December  2018, 11(6): 1277-1299. doi: 10.3934/krm.2018050

Lagrangian solutions to the Vlasov-Poisson system with a point charge

1. 

Departement Mathematik und Informatik, Universität Basel, Spiegelgasse 1, CH-4051 Basel, Switzerland

2. 

Institut für Mathematik, Universität Zürich, Winterthurerstrasse 190, CH-8057 Zürich, Switzerland

* Corresponding author: Chiara Saffirio

Received  May 2017 Revised  November 2017 Published  June 2018

We consider the Cauchy problem for the repulsive Vlasov-Poisson system in the three dimensional space, where the initial datum is the sum of a diffuse density, assumed to be bounded and integrable, and a point charge. Under some decay assumptions for the diffuse density close to the point charge, under bounds on the total energy, and assuming that the initial total diffuse charge is strictly less than one, we prove existence of global Lagrangian solutions. Our result extends the Eulerian theory of [17], proving that solutions are transported by the flow trajectories. The proof is based on the ODE theory developed in [8] in the setting of vector fields with anisotropic regularity, where some components of the gradient of the vector field is a singular integral of a measure.

Citation: Gianluca Crippa, Silvia Ligabue, Chiara Saffirio. Lagrangian solutions to the Vlasov-Poisson system with a point charge. Kinetic and Related Models, 2018, 11 (6) : 1277-1299. doi: 10.3934/krm.2018050
References:
[1]

L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math., 158 (2004), 227-260.  doi: 10.1007/s00222-004-0367-2.

[2]

L. AmbrosioM. Colombo and A. Figalli, Existence and uniqueness of maximal regular flows for non-smooth vector fields, Arch. Rational Mech. Anal., 218 (2015), 1043-1081.  doi: 10.1007/s00205-015-0875-9.

[3]

L. AmbrosioM. Colombo and A. Figalli, On the Lagrangian structure of transport equations: The Vlasov-Poisson system, Duke Math. J., 166 (2017), 3505-3568.  doi: 10.1215/00127094-2017-0032.

[4]

L. Ambrosio and G. Crippa, Continuity equations and ODE flows with non-smooth velocity, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 144 (2014), 1191-1244.  doi: 10.1017/S0308210513000085.

[5]

A. A. Arsenev, Global existence of a weak solution of Vlasov's system of equations, U. S. S. R. Comput. Math. Math. Phys., 15 (1975), 136-147,276. 

[6]

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.

[7]

A. BohunF. Bouchut and G. Crippa, Lagrangian solutions to the Vlasov-Poisson system with $L^1$ density, J. Differential Equations, 260 (2016), 3576-3597.  doi: 10.1016/j.jde.2015.10.041.

[8]

A. BohunF. Bouchut and G. Crippa, Lagrangian flows for vector fields with anisotropic regularity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1409-1429.  doi: 10.1016/j.anihpc.2015.05.005.

[9]

A. BohunF. Bouchut and G. Crippa, Lagrangian solutions to the 2D Euler system with $L^1$ vorticity and infinite energy, Nonlinear Analysis: Theory, Methods & Applications, 132 (2016), 160-172.  doi: 10.1016/j.na.2015.11.004.

[10]

F. Bouchut and G. Crippa, Lagrangian flows for vector fields with gradient given by a singular integral, J. Hyper. Differential Equations, 10 (2013), 235-282.  doi: 10.1142/S0219891613500100.

[11]

S. Caprino and C. Marchioro, On the plasma-charge model, Kinet. Relat. Models, 3 (2010), 241-254.  doi: 10.3934/krm.2010.3.241.

[12]

S. CaprinoC. MarchioroE. Miot and M. Pulvirenti, On the 2D attractive plasma-charge model, Comm. Partial Differential Equations, 37 (2012), 1237-1272.  doi: 10.1080/03605302.2011.653032.

[13]

F. Castella, Propagation of space moments in the Vlasov-Poisson Equation and further results, Ann. Inst. Henri Poincaré, 16 (1999), 503-533.  doi: 10.1016/S0294-1449(99)80026-2.

[14]

Z. Chen and X. Zhang, Sub-linear estimate of large velocity in a collisionless plasma, Commun. Math. Sciences, 12 (2014), 279-291.  doi: 10.4310/CMS.2014.v12.n2.a4.

[15]

G. Crippa and C. De Lellis, Estimates and regularity results for the DiPerna-Lions flow, J. Reine Angew. Math., 616 (2008), 15-46.  doi: 10.1515/CRELLE.2008.016.

[16]

G. CrippaM. C. Lopes FilhoE. Miot and H. J. Nussenzveig Lopes, Flows of vector fields with point singularities and the vortex-wave system, Discrete Contin. Dyn. Syst., 36 (2016), 2405-2417.  doi: 10.3934/dcds.2016.36.2405.

[17]

L. DesvillettesE. Miot and C. Saffirio, Polynomial propagation of moments and global existence for a Vlasov-Poisson system with a point charge, Ann. Inst. H. Poincaré (C) Anal. Non Linéaire, 32 (2015), 373-400.  doi: 10.1016/j.anihpc.2014.01.001.

[18]

R. J. Di Perna and P.-L. Lions, Ordinary differential equations, transport equations and Sobolev spaces, Invent. Math., 98 (1989), 511-547.  doi: 10.1007/BF01393835.

[19]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.

[20]

L. Grafakos, Classical Fourier analysis, Graduate Texts in Mathematics, 249, Third Edition, Springer, New York, 2014. doi: 10.1007/978-1-4939-1194-3.

[21]

T. Holding and E. Miot, Uniqueness and stability for the Vlasov-Poisson system with spatial density in Orlicz spaces, accepted for publication in Commun. Contemp. Math., arXiv: 1703.03046v1.

[22]

S. V. Iordanskii, The Cauchy problem for the kinetic equation of plasma, Trudy Mat. Inst. Steklov., 60 (1961), 181-194. 

[23]

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.

[24]

G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl., 86 (2006), 68-79.  doi: 10.1016/j.matpur.2006.01.005.

[25]

C. MarchioroE. Miot and M. Pulvirenti, The Cauchy problem for the $3-D$ Vlasov-Poisson system with point charges, Arch. Ration. Mech. Anal., 201 (2011), 1-26.  doi: 10.1007/s00205-010-0388-5.

[26]

E. Miot, A uniqueness criterion for unbounded solutions to the Vlasov-Poisson system, Commun. Math. Phys., 345 (2016), 469-482.  doi: 10.1007/s00220-016-2707-7.

[27]

S. Okabe and T. Ukai, On classical solutions in the large in time of the two-dimensional Vlasov equation, Osaka J. Math., 15 (1978), 245-261. 

[28]

K. Pfaffelmoser, Global existence of the Vlasov-Poisson system in three dimensions for general initial data, J. Differ. Equ., 95 (1992), 281-303.  doi: 10.1016/0022-0396(92)90033-J.

[29]

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

[30]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.

[31]

S. Wollman, Global in time solution to the three-dimensional Vlasov-Poisson system, J. Math. Anal. Appl., 176 (1996), 76-91.  doi: 10.1006/jmaa.1993.1200.

show all references

References:
[1]

L. Ambrosio, Transport equation and Cauchy problem for BV vector fields, Invent. Math., 158 (2004), 227-260.  doi: 10.1007/s00222-004-0367-2.

[2]

L. AmbrosioM. Colombo and A. Figalli, Existence and uniqueness of maximal regular flows for non-smooth vector fields, Arch. Rational Mech. Anal., 218 (2015), 1043-1081.  doi: 10.1007/s00205-015-0875-9.

[3]

L. AmbrosioM. Colombo and A. Figalli, On the Lagrangian structure of transport equations: The Vlasov-Poisson system, Duke Math. J., 166 (2017), 3505-3568.  doi: 10.1215/00127094-2017-0032.

[4]

L. Ambrosio and G. Crippa, Continuity equations and ODE flows with non-smooth velocity, Proceedings of the Royal Society of Edinburgh: Section A Mathematics, 144 (2014), 1191-1244.  doi: 10.1017/S0308210513000085.

[5]

A. A. Arsenev, Global existence of a weak solution of Vlasov's system of equations, U. S. S. R. Comput. Math. Math. Phys., 15 (1975), 136-147,276. 

[6]

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.

[7]

A. BohunF. Bouchut and G. Crippa, Lagrangian solutions to the Vlasov-Poisson system with $L^1$ density, J. Differential Equations, 260 (2016), 3576-3597.  doi: 10.1016/j.jde.2015.10.041.

[8]

A. BohunF. Bouchut and G. Crippa, Lagrangian flows for vector fields with anisotropic regularity, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 1409-1429.  doi: 10.1016/j.anihpc.2015.05.005.

[9]

A. BohunF. Bouchut and G. Crippa, Lagrangian solutions to the 2D Euler system with $L^1$ vorticity and infinite energy, Nonlinear Analysis: Theory, Methods & Applications, 132 (2016), 160-172.  doi: 10.1016/j.na.2015.11.004.

[10]

F. Bouchut and G. Crippa, Lagrangian flows for vector fields with gradient given by a singular integral, J. Hyper. Differential Equations, 10 (2013), 235-282.  doi: 10.1142/S0219891613500100.

[11]

S. Caprino and C. Marchioro, On the plasma-charge model, Kinet. Relat. Models, 3 (2010), 241-254.  doi: 10.3934/krm.2010.3.241.

[12]

S. CaprinoC. MarchioroE. Miot and M. Pulvirenti, On the 2D attractive plasma-charge model, Comm. Partial Differential Equations, 37 (2012), 1237-1272.  doi: 10.1080/03605302.2011.653032.

[13]

F. Castella, Propagation of space moments in the Vlasov-Poisson Equation and further results, Ann. Inst. Henri Poincaré, 16 (1999), 503-533.  doi: 10.1016/S0294-1449(99)80026-2.

[14]

Z. Chen and X. Zhang, Sub-linear estimate of large velocity in a collisionless plasma, Commun. Math. Sciences, 12 (2014), 279-291.  doi: 10.4310/CMS.2014.v12.n2.a4.

[15]

G. Crippa and C. De Lellis, Estimates and regularity results for the DiPerna-Lions flow, J. Reine Angew. Math., 616 (2008), 15-46.  doi: 10.1515/CRELLE.2008.016.

[16]

G. CrippaM. C. Lopes FilhoE. Miot and H. J. Nussenzveig Lopes, Flows of vector fields with point singularities and the vortex-wave system, Discrete Contin. Dyn. Syst., 36 (2016), 2405-2417.  doi: 10.3934/dcds.2016.36.2405.

[17]

L. DesvillettesE. Miot and C. Saffirio, Polynomial propagation of moments and global existence for a Vlasov-Poisson system with a point charge, Ann. Inst. H. Poincaré (C) Anal. Non Linéaire, 32 (2015), 373-400.  doi: 10.1016/j.anihpc.2014.01.001.

[18]

R. J. Di Perna and P.-L. Lions, Ordinary differential equations, transport equations and Sobolev spaces, Invent. Math., 98 (1989), 511-547.  doi: 10.1007/BF01393835.

[19]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998.

[20]

L. Grafakos, Classical Fourier analysis, Graduate Texts in Mathematics, 249, Third Edition, Springer, New York, 2014. doi: 10.1007/978-1-4939-1194-3.

[21]

T. Holding and E. Miot, Uniqueness and stability for the Vlasov-Poisson system with spatial density in Orlicz spaces, accepted for publication in Commun. Contemp. Math., arXiv: 1703.03046v1.

[22]

S. V. Iordanskii, The Cauchy problem for the kinetic equation of plasma, Trudy Mat. Inst. Steklov., 60 (1961), 181-194. 

[23]

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.

[24]

G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl., 86 (2006), 68-79.  doi: 10.1016/j.matpur.2006.01.005.

[25]

C. MarchioroE. Miot and M. Pulvirenti, The Cauchy problem for the $3-D$ Vlasov-Poisson system with point charges, Arch. Ration. Mech. Anal., 201 (2011), 1-26.  doi: 10.1007/s00205-010-0388-5.

[26]

E. Miot, A uniqueness criterion for unbounded solutions to the Vlasov-Poisson system, Commun. Math. Phys., 345 (2016), 469-482.  doi: 10.1007/s00220-016-2707-7.

[27]

S. Okabe and T. Ukai, On classical solutions in the large in time of the two-dimensional Vlasov equation, Osaka J. Math., 15 (1978), 245-261. 

[28]

K. Pfaffelmoser, Global existence of the Vlasov-Poisson system in three dimensions for general initial data, J. Differ. Equ., 95 (1992), 281-303.  doi: 10.1016/0022-0396(92)90033-J.

[29]

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

[30]

E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.

[31]

S. Wollman, Global in time solution to the three-dimensional Vlasov-Poisson system, J. Math. Anal. Appl., 176 (1996), 76-91.  doi: 10.1006/jmaa.1993.1200.

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