December  2020, 13(6): 1193-1218. doi: 10.3934/krm.2020043

Derivative estimates for screened Vlasov-Poisson system around Penrose-stable equilibria

Department of Mathematics, Pennsylvania State University, State College, PA 16802, USA

Received  June 2020 Published  September 2020

Fund Project: The author would like to thank Toan T. Nguyen for his many insightful discussions on the subject. The research was supported by the NSF under grant DMS-1764119

In this paper, we establish derivative estimates for the Vlasov-Poisson system with screening interactions around Penrose-stable equilibria on the phase space $ {\mbox{Re }}^d_x\times {\mbox{Re }}_v^d $, with dimension $ d\ge 3 $. In particular, we establish the optimal decay estimates for higher derivatives of the density of the perturbed system, precisely like the free transport, up to a log correction in time. This extends the recent work [13] by Han-Kwan, Nguyen and Rousset to higher derivatives of the density. The proof makes use of several key observations from [13] on the structure of the forcing term in the linear problem, with induction arguments to classify all the terms appearing in the derivative estimates.

Citation: Trinh T. Nguyen. Derivative estimates for screened Vlasov-Poisson system around Penrose-stable equilibria. Kinetic & Related Models, 2020, 13 (6) : 1193-1218. doi: 10.3934/krm.2020043
References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, volume 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

R. Balescu, Statistical Mechanics of Charged Particles, Monographs in Statistical Physics and Thermodynamics, Vol. 4. Interscience Publishers John Wiley & Sons, Ltd. London-New York-Sydney, 1963.  Google Scholar

[3]

C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in $3$ space variables with small initial data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 101-118.  doi: 10.1016/S0294-1449(16)30405-X.  Google Scholar

[4]

J. Bedrossian, N. Masmoudi and C. Mouhot, Landau damping: Paraproducts and Gevrey regularity, Ann. PDE, 2 (2016), Art. 4, 71 pp. doi: 10.1007/s40818-016-0008-2.  Google Scholar

[5]

J. BedrossianN. Masmoudi and C. Mouhot, Landau damping in finite regularity for unconfined systems with screened interactions, Comm. Pure Appl. Math., 71 (2018), 537-576.  doi: 10.1002/cpa.21730.  Google Scholar

[6] T. J. M. Boyd and J. J. Sanderson, The Physics of Plasmas, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511755750.  Google Scholar
[7]

P.-H. Chavanis, Statistical mechanics of violent relaxation in stellar systems, In Multiscale problems in science and technology (Dubrovnik, 2000), Springer, Berlin, 2002, pages 85–116.  Google Scholar

[8]

S.-H. ChoiS.-Y. Ha and H. Lee, Dispersion estimates for the two-dimensional Vlasov-Yukawa system with small data, J. Differential Equations, 250 (2011), 515-550.  doi: 10.1016/j.jde.2010.10.005.  Google Scholar

[9]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.  Google Scholar

[10]

D. Han-Kwan, Quasineutral limit of the Vlasov-Poisson system with massless electrons, Comm. Partial Differential Equations, 36 (2011), 1385-1425.  doi: 10.1080/03605302.2011.555804.  Google Scholar

[11]

D. Han-Kwan and M. Iacobelli, The quasineutral limit of the Vlasov-Poisson equation in {W}asserstein metric, Commun. Math. Sci., 15 (2017), 481-509.  doi: 10.4310/CMS.2017.v15.n2.a8.  Google Scholar

[12]

D. Han-KwanT. T. Rousset and F. Nguyenand, Long time estimates for the Vlasov-Maxwell system in the non-relativistic limit, Comm. Math. Phys., 363 (2018), 389-434.  doi: 10.1007/s00220-018-3208-7.  Google Scholar

[13]

D. Han-Kwan, T. T. Nguyen and F. Rousset, Asymptotic stability of equilibria for screened vlasov-poisson systems via pointwise dispersive estimates, 2019. Google Scholar

[14]

D. Han-Kwan and F. Rousset, Quasineutral limit for Vlasov-Poisson with {P}enrose stable data, Ann. Sci. Éc. Norm. Supér., 49 (2016), 1445–1495. doi: 10.24033/asens.2313.  Google Scholar

[15]

E. Horst, On the asymptotic growth of the solutions of the Vlasov-Poisson system, Math. Methods Appl. Sci., 16 (1993), 75-86.  doi: 10.1002/mma.1670160202.  Google Scholar

[16]

H. J. HwangA. Rendall and J. J. L. Velázquez, Optimal gradient estimates and asymptotic behaviour for the Vlasov-Poisson system with small initial data, Arch. Ration. Mech. Anal., 200 (2011), 313-360.  doi: 10.1007/s00205-011-0405-3.  Google Scholar

[17]

P.-L. Lions and B. Perthame, Propagation of moments and regularity for the $3$-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430.  doi: 10.1007/BF01232273.  Google Scholar

[18]

C. Mouhot and C. Villani, On Landau damping, Acta Math., 207 (2011), 29-201.  doi: 10.1007/s11511-011-0068-9.  Google Scholar

[19]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations, 95 (1992), 281-303.  doi: 10.1016/0022-0396(92)90033-J.  Google Scholar

[20]

J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. Partial Differential Equations, 16 (1991), 1313-1335.  doi: 10.1080/03605309108820801.  Google Scholar

[21]

J. Smulevici, Small data solutions of the Vlasov-Poisson system and the vector field method, Ann. PDE, 2 (2016), Art. 11, 55 pp. doi: 10.1007/s40818-016-0016-2.  Google Scholar

[22]

X. Wang, Decay estimates for the $3d$ relativistic and non-relativistic vlasov-poisson systems, 2018. Google Scholar

show all references

References:
[1]

H. Bahouri, J.-Y. Chemin and R. Danchin, Fourier Analysis and Nonlinear Partial Differential Equations, volume 343 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-16830-7.  Google Scholar

[2]

R. Balescu, Statistical Mechanics of Charged Particles, Monographs in Statistical Physics and Thermodynamics, Vol. 4. Interscience Publishers John Wiley & Sons, Ltd. London-New York-Sydney, 1963.  Google Scholar

[3]

C. Bardos and P. Degond, Global existence for the Vlasov-Poisson equation in $3$ space variables with small initial data, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2 (1985), 101-118.  doi: 10.1016/S0294-1449(16)30405-X.  Google Scholar

[4]

J. Bedrossian, N. Masmoudi and C. Mouhot, Landau damping: Paraproducts and Gevrey regularity, Ann. PDE, 2 (2016), Art. 4, 71 pp. doi: 10.1007/s40818-016-0008-2.  Google Scholar

[5]

J. BedrossianN. Masmoudi and C. Mouhot, Landau damping in finite regularity for unconfined systems with screened interactions, Comm. Pure Appl. Math., 71 (2018), 537-576.  doi: 10.1002/cpa.21730.  Google Scholar

[6] T. J. M. Boyd and J. J. Sanderson, The Physics of Plasmas, Cambridge University Press, Cambridge, 2003.  doi: 10.1017/CBO9780511755750.  Google Scholar
[7]

P.-H. Chavanis, Statistical mechanics of violent relaxation in stellar systems, In Multiscale problems in science and technology (Dubrovnik, 2000), Springer, Berlin, 2002, pages 85–116.  Google Scholar

[8]

S.-H. ChoiS.-Y. Ha and H. Lee, Dispersion estimates for the two-dimensional Vlasov-Yukawa system with small data, J. Differential Equations, 250 (2011), 515-550.  doi: 10.1016/j.jde.2010.10.005.  Google Scholar

[9]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.  Google Scholar

[10]

D. Han-Kwan, Quasineutral limit of the Vlasov-Poisson system with massless electrons, Comm. Partial Differential Equations, 36 (2011), 1385-1425.  doi: 10.1080/03605302.2011.555804.  Google Scholar

[11]

D. Han-Kwan and M. Iacobelli, The quasineutral limit of the Vlasov-Poisson equation in {W}asserstein metric, Commun. Math. Sci., 15 (2017), 481-509.  doi: 10.4310/CMS.2017.v15.n2.a8.  Google Scholar

[12]

D. Han-KwanT. T. Rousset and F. Nguyenand, Long time estimates for the Vlasov-Maxwell system in the non-relativistic limit, Comm. Math. Phys., 363 (2018), 389-434.  doi: 10.1007/s00220-018-3208-7.  Google Scholar

[13]

D. Han-Kwan, T. T. Nguyen and F. Rousset, Asymptotic stability of equilibria for screened vlasov-poisson systems via pointwise dispersive estimates, 2019. Google Scholar

[14]

D. Han-Kwan and F. Rousset, Quasineutral limit for Vlasov-Poisson with {P}enrose stable data, Ann. Sci. Éc. Norm. Supér., 49 (2016), 1445–1495. doi: 10.24033/asens.2313.  Google Scholar

[15]

E. Horst, On the asymptotic growth of the solutions of the Vlasov-Poisson system, Math. Methods Appl. Sci., 16 (1993), 75-86.  doi: 10.1002/mma.1670160202.  Google Scholar

[16]

H. J. HwangA. Rendall and J. J. L. Velázquez, Optimal gradient estimates and asymptotic behaviour for the Vlasov-Poisson system with small initial data, Arch. Ration. Mech. Anal., 200 (2011), 313-360.  doi: 10.1007/s00205-011-0405-3.  Google Scholar

[17]

P.-L. Lions and B. Perthame, Propagation of moments and regularity for the $3$-dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430.  doi: 10.1007/BF01232273.  Google Scholar

[18]

C. Mouhot and C. Villani, On Landau damping, Acta Math., 207 (2011), 29-201.  doi: 10.1007/s11511-011-0068-9.  Google Scholar

[19]

K. Pfaffelmoser, Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data, J. Differential Equations, 95 (1992), 281-303.  doi: 10.1016/0022-0396(92)90033-J.  Google Scholar

[20]

J. Schaeffer, Global existence of smooth solutions to the Vlasov-Poisson system in three dimensions, Comm. Partial Differential Equations, 16 (1991), 1313-1335.  doi: 10.1080/03605309108820801.  Google Scholar

[21]

J. Smulevici, Small data solutions of the Vlasov-Poisson system and the vector field method, Ann. PDE, 2 (2016), Art. 11, 55 pp. doi: 10.1007/s40818-016-0016-2.  Google Scholar

[22]

X. Wang, Decay estimates for the $3d$ relativistic and non-relativistic vlasov-poisson systems, 2018. Google Scholar

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