On linear damping around inhomogeneous stationary states of the Vlasov-HMF model. (English) Zbl 1512.35578
Summary: We study the dynamics of perturbations around an inhomogeneous stationary state of the Vlasov-HMF (Hamiltonian Mean-Field) model, satisfying a linearized stability criterion (Penrose criterion). We consider solutions of the linearized equation around the steady state, and prove the algebraic decay in time of the Fourier modes of their density. We prove moreover that these solutions exhibit a scattering behavior to a modified state, implying a linear damping effect with an algebraic rate of damping.
References:
| [1] | Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Function. National Bureau of Standards Applied Mathematics Series, vol. 55 (1964) · Zbl 0171.38503 |
| [2] | Barré, J.; Bouchet, F.; Dauxois, T.; Ruffo, S.; Yamaguchi, YY, Stability criteria of the Vlasov equation and quasi stationary states of the HMF model, Physica A, 337, 36 (2004) · doi:10.1016/j.physa.2004.01.041 |
| [3] | Barré, J.; Yamaguchi, YY, On the neighborhood of an inhomogeneous stationary solutions of the Vlasov equation—case of an attractive cosine potential, J. Math. Phys., 56, 081502 (2015) · Zbl 1332.35361 · doi:10.1063/1.4927689 |
| [4] | Barré, J.; Olivetti, A.; Yamaguchi, YY, Algebraic damping in the one-dimensional Vlasov equation, J. Phys. A Math. Theor., 44, 405502 (2011) · Zbl 1432.35198 · doi:10.1088/1751-8113/44/40/405502 |
| [5] | Barré, J.; Bouchet, F.; Dauxois, T.; Ruffo, S.; Yamaguchi, YY, The Vlasov equation and the Hamiltonian mean-field model, Physica A, 365, 177 (2005) · doi:10.1016/j.physa.2006.01.005 |
| [6] | Barré, J., Olivetti, A., Yamaguchi, Y.Y.: Dynamics of perturbations around inhomogeneous backgrounds in the HMF model. J. Stat. Mech. P08002 (2010) |
| [7] | Bedrossian, J., Suppression of plasma echoes and Landau damping in Sobolev spaces by weak collisions in a Vlasov-Fokker-Planck equation, Ann. PDE, 3, 19 (2017) · Zbl 1397.35024 · doi:10.1007/s40818-017-0036-6 |
| [8] | Bedrossian, J., Masmoudi, N., Mouhot, C.: Landau damping, paraproducts and Gevrey regularity. Ann. PDE 2(1), 1-71 (2016) · Zbl 1402.35058 |
| [9] | Bedrossian, J., Masmoudi, N., Mouhot, C.: Landau damping in finite regularity for unconfined systems with screened interactions. Commun. Pure Appl. Math. 71(3), 537-576 (2018) · Zbl 1384.35127 |
| [10] | Bedrossian, J., Masmoudi, N., Mouhot, C.: Linearized wave-damping structure of Vlasov-Poisson in \({{R}}^3\). arXiv:2007.08580 (2020) |
| [11] | Benedetto, D.; Caglioti, E.; Montemagno, U., Exponential dephasing of oscillators in the kinetic Kuramoto model, J. Stat. Phys., 162, 813-823 (2016) · Zbl 1339.35328 · doi:10.1007/s10955-015-1426-3 |
| [12] | Byrd, PF; Friedman, MD, Handbook of Elliptic Integrals for Engineers and Scientists (1971), Berlin: Springer, Berlin · Zbl 0213.16602 · doi:10.1007/978-3-642-65138-0 |
| [13] | Caglioti, E.; Rousset, F., Long time estimates in the mean field limit, Arch. Ration. Mech. Anal., 190, 3, 517-547 (2008) · Zbl 1155.76063 · doi:10.1007/s00205-008-0157-x |
| [14] | Caglioti, E.; Rousset, F., Quasi-stationary states for particle systems in the mean-field limit, J. Stat. Phys., 129, 2, 241-263 (2007) · Zbl 1208.82034 · doi:10.1007/s10955-007-9390-1 |
| [15] | Campa, A., Chavanis, P.H.: A dynamical stability criterion for inhomogeneous quasi-stationary states in long-range systems. J. Stat. Mech. P06001,(2010) · Zbl 1202.82055 |
| [16] | Campa, A.; Chavanis, PH, Inhomogeneous Tsallis distributions in the HMF model, Eur. Phys. J. B, 76, 581-611 (2010) · Zbl 1202.82055 · doi:10.1140/epjb/e2010-00243-x |
| [17] | Chavanis, PH, Lynden-Bell and Tsallis distributions in the HMF model, Eur. Phys. J. B, 53, 487 (2006) · doi:10.1140/epjb/e2006-00405-5 |
| [18] | Chavanis, PH; Delfini, L., Dynamical stability of systems with long-range interactions: application of the Nyquist method to the HMF model, Eur. Phys. J. B, 69, 389-429 (2009) · doi:10.1140/epjb/e2009-00180-9 |
| [19] | Chavanis, PH; Vatteville, J.; Bouchet, F., Dynamics and thermodynamics of a simple model similar to self-gravitating systems: the HMF model, Eur. Phys. J. B, 46, 61-99 (2005) · doi:10.1140/epjb/e2005-00234-0 |
| [20] | Després, B., Scattering structure and Landau damping for linearized Vlasov equations with inhomogeneous Boltzmannian states, Ann. Henri Poincaré, 20, 2767-2818 (2019) · Zbl 1420.35411 · doi:10.1007/s00023-019-00818-y |
| [21] | Dietert, H., Stability and Bifurcation for the Kuramoto model, J. Math. Pures Appl., 105, 451-489 (2016) · Zbl 1339.35318 · doi:10.1016/j.matpur.2015.11.001 |
| [22] | Faou, E.; Rousset, F., Landau damping in Sobolev spaces for the Vlasov-HMF model, Arch. Ration. Mech. Anal., 219, 887-902 (2016) · Zbl 1333.35291 · doi:10.1007/s00205-015-0911-9 |
| [23] | Fernandez, B.; Gérard-Varet, D.; Giacomin, G., Landau damping in the Kuramoto model, Ann. Henri Poincaré, 17, 7, 1793-1823 (2016) · Zbl 1347.34052 · doi:10.1007/s00023-015-0450-9 |
| [24] | Grenier, E., Nguyen, T., Rodnianski, I.: Landau damping for analytic and Gevrey data. arxiv:2004.05979 (2020) |
| [25] | Gripenberg, G.; Londen, SO; Staffans, O., Volterra Integral and Functional Equations (1990), Cambridge: Cambridge University Press, Cambridge · Zbl 0695.45002 · doi:10.1017/CBO9780511662805 |
| [26] | Hairer, E.; Lubich, C.; Wanner, G., Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations (2006), Berlin: Springer, Berlin · Zbl 1094.65125 |
| [27] | Han-Kwan, D., Nguyen, T., Rousset, F.: Asymptotic stability of equilibria for screened Vlasov-Poisson systems via pointwise dispersive estimates. arxiv:1906.05723 (2019) |
| [28] | Han-Kwan, D., Nguyen, T., Rousset, F.: On the linearized Vlasov-Poisson system on the whole space around stable homogeneous equilibria. arxiv:2007.07787 (2020) |
| [29] | Laforgia, A.; Nataline, P., Some inequalities for modified Bessel functions, J. Inequal. Appl., 2010, 253035 (2010) · Zbl 1187.33002 · doi:10.1155/2010/253035 |
| [30] | Landau, L.: On the vibration of the electronic plasma. J. Phys. USSR 10(25) (1946). English translation in JETP 16, 574. Reproducted in Collected papers of L.D. Landau, edited with an introduction by D. ter Haar, Pergamon Press, 1965, 445-460; and in Men of Physics: L.D. Landau, Vol 2, Pergamon Press, D. ter Haar, ed. (1965) |
| [31] | Lemou, M.; Luz, AM; Méhats, F., Nonlinear stability criteria for the HMF model, Arch. Ration. Mech. Anal., 224, 353-380 (2017) · Zbl 1371.35294 · doi:10.1007/s00205-017-1077-4 |
| [32] | Milne, SC, Infinite Families of Exact Sums of Squares Formulae, Jacobi Elliptic Functions, Continued Fractions, and Schur Functions, Developments in Mathematics Series (2002), Berlin: Springer, Berlin · Zbl 1125.11316 · doi:10.1007/978-1-4757-5462-9 |
| [33] | Mouhot, C.; Villani, C., On Landau damping, Acta Math., 207, 1, 29-201 (2011) · Zbl 1239.82017 · doi:10.1007/s11511-011-0068-9 |
| [34] | Paley, REAC; Wiener, N., Fourier Transforms in the Complex Domain, Colloquium Publications (1934), Providence: American Mathematical Society, Providence · Zbl 0011.01601 |
| [35] | Tristani, I., Landau damping for the linearized Vlasov Poisson equation in a weakly collisional regime, J. Stat. Phys., 169, 107-125 (2017) · Zbl 1379.35317 · doi:10.1007/s10955-017-1848-1 |
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.
