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Global existence theorem for a model governing the motion of two cell populations
Derivative estimates for screened Vlasov-Poisson system around Penrose-stable equilibria
Department of Mathematics, Pennsylvania State University, State College, PA 16802, USA |
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 [
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. |
[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. |
[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. |
[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. |
[5] |
J. Bedrossian, N. 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. |
[6] |
T. J. M. Boyd and J. J. Sanderson, The Physics of Plasmas, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511755750.![]() ![]() ![]() |
[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. |
[8] |
S.-H. Choi, S.-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. |
[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. |
[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. |
[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. |
[12] |
D. Han-Kwan, T. 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. |
[13] |
D. Han-Kwan, T. T. Nguyen and F. Rousset, Asymptotic stability of equilibria for screened vlasov-poisson systems via pointwise dispersive estimates, 2019. |
[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. |
[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. |
[16] |
H. J. Hwang, A. 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. |
[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. |
[18] |
C. Mouhot and C. Villani,
On Landau damping, Acta Math., 207 (2011), 29-201.
doi: 10.1007/s11511-011-0068-9. |
[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. |
[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. |
[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. |
[22] |
X. Wang, Decay estimates for the $3d$ relativistic and non-relativistic vlasov-poisson systems, 2018. |
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. |
[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. |
[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. |
[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. |
[5] |
J. Bedrossian, N. 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. |
[6] |
T. J. M. Boyd and J. J. Sanderson, The Physics of Plasmas, Cambridge University Press, Cambridge, 2003.
doi: 10.1017/CBO9780511755750.![]() ![]() ![]() |
[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. |
[8] |
S.-H. Choi, S.-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. |
[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. |
[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. |
[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. |
[12] |
D. Han-Kwan, T. 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. |
[13] |
D. Han-Kwan, T. T. Nguyen and F. Rousset, Asymptotic stability of equilibria for screened vlasov-poisson systems via pointwise dispersive estimates, 2019. |
[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. |
[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. |
[16] |
H. J. Hwang, A. 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. |
[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. |
[18] |
C. Mouhot and C. Villani,
On Landau damping, Acta Math., 207 (2011), 29-201.
doi: 10.1007/s11511-011-0068-9. |
[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. |
[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. |
[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. |
[22] |
X. Wang, Decay estimates for the $3d$ relativistic and non-relativistic vlasov-poisson systems, 2018. |
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