This paper investigates nonlinear Landau damping in the 3D Vlasov-Poisson (VP) system. We study the asymptotic stability of the Poisson equilibrium $\mu(v)=\frac{1}{\pi^2(1+|v|^2)^2}$ under small perturbations. Building on the foundational work of Ionescu, Pausader, Wang and Widmayer [28], we provide a streamlined proof of nonlinear Landau damping for the 3D unscreened VP system. Our analysis leverages sharp decay estimates, novel decomposition techniques to demonstrate the stabilization of the particle distribution and the decay of electric field. These results reveal the free transport-like behavior for the perturbed density $\rho(t,x)$, and enhance the understanding of Landau damping in an unconfined setting near stable equilibria.
Quoc-Hung NGUYEN
,
Dongyi WEI
,
Zhifei ZHANG
. A NEW PROOF OF NONLINEAR LANDAU DAMPING FOR THE 3D VLASOV-POISSON SYSTEM NEAR POISSON EQUILIBRIUM[J]. Acta mathematica scientia, Series B, 2025
, 45(6)
: 2669
-2684
.
DOI: 10.1007/s10473-025-0616-6
[1] Arsenév A. Global existence of a weak solution of Vlasov system of equations. USSR Comp Math Math Phys,1975, 15: 131-143
[2] Bardos C, Degond P. Global existence for the Vlasov-Poisson equation in 3 space variables with small initial data. Annales de l'Institut Henri Poincaré C Analyse nonlinéaire,1985, 2: 101-118
[3] Bedrossian J, Masmoudi N. Inviscid damping and the asymptotic stability of planar shear flows in the $2D$ Euler equations. Publ Math Inst Hautes études Sci,2015, 122: 195-300
[4] Bedrossian J. Nonlinear echoes and Landau damping with insufficient regularity. Tunis J of Math, 2021, 3: 121-205
[5] Bedrossian J, Masmoudi N, Mouhot C. Linearized wave-damping structure of Vlasov-Poisson in $\mathbb{R}^3$. SIAM J Math Anal, 2022, 54: 4379-4406
[6] Bedrossian J, Masmoudi N, Mouhot C. Landau damping in finite regularity for unconfined systems with screened interactions. Comm Pure Appl Math, 2018, 71: 537-576
[7] Bedrossian J, Masmoudi N, Mouhot C. Landau damping: Paraproducts and Gevrey regularity. Ann PDE, 2016, 2: Art 4
[8] Bouchut F. Global weak solution of the Vlasov-Poisson system for small electrons mass. Comm Partial Differential Equations, 1991, 16: 1337-1365
[9] Chen Q, Wei D, Zhang P, Zhang Z.Nonlinear inviscid damping for 2-D inhomogeneous incompressible Euler equations. J Eur Math Soc, 2025. DOI: 10.4171/JEMS/1608
[10] Choi S H, Ha S Y, Lee H. Dispersion estimates for the two-dimensional Vlasov-Yukawa system with small data. J Differential Equations, 2011, 250: 515-550
[11] Flynn P, Ouyang Z, Pausader B, Widmayer K. Scattering map for the Vlasov-Poisson system. Peking Math J, 2023, 6: 365-392
[12] Glassey R T.The Cauchy Problem in Kinetic Theory. Philadelphia, PA: Society for Industrial and Applied Mathematics, 1996
[13] Grenier E, Nguyen T T, Rodnianski I. Landau damping for analytic and Gevrey data. Math Res Lett, 2021, 28: 1679-1702
[14] Grenier E, Nguyen T T, Rodnianski I. Plasma echoes near stable Penrose data. SIAM J Math Anal, 2022, 54:
[15] Griffin-Pickering M, Iacobelli M.Global well-posedness for the Vlasov-Poisson system with massless electrons in the 3-dimensional torus. Comm Partial Differential Equations, 2021, 46: 1892-1939
[16] Griffin-Pickering M, Iacobelli M. Global strong solutions in $\mathbb{R}^3$ for ionic Vlasov-Poisson systems. Kinet Relat Models, 2021, 14: 571-597
[17] Han-Kwan D, Nguyen T T, Rousset F. Asymptotic stability of equilibria for screened Vlasov-Poisson systems via pointwise dispersive estimates. Ann PDE, 2021, 7: Paper 18
[18] Han-Kwan D, Nguyen T, Rousset F. On the linearized Vlasov-Poisson system on the whole space around stable homogeneous equilibria. Commun Math Phys, 2021, 387: 1405-1440
[19] Horst E. On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation I general theory. Math Meth Appl Sci, 1981, 3: 229-248
[20] Horst E. On the classical solutions of the initial value problem for the unmodified non-linear Vlasov equation II special cases. Math Meth Appl Sci, 1982, 4: 19-32
[21] Horst E. On the asymptotic growth of the solutions of the Vlasov-Poisson system. Mathematical Methods in the Applied Sciences, 1993, 16: 75-86
[22] Huang L, Nguyen Q H, Xu Y.Sharp estimates for screened Vlasov-Poisson system around Penrose-stable equilibria in $\mathbb{R}^d $, $ d\geq3. $ arXiv: 2205.10261
[23] Huang L, Nguyen Q H, Xu Y.Nonlinear Landau damping for the 2d Vlasov-Poisson system with massless electrons around Penrose-stable equilibrium. arXiv: 2206.11744
[24] Hwang H J, Rendall A, Velázquez J L. Optimal gradient estimates and asymptotic behaviour for the Vlasov-Poisson system with small initial data. Arch Ration Mech Anal,2011, 200: 313-360
[25] Ionescu A D, Jia H.Inviscid damping near the Couette flow in a channel. Commun Math Phys, 2020, 374: 2015-2096
[26] Ionescu A D, Jia H. Nonlinear inviscid damping near monotonic shear flows. Acta Math, 2023, 230: 321-399
[27] Ionescu A, Pausader B, Wang X, Widmayer K. On the asymptotic behavior of solutions to the Vlasov-Poisson system. Int Math Res Not, 2022, 2022: 8865-8889
[28] Ionescu A, Pausader B, Wang X, Widmayer K. Nonlinear Landau damping for the Vlasov-Poisson system in $\mathbb{R}^ 3$: The Poisson equilibrium. Ann PDE, 2024, 10: Paper 2
[29] Lions P L, Perthame B. Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system. Invent Math, 1991, 105: 415-430
[30] Mouhot C, Villani C. On Landau damping. Acta Math, 2011, 207: 29-201
[31] Masmoudi N, Zhao W. Nonlinear inviscid damping for a class of monotone shear flows in finite channel. Ann of Math, 2024, 199: 1093-1175
[32] Pfaffelmoser K. Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data. J Differential Equations, 1992, 95: 281-303
[33] Schaeffer J. Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions. Comm Partial Differential Equations, 1991, 16: 1313-1335
[34] Smulevici J. Small data solutions of the Vlasov-Poisson system and the vector field method. Ann PDE, 2016, 2: Art 11
[35] Wang X.Decay estimates for the 3D relativistic and non-relativistic Vlasov-Poisson systems. arXiv: 1805.10837
