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Glassey-Strauss representation of Vlasov-Maxwell systems in a Half Space
Phase mixing for solutions to 1D transport equation in a confining potential
Department of Mathematics, Stanford University, 450 Jane Stanford Way, Bldg 380, Stanford, CA 94305, USA |
$ \Phi $ |
$ \begin{equation*} {\partial}_t f + v {\partial}_x f - {\partial}_x \Phi {\partial}_v f = 0. \end{equation*} $ |
$ \Phi = \frac {x^2}2 + \frac { \varepsilon x^4}2 $ |
$ \varepsilon >0 $ |
$ {\partial}_t \varphi : = - \Delta^{-1} \int_{ \mathbb{R}} {\partial}_t f \, \mathrm{d} v $ |
$ O({\langle} t{\rangle}^{-2}) $ |
$ 1 $ |
$ \Phi $ |
References:
[1] |
J. Bedrossian, Suppression of plasma echoes and Landau damping in Sobolev spaces by weak collisions in a Vlasov-Fokker-Planck equation, Ann. PDE, 3 (2017), Paper No. 19, 66 pp.
doi: 10.1007/s40818-017-0036-6. |
[2] |
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. |
[3] |
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. |
[4] |
J. Bedrossian, N. Masmoudi and C. Mouhot, Linearized wave-damping structure of Vlasov–Poisson in $\mathbb R^3$, arXiv preprint, arXiv: 2007.08580. |
[5] |
J. Bedrossian and F. Wang,
The linearized Vlasov and Vlasov-Fokker-Planck equations in a uniform magnetic field, J. Stat. Phys., 178 (2020), 552-594.
doi: 10.1007/s10955-019-02441-x. |
[6] |
L. Bigorgne, Asymptotic properties of small data solutions of the vlasov-maxwell system in high dimensions, arXiv: 1712.09698, preprint. |
[7] |
L. Bigorgne, D. Fajman, J. Joudioux, J. Smulevici and M. Thaller, Asymptotic stability of Minkowski space-time with non-compactly supported massless Vlasov matter, Arch. Ration. Mech. Anal., 242 (2021), 1–147. arXiv: 2003.03346.
doi: 10.1007/s00205-021-01639-2. |
[8] |
J. Binney and S. Tremaine, Galactic Dynamics, Princeton university press, 2011.
![]() |
[9] |
K. Carrapatoso, J. Dolbeault, F. Hérau, S. Mischler, C. Mouhot and C. Schmeiser, Special modes and hypocoercivity for linear kinetic equations with several conservation laws and a confining potential, 2021. |
[10] |
S. Chaturvedi, Stability of vacuum for the Landau equation with hard potentials, arXiv: 2001.07208, preprint. |
[11] |
S. Chaturvedi, Stability of vacuum for the Boltzmann equation with moderately soft potentials, Ann. PDE, 7 (2021), Paper No. 15,104 pp.
doi: 10.1007/s40818-021-00103-4. |
[12] |
S. Chaturvedi, J. Luk and T. T. Nguyen, The Vlasov–Poisson–Landau system in the weakly collisional regime, arXiv preprint, arXiv: 2104.05692. |
[13] |
J. Dolbeault, C. Mouhot and C. Schmeiser,
Hypocoercivity for kinetic equations with linear relaxation terms, C. R. Math. Acad. Sci. Paris, 347 (2009), 511-516.
doi: 10.1016/j.crma.2009.02.025. |
[14] |
J. Dolbeault, C. Mouhot and C. Schmeiser,
Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015), 3807-3828.
doi: 10.1090/S0002-9947-2015-06012-7. |
[15] |
P. Dominguez-Fernández, E. Jiménez-Vázquez, M. Alcubierre, E. Montoya and D. Núñez, Description of the evolution of inhomogeneities on a dark matter halo with the Vlasov equation, Gen. Relativity Gravitation, 49 (2017), Paper No. 123, 35 pp.
doi: 10.1007/s10714-017-2286-8. |
[16] |
R. Duan,
Hypocoercivity of linear degenerately dissipative kinetic equations, Nonlinearity, 24 (2011), 2165-2189.
doi: 10.1088/0951-7715/24/8/003. |
[17] |
R. Duan and W.-X. Li,
Hypocoercivity for the linear Boltzmann equation with confining forces, J. Stat. Phys., 148 (2012), 306-324.
doi: 10.1007/s10955-012-0545-3. |
[18] |
D. Fajman, J. Joudioux and J. Smulevici, The stability of the Minkowski space for the Einstein–Vlasov system, Anal. PDE, 14 (2021), 425–531. arXiv: 1707.06141.
doi: 10.2140/apde.2021.14.425. |
[19] |
D. Fajman, J. Joudioux and J. Smulevici,
A vector field method for relativistic transport equations with applications, Anal. PDE, 10 (2017), 1539-1612.
doi: 10.2140/apde.2017.10.1539. |
[20] |
E. Faou, R. Horsin and F. Rousset,
On linear damping around inhomogeneous stationary states of the Vlasov-HMF model, J. Dynam. Differential Equations, 33 (2021), 1531-1577.
doi: 10.1007/s10884-021-10044-y. |
[21] |
E. Faou and F. Rousset,
Landau damping in Sobolev spaces for the Vlasov–HMF model, Arch. Ration. Mech. Anal., 219 (2016), 887-902.
doi: 10.1007/s00205-015-0911-9. |
[22] |
R. Glassey and J. Schaeffer,
Time decay for solutions to the linearized Vlasov equation, Transport Theory Statist. Phys., 23 (1994), 411-453.
doi: 10.1080/00411459408203873. |
[23] |
R. Glassey and J. Schaeffer,
On time decay rates in Landau damping, Comm. Partial Differential Equations, 20 (1995), 647-676.
doi: 10.1080/03605309508821107. |
[24] |
E. Grenier, T. T. Nguyen and I. Rodnianski, Landau damping for analytic and Gevrey data, arXiv preprint, arXiv: 2004.05979. |
[25] |
D. Han-Kwan, T. T. Nguyen and F. Rousset, Asymptotic stability of equilibria for screened Vlasov–Poisson systems via pointwise dispersive estimates, Ann. PDE, 7 (2021), Paper No. 18, 37 pp. arXiv: 1906.05723.
doi: 10.1007/s40818-021-00110-5. |
[26] |
D. Han-Kwan, T. T. Nguyen and F. Rousset, On the linearized Vlasov-Poisson system on the whole space around stable homogeneous equilibria, Comm. Math. Phys., 387 (2021), 1405–1440. arXiv: 2007.07787.
doi: 10.1007/s00220-021-04228-2. |
[27] |
L. Landau,
On the vibrations of the electronic plasma, Acad. Sci. USSR. J. Phys., 10 (1946), 25-34.
|
[28] |
H. Lindblad and M. Taylor,
Global stability of Minkowski space for the Einstein-Vlasov system in the harmonic gauge, Arch. Ration. Mech. Anal., 235 (2020), 517-633.
doi: 10.1007/s00205-019-01425-1. |
[29] |
J. Luk, Stability of vacuum for the Landau equation with moderately soft potentials, Ann. PDE, 5 (2019), Paper No. 11,101 pp.
doi: 10.1007/s40818-019-0067-2. |
[30] |
C. Mouhot and C. Villani,
On Landau damping, Acta Math., 207 (2011), 29-201.
doi: 10.1007/s11511-011-0068-9. |
[31] |
P. Rioseco and O. Sarbach, Phase space mixing in an external gravitational central potential, Classical Quantum Gravity, 37 (2020), 195027, 42 pp.
doi: 10.1088/1361-6382/ababb3. |
[32] |
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. |
[33] |
M. Taylor, The global nonlinear stability of Minkowski space for the massless Einstein-Vlasov system, Ann. PDE, 3 (2017), Art. 9,177 pp.
doi: 10.1007/s40818-017-0026-8. |
[34] |
I. Tristani,
Landau damping for the linearized Vlasov Poisson equation in a weakly collisional regime, J. Stat. Phys., 169 (2017), 107-125.
doi: 10.1007/s10955-017-1848-1. |
[35] |
C. Villani, Landau damping, Proceedings of the International Congress of Mathematicians 2010 (ICM 2010), (2011), 640–654.
doi: 10.1142/9789814324359_0028. |
[36] |
W. W. Y. Wong,
A commuting-vector-field approach to some dispersive estimates, Arch. Math. (Basel), 110 (2018), 273-289.
doi: 10.1007/s00013-017-1114-4. |
[37] |
B. Young, Landau damping in relativistic plasmas, J. Math. Phys., 57 (2016), 021502, 68 pp.
doi: 10.1063/1.4939275. |
show all references
References:
[1] |
J. Bedrossian, Suppression of plasma echoes and Landau damping in Sobolev spaces by weak collisions in a Vlasov-Fokker-Planck equation, Ann. PDE, 3 (2017), Paper No. 19, 66 pp.
doi: 10.1007/s40818-017-0036-6. |
[2] |
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. |
[3] |
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. |
[4] |
J. Bedrossian, N. Masmoudi and C. Mouhot, Linearized wave-damping structure of Vlasov–Poisson in $\mathbb R^3$, arXiv preprint, arXiv: 2007.08580. |
[5] |
J. Bedrossian and F. Wang,
The linearized Vlasov and Vlasov-Fokker-Planck equations in a uniform magnetic field, J. Stat. Phys., 178 (2020), 552-594.
doi: 10.1007/s10955-019-02441-x. |
[6] |
L. Bigorgne, Asymptotic properties of small data solutions of the vlasov-maxwell system in high dimensions, arXiv: 1712.09698, preprint. |
[7] |
L. Bigorgne, D. Fajman, J. Joudioux, J. Smulevici and M. Thaller, Asymptotic stability of Minkowski space-time with non-compactly supported massless Vlasov matter, Arch. Ration. Mech. Anal., 242 (2021), 1–147. arXiv: 2003.03346.
doi: 10.1007/s00205-021-01639-2. |
[8] |
J. Binney and S. Tremaine, Galactic Dynamics, Princeton university press, 2011.
![]() |
[9] |
K. Carrapatoso, J. Dolbeault, F. Hérau, S. Mischler, C. Mouhot and C. Schmeiser, Special modes and hypocoercivity for linear kinetic equations with several conservation laws and a confining potential, 2021. |
[10] |
S. Chaturvedi, Stability of vacuum for the Landau equation with hard potentials, arXiv: 2001.07208, preprint. |
[11] |
S. Chaturvedi, Stability of vacuum for the Boltzmann equation with moderately soft potentials, Ann. PDE, 7 (2021), Paper No. 15,104 pp.
doi: 10.1007/s40818-021-00103-4. |
[12] |
S. Chaturvedi, J. Luk and T. T. Nguyen, The Vlasov–Poisson–Landau system in the weakly collisional regime, arXiv preprint, arXiv: 2104.05692. |
[13] |
J. Dolbeault, C. Mouhot and C. Schmeiser,
Hypocoercivity for kinetic equations with linear relaxation terms, C. R. Math. Acad. Sci. Paris, 347 (2009), 511-516.
doi: 10.1016/j.crma.2009.02.025. |
[14] |
J. Dolbeault, C. Mouhot and C. Schmeiser,
Hypocoercivity for linear kinetic equations conserving mass, Trans. Amer. Math. Soc., 367 (2015), 3807-3828.
doi: 10.1090/S0002-9947-2015-06012-7. |
[15] |
P. Dominguez-Fernández, E. Jiménez-Vázquez, M. Alcubierre, E. Montoya and D. Núñez, Description of the evolution of inhomogeneities on a dark matter halo with the Vlasov equation, Gen. Relativity Gravitation, 49 (2017), Paper No. 123, 35 pp.
doi: 10.1007/s10714-017-2286-8. |
[16] |
R. Duan,
Hypocoercivity of linear degenerately dissipative kinetic equations, Nonlinearity, 24 (2011), 2165-2189.
doi: 10.1088/0951-7715/24/8/003. |
[17] |
R. Duan and W.-X. Li,
Hypocoercivity for the linear Boltzmann equation with confining forces, J. Stat. Phys., 148 (2012), 306-324.
doi: 10.1007/s10955-012-0545-3. |
[18] |
D. Fajman, J. Joudioux and J. Smulevici, The stability of the Minkowski space for the Einstein–Vlasov system, Anal. PDE, 14 (2021), 425–531. arXiv: 1707.06141.
doi: 10.2140/apde.2021.14.425. |
[19] |
D. Fajman, J. Joudioux and J. Smulevici,
A vector field method for relativistic transport equations with applications, Anal. PDE, 10 (2017), 1539-1612.
doi: 10.2140/apde.2017.10.1539. |
[20] |
E. Faou, R. Horsin and F. Rousset,
On linear damping around inhomogeneous stationary states of the Vlasov-HMF model, J. Dynam. Differential Equations, 33 (2021), 1531-1577.
doi: 10.1007/s10884-021-10044-y. |
[21] |
E. Faou and F. Rousset,
Landau damping in Sobolev spaces for the Vlasov–HMF model, Arch. Ration. Mech. Anal., 219 (2016), 887-902.
doi: 10.1007/s00205-015-0911-9. |
[22] |
R. Glassey and J. Schaeffer,
Time decay for solutions to the linearized Vlasov equation, Transport Theory Statist. Phys., 23 (1994), 411-453.
doi: 10.1080/00411459408203873. |
[23] |
R. Glassey and J. Schaeffer,
On time decay rates in Landau damping, Comm. Partial Differential Equations, 20 (1995), 647-676.
doi: 10.1080/03605309508821107. |
[24] |
E. Grenier, T. T. Nguyen and I. Rodnianski, Landau damping for analytic and Gevrey data, arXiv preprint, arXiv: 2004.05979. |
[25] |
D. Han-Kwan, T. T. Nguyen and F. Rousset, Asymptotic stability of equilibria for screened Vlasov–Poisson systems via pointwise dispersive estimates, Ann. PDE, 7 (2021), Paper No. 18, 37 pp. arXiv: 1906.05723.
doi: 10.1007/s40818-021-00110-5. |
[26] |
D. Han-Kwan, T. T. Nguyen and F. Rousset, On the linearized Vlasov-Poisson system on the whole space around stable homogeneous equilibria, Comm. Math. Phys., 387 (2021), 1405–1440. arXiv: 2007.07787.
doi: 10.1007/s00220-021-04228-2. |
[27] |
L. Landau,
On the vibrations of the electronic plasma, Acad. Sci. USSR. J. Phys., 10 (1946), 25-34.
|
[28] |
H. Lindblad and M. Taylor,
Global stability of Minkowski space for the Einstein-Vlasov system in the harmonic gauge, Arch. Ration. Mech. Anal., 235 (2020), 517-633.
doi: 10.1007/s00205-019-01425-1. |
[29] |
J. Luk, Stability of vacuum for the Landau equation with moderately soft potentials, Ann. PDE, 5 (2019), Paper No. 11,101 pp.
doi: 10.1007/s40818-019-0067-2. |
[30] |
C. Mouhot and C. Villani,
On Landau damping, Acta Math., 207 (2011), 29-201.
doi: 10.1007/s11511-011-0068-9. |
[31] |
P. Rioseco and O. Sarbach, Phase space mixing in an external gravitational central potential, Classical Quantum Gravity, 37 (2020), 195027, 42 pp.
doi: 10.1088/1361-6382/ababb3. |
[32] |
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. |
[33] |
M. Taylor, The global nonlinear stability of Minkowski space for the massless Einstein-Vlasov system, Ann. PDE, 3 (2017), Art. 9,177 pp.
doi: 10.1007/s40818-017-0026-8. |
[34] |
I. Tristani,
Landau damping for the linearized Vlasov Poisson equation in a weakly collisional regime, J. Stat. Phys., 169 (2017), 107-125.
doi: 10.1007/s10955-017-1848-1. |
[35] |
C. Villani, Landau damping, Proceedings of the International Congress of Mathematicians 2010 (ICM 2010), (2011), 640–654.
doi: 10.1142/9789814324359_0028. |
[36] |
W. W. Y. Wong,
A commuting-vector-field approach to some dispersive estimates, Arch. Math. (Basel), 110 (2018), 273-289.
doi: 10.1007/s00013-017-1114-4. |
[37] |
B. Young, Landau damping in relativistic plasmas, J. Math. Phys., 57 (2016), 021502, 68 pp.
doi: 10.1063/1.4939275. |
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