# American Institute of Mathematical Sciences

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 & 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. Google Scholar [2] L. Ambrosio, M. 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. Google Scholar [3] L. Ambrosio, M. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] A. Bohun, F. 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. Google Scholar [8] A. Bohun, F. 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. Google Scholar [9] A. Bohun, F. 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. Google Scholar [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. Google Scholar [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. Google Scholar [12] S. Caprino, C. Marchioro, E. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [16] G. Crippa, M. C. Lopes Filho, E. 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. Google Scholar [17] L. Desvillettes, E. 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. Google Scholar [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. Google Scholar [19] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998. Google Scholar [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. Google Scholar [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.Google Scholar [22] S. V. Iordanskii, The Cauchy problem for the kinetic equation of plasma, Trudy Mat. Inst. Steklov., 60 (1961), 181-194. Google Scholar [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. Google Scholar [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. Google Scholar [25] C. Marchioro, E. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [30] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. Google Scholar [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. Google Scholar

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. Google Scholar [2] L. Ambrosio, M. 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. Google Scholar [3] L. Ambrosio, M. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [7] A. Bohun, F. 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. Google Scholar [8] A. Bohun, F. 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. Google Scholar [9] A. Bohun, F. 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. Google Scholar [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. Google Scholar [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. Google Scholar [12] S. Caprino, C. Marchioro, E. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [16] G. Crippa, M. C. Lopes Filho, E. 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. Google Scholar [17] L. Desvillettes, E. 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. Google Scholar [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. Google Scholar [19] L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 1998. Google Scholar [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. Google Scholar [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.Google Scholar [22] S. V. Iordanskii, The Cauchy problem for the kinetic equation of plasma, Trudy Mat. Inst. Steklov., 60 (1961), 181-194. Google Scholar [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. Google Scholar [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. Google Scholar [25] C. Marchioro, E. 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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [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. Google Scholar [30] E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970. Google Scholar [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. Google Scholar
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