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June  2022, 15(3): 403-416. doi: 10.3934/krm.2022002

Phase mixing for solutions to 1D transport equation in a confining potential

Sanchit Chaturvedi and  Jonathan Luk ,

Department of Mathematics, Stanford University, 450 Jane Stanford Way, Bldg 380, Stanford, CA 94305, USA

* Corresponding author: Jonathan Luk

Received  September 2021 Revised  December 2021 Published  June 2022 Early access  January 2022

Fund Project: The authors are supported by NSF grant DMS-2005435

Consider the linear transport equation in 1D under an external confining potential
$ \Phi $
:
$ \begin{equation*} {\partial}_t f + v {\partial}_x f - {\partial}_x \Phi {\partial}_v f = 0. \end{equation*} $
For
$ \Phi = \frac {x^2}2 + \frac { \varepsilon x^4}2 $
 (with
$ \varepsilon >0 $
 small), we prove phase mixing and quantitative decay estimates for
$ {\partial}_t \varphi : = - \Delta^{-1} \int_{ \mathbb{R}} {\partial}_t f \, \mathrm{d} v $
, with an inverse polynomial decay rate
$ O({\langle} t{\rangle}^{-2}) $
. In the proof, we develop a commuting vector field approach, suitably adapted to this setting. We will explain why we hope this is relevant for the nonlinear stability of the zero solution for the Vlasov–Poisson system in
$ 1 $
D under the external potential
$ \Phi $
.
Keywords: Vlasov equation, confining potential, phase mixing, vector field method.
Mathematics Subject Classification: Primary: 35B40, 82C40, 35Q83.
Citation: Sanchit Chaturvedi, Jonathan Luk. Phase mixing for solutions to 1D transport equation in a confining potential. Kinetic and Related Models, 2022, 15 (3) : 403-416. doi: 10.3934/krm.2022002
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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