl 0764.35024 |
| [35] | Dean, Harold; Victory, jun; O’Dwyer, Brian P., On classical solutions of Vlasov-Poisson Fokker-Planck systems, Indiana Univ. Math. J., 39, 1, 105-156, 1990 · Zbl 0674.60097 |
| [36] | Villani, Cédric, Hypocoercivity, Mem. Am. Math. Soc., vol. 950, 2009, American Mathematical Society (AMS): American Mathematical Society (AMS) Providence, RI · Zbl 1197.35004 |
| [37] | Warnock, Robert; Bane, Karl, Numerical solution of the haïssinski equation for the equilibrium state of a stored electron beam, Phys. Rev. Accel. Beams, 21, 12, Article 124401 pp., 2018 |
| [38] | Warnock, Robert L.; Ellison, James A., A general method for propagation of the phase space distribution, with application to the saw-tooth instability, (The Physics of High Brightness Beams, 2000, World Scientific), 322-348 |
| [39] | Wiedemann, Helmut, Particle Accelerator Physics, 2015, Springer Nature |
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.
