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Recent Advances in Kinetic Equations and Applications pp 211–231Cite as

Recent Developments on Quasineutral Limits for Vlasov-Type Equations

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Part of the book series: Springer INdAM Series ((SINDAMS,volume 48))

Abstract

Kinetic equations of Vlasov type are in widespread use as models in plasma physics. A well known example is the Vlasov-Poisson system for collisionless, unmagnetised plasma. In these notes, we discuss recent progress on the quasineutral limit in which the Debye length of the plasma tends to zero, an approximation widely assumed in applications. The models formally obtained from Vlasov-Poisson systems in this limit can be seen as kinetic formulations of the Euler equations. However, rigorous results on this limit typically require a structural or strong regularity condition. Here we present recent results for a variant of the Vlasov-Poisson system, modelling ions in a regime of massless electrons. We discuss the quasineutral limit from this system to the kinetic isothermal Euler system, in a setting with rough initial data. Then, we consider the connection between the quasineutral limit and the problem of deriving these models from particle systems. We begin by presenting a recent result on the derivation of the Vlasov-Poisson system with massless electrons from a system of extended charges. Finally, we discuss a combined limit in which the kinetic isothermal Euler system is derived.

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References

  1. Baradat, A.: Nonlinear instability in Vlasov type equations around rough velocity profiles. Annales de l’Institut Henri Poincaré C, Analyse non linéaire 37(3), 489–547 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bardos, C.: About a Variant of the 1d Vlasov equation, dubbed “Vlasov-Dirac-Benney equation”. In: Séminaire Laurent Schwartz—Équations aux dérivées partielles et applications. Année 2012–2013., Sémin. Équ. Dériv. Partielles, pp. 1–21. École Polytechnique, Centre de Mathématiques, Palaiseau (2014)

    Google Scholar 

  3. Bardos, C., Besse, N.: The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits. Kinet. Relat. Models 6(4), 893–917 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bardos, C., Besse, N.: Hamiltonian structure, fluid representation and stability for the Vlasov-Dirac-Benney equation. In: Hamiltonian Partial Differential Equations and Applications. Fields Institute Communications, vol. 75, pp. 1–30. Fields Institute Research Mathematical Science, Toronto (2015)

    Google Scholar 

  5. Bardos, C., Besse, N.: Semi-classical limit of an infinite dimensional system of nonlinear Schrödinger equations. Bull. Inst. Math. Acad. Sin. (N.S.) 11(1), 43–61 (2016)

    Google Scholar 

  6. Bardos, C., Nouri, A.: A Vlasov equation with Dirac potential used in fusion plasmas. J. Math. Phys. 53(11), 115621 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bardos, C., Golse, F., Nguyen, T.T., Sentis, R.: The Maxwell-Boltzmann approximation for ion kinetic modeling. Phys. D 376/377, 94–107 (2018)

    Google Scholar 

  8. Bellan, P.M.: Fundamentals of Plasma Physics. Cambridge University, Cambridge (2008)

    Google Scholar 

  9. Berk, H.L., Nielsen, C.E., Roberts, K.V.: Phase space hydrodynamics of equivalent nonlinear systems: experimental and computational observations. Phys. Fluids 13(4), 980–995 (1970)

    Article  MATH  Google Scholar 

  10. Bonhomme, G., Pierre, T., Leclert, G., Trulsen, J.: Ion phase space vortices in ion beam-plasma systems and their relation with the ion acoustic instability: numerical and experimental results. Plasma Phys. Controlled Fusion 33(5), 507–520 (1991)

    Article  Google Scholar 

  11. Bossy, M., Fontbona, J., Jabin, P.E., Jabir, J.F.: Local existence of analytical solutions to an incompressible Lagrangian stochastic model in a periodic domain. Comm. Partial Differential Equations 38(7), 1141–1182 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  12. Bouchut, F.: Global weak solution of the Vlasov-Poisson system for small electrons mass. Comm. Partial Differential Equations 16(8–9), 1337–1365 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. Bouchut, F., Dolbeault, J.: On long time asymptotics of the Vlasov-Fokker-Planck equation and of the Vlasov-Poisson-Fokker-Planck system with Coulombic and Newtonian potentials. Differential Integral Equations 8(3), 487–514 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  14. Braun, W., Hepp, K.: The Vlasov dynamics and its fluctuations in the 1∕N limit of interacting classical particles. Comm. Math. Phys. 56(2), 101–113 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  15. Brenier, Y.: Une formulation de type Vlassov–Poisson pour les équations d’Euler des fluides parfaits incompressibles. [Rapport de recherche] RR-1070, INRIA (1989)

    Google Scholar 

  16. Brenier, Y.: Minimal geodesics on groups of volume-preserving maps and generalized solutions of the euler equations. Comm. Pure Appl. Math. 52(4), 411–452 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. Brenier, Y.: Convergence of the Vlasov–Poisson system to the incompressible Euler equations. Comm. Partial Differential Equations 25(3–4), 737–754 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Brenier, Y., Grenier, E.: Limite singulière du système de Vlasov-Poisson dans le régime de quasi neutralité: le cas indépendant du temps. C. R. Acad. Sci. Paris Sér. I Math. 318(2), 121–124 (1994)

    MathSciNet  MATH  Google Scholar 

  19. Carles, R., Nouri, A.: Monokinetic solutions to a singular Vlasov equation from a semiclassical perspective. Asymptot. Anal. 102(1–2), 99–117 (2017)

    MathSciNet  MATH  Google Scholar 

  20. Chen, F.F.: Introduction to Plasma Physics and Controlled Fusion, 3rd edn. Springer, New York (2016)

    Book  Google Scholar 

  21. Dobrushin, R.L.: Vlasov equations. Funktsional. Anal. i Prilozhen. 13(2), 48–58 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ferriere, G.: Convergence rate in Wasserstein distance and semiclassical limit for the defocusing logarithmic Schrödinger equation (2019). Preprint, arXiv:1903.04309

    Google Scholar 

  23. Golse, F.: On the dynamics of large particle systems in the mean field limit. In: Muntean, A., Rademacher, J., Zagaris, A. (eds.) Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity. Lecture Notes in Application Mathematical Mechanical, vol. 3, pp. 1–144. Springer, New York (2016)

    Google Scholar 

  24. Golse, F., Saint-Raymond, L.: The Vlasov-Poisson system with strong magnetic field in quasineutral regime. Math. Models Methods Appl. Sci. 13(5), 661–714 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  25. Grenier, E.: Defect measures of the Vlasov-Poisson system in the quasineutral regime. Comm. Partial Differential Equations 20(7–8), 1189–1215 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  26. Grenier, E.: Oscillations in quasineutral plasmas. Comm. Partial Differential Equations 21(3–4), 363–394 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  27. Grenier, E.: Limite quasineutre en dimension 1. In: Journées “Équations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1999), pp. Exp. No. II, 8. University of Nantes, Nantes (1999)

    Google Scholar 

  28. Griffin-Pickering, M., Iacobelli, M.: Global well-posedness for the Vlasov-Poisson system with massless electrons in the 3-dimensional torus. ArXiv:1810.06928

    Google Scholar 

  29. Griffin-Pickering, M., Iacobelli, M.: A mean field approach to the quasi-neutral limit for the Vlasov–Poisson equation. SIAM J. Math. Anal. 50(5), 5502–5536 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  30. Griffin-Pickering, M., Iacobelli, M.: Singular limits for plasmas with thermalised electrons. J. Math. Pures Appl. 135, 199–255 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  31. Gurevich, A.V., Pitaevsky, L.P.: Non-linear dynamics of a rarefied ionized gas. Prog. Aerosp. Sci. 16(3), 227–272 (1975)

    Article  Google Scholar 

  32. Gurevich, A., Pariiskaya, L., Pitaevskii, L.: Self-similar motion of rarefied plasma. Soviet Phys. JETP 22(2), 449–454 (1966)

    Google Scholar 

  33. Gurevich, A., Pariiskaya, L., Pitaevskii, L.: Self-similar motion of a low-density plasma II. Soviet Phys. JETP 27(3), 476–482 (1968)

    Google Scholar 

  34. Han-Kwan, D.: Quasineutral limit of the Vlasov–Poisson system with massless electrons. Comm. Partial Differential Equations 36(8), 1385–1425 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Han-Kwan, D., Hauray, M.: Stability issues in the quasineutral limit of the one-dimensional Vlasov-Poisson equation. Comm. Math. Phys. 334(2), 1101–1152 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. Han-Kwan, D., Iacobelli, M.: Quasineutral limit for Vlasov-Poisson via Wasserstein stability estimates in higher dimension. J. Differential Equations 263(1), 1–25 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  37. Han-Kwan, D., Iacobelli, M.: The quasineutral limit of the Vlasov-Poisson equation in Wasserstein metric. Commun. Math. Sci. 15(2), 481–509 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  38. Han-Kwan, D., Iacobelli, M.: From Newton’s second law to Euler’s equations of perfect fluids (2020). Preprint, arXiv:2006.14924

    Google Scholar 

  39. Han-Kwan, D., Nguyen, T.T.: Ill-posedness of the hydrostatic Euler and singular Vlasov equations. Arch. Ration. Mech. Anal. 221(3), 1317–1344 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  40. Han-Kwan, D., Rousset, F.: Quasineutral limit for Vlasov-Poisson with Penrose stable data. Ann. Sci. Éc. Norm. Supér. (4) 49(6), 1445–1495 (2016)

    Google Scholar 

  41. Hauray, M.: Mean field limit for the one dimensional Vlasov-Poisson equation. In: Séminaire Laurent Schwartz—Équations aux dérivées partielles et applications. Année 2012–2013, Exp. No. XXI, Sémin. Équ. Dériv. Partielles École Polytechnic, Palaiseau (2014)

    Google Scholar 

  42. Hauray, M., Jabin, P.E.: N-particles approximation of the Vlasov equations with singular potential. Arch. Ration. Mech. Anal. 183(3), 489–524 (2007)

    Google Scholar 

  43. Hauray, M., Jabin, P.E.: Particle approximation of Vlasov equations with singular forces: propagation of chaos. Ann. Sci. Éc. Norm. Supér. (4) 48(4), 891–940 (2015)

    Google Scholar 

  44. Herda, M.: On massless electron limit for a multispecies kinetic system with external magnetic field. J. Differential Equations 260(11), 7861–7891 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  45. Jabin, P.E.: A review of the mean field limits for Vlasov equations. Kinet. Relat. Models 7(4), 661 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  46. Jabin, P., Nouri, A.: Analytic solutions to a strongly nonlinear Vlasov equation. C.R. Acad. Sci. Paris, Sér. 1 349, 541–546 (2011)

    Google Scholar 

  47. Lazarovici, D.: The Vlasov-Poisson dynamics as the mean field limit of extended charges. Comm. Math. Phys. 347(1), 271–289 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  48. Lazarovici, D., Pickl, P.: A mean field limit for the Vlasov-Poisson system. Arch. Ration. Mech. Anal. 225(3), 1201–1231 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  49. Lions, P.L., Perthame, B.: Propagation of moments and regularity for the 3-dimensional Vlasov-Poisson system. Invent. Math. 105(2), 415–430 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  50. Loeper, G.: Uniqueness of the solution to the Vlasov-Poisson system with bounded density. J. Math. Pures Appl. (9) 86(1), 68–79 (2006)

    Google Scholar 

  51. Masmoudi, N.: From Vlasov–Poisson system to the incompressible Euler system. Comm. Partial Differential Equations 26(9–10) (2001)

    Google Scholar 

  52. Mason, R.J.: Computer simulation of ion-acoustic shocks. The diaphragm problem. Phys. Fluids 14(9), 1943–1958 (1971)

    Article  Google Scholar 

  53. Medvedev, Y.V.: Ion front in an expanding collisionless plasma. Plasma Phys. Controlled Fusion 53(12), 125007 (2011)

    Article  Google Scholar 

  54. Mouhot, C., Villani, C.: On Landau damping. Acta Math. 207(1), 29–201 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  55. Neunzert, H., Wick, J.: Die Approximation der Lösung von Integro-Differentialgleichungen durch endliche Punktmengen. In: Numerische Behandlung nichtlinearer Integrodifferential-und Differentialgleichungen. Lecture Notes in Mathematical, vol. 395, pp. 275–290. Springer, Berlin (1974)

    Google Scholar 

  56. Penrose, O.: Electrostatic Instabilities of a Uniform Non-Maxwellian Plasma. Phys. Fluids 3(2), 258–265 (1960)

    Article  MATH  Google Scholar 

  57. Pfaffelmoser, K.: Global classical solutions of the Vlasov-Poisson system in three dimensions for general initial data. J. Differential Equations 95(2), 281–303 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  58. Sakanaka, P., Chu, C., Marshall, T.: Formation of ion-acoustic collisionless shocks. Phys. Fluids 14(611) (1971)

    Google Scholar 

  59. Serfaty, S.: Mean field limit for Coulomb-type flows. Duke Math. J. 169(15), 2887–2935 (2020). Appendix with M. Duerinckx

    Google Scholar 

  60. Ukai, S., Okabe, T.: On classical solutions in the large in time of two-dimensional Vlasov’s equation. Osaka J. Math. 15(2), 245–261 (1978)

    MathSciNet  MATH  Google Scholar 

  61. Zakharov, V.E.: Benney equations and quasiclassical approximation in the inverse problem method. Funktsional. Anal. i Prilozhen. 14(2), 15–24 (1980)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematical Sciences, Durham University, Durham, UK

    Megan Griffin-Pickering

  2. ETH Zürich, Zürich, Switzerland

    Mikaela Iacobelli

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  1. Megan Griffin-Pickering
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  2. Mikaela Iacobelli
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Correspondence to Mikaela Iacobelli .

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  1. De Vinci Pôle Universitaire Research Center, Courbevoie, France

    Francesco Salvarani

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Griffin-Pickering, M., Iacobelli, M. (2021). Recent Developments on Quasineutral Limits for Vlasov-Type Equations. In: Salvarani, F. (eds) Recent Advances in Kinetic Equations and Applications. Springer INdAM Series, vol 48. Springer, Cham. https://doi.org/10.1007/978-3-030-82946-9_9

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