About a Variant of the 1 d Vlasov equation, dubbed “Vlasov-Dirac-Benney Equation"

Claude Bardos[1]

  • [1] Laboratoire Jacques-Louis Lions Paris France

Séminaire Laurent Schwartz — EDP et applications (2012-2013)

  • Volume: 2012-2013, page 1-21
  • ISSN: 2266-0607

Abstract

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This is a report on project initiated with Anne Nouri [3], presently in progress, with the collaboration of Nicolas Besse [2] ([2] is mainly the material of this report) . It concerns a version of the Vlasov equation where the self interacting potential is replaced by a Dirac mass. Emphasis is put on the relations between the linearized version, the full non linear problem and also on natural connections with several other equations of mathematical physic.

How to cite

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Bardos, Claude. "About a Variant of the $1d$ Vlasov equation, dubbed “Vlasov-Dirac-Benney Equation"." Séminaire Laurent Schwartz — EDP et applications 2012-2013 (2012-2013): 1-21. <http://eudml.org/doc/275677>.

@article{Bardos2012-2013,
abstract = {This is a report on project initiated with Anne Nouri [3], presently in progress, with the collaboration of Nicolas Besse [2] ([2] is mainly the material of this report) . It concerns a version of the Vlasov equation where the self interacting potential is replaced by a Dirac mass. Emphasis is put on the relations between the linearized version, the full non linear problem and also on natural connections with several other equations of mathematical physic.},
affiliation = {Laboratoire Jacques-Louis Lions Paris France},
author = {Bardos, Claude},
journal = {Séminaire Laurent Schwartz — EDP et applications},
language = {eng},
pages = {1-21},
publisher = {Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique},
title = {About a Variant of the $1d$ Vlasov equation, dubbed “Vlasov-Dirac-Benney Equation"},
url = {http://eudml.org/doc/275677},
volume = {2012-2013},
year = {2012-2013},
}

TY - JOUR
AU - Bardos, Claude
TI - About a Variant of the $1d$ Vlasov equation, dubbed “Vlasov-Dirac-Benney Equation"
JO - Séminaire Laurent Schwartz — EDP et applications
PY - 2012-2013
PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique
VL - 2012-2013
SP - 1
EP - 21
AB - This is a report on project initiated with Anne Nouri [3], presently in progress, with the collaboration of Nicolas Besse [2] ([2] is mainly the material of this report) . It concerns a version of the Vlasov equation where the self interacting potential is replaced by a Dirac mass. Emphasis is put on the relations between the linearized version, the full non linear problem and also on natural connections with several other equations of mathematical physic.
LA - eng
UR - http://eudml.org/doc/275677
ER -

References

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  1. V. I. Arnold, On an a priori estimate in the theory of hydrodynamical stability;, Amer. Math. Soc. Transl. 79 (1969) 267–269. Zbl0191.56303
  2. C. Bardos and N. Besse, The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in Fluid Mechanics and Semi-classical Limits, Submitted to Kinetic Theory and Related Models. Zbl1292.35293
  3. C. Bardos and A. Nouri, A Vlasov equation with Dirac potential used in fusion plasmas, J. Math. Phys. 53 (2012), no. 11, 115621–115637. Zbl06245443MR3026566
  4. D. J. Benney, Some properties of long nonlinear waves, Stud. Appl. Math. 53 (1973), 45–50. Zbl0259.35011
  5. N. Besse, On the Waterbag Continuum, Arch. Rat. Mech. Anal. 199 (2011), no. 2, 453–491. Zbl1229.35297MR2763031
  6. N. Besse, F. Berthelin, Y. Brenier, P. Bertrand, The multi-water-bag equations for collision less kinetic modelization, Kin. Relat. Models 2 (2009), 39-90. Zbl1185.35292MR2472149
  7. Y. Brenier, Une formulation de type Vlasov-Poisson pour les equations d’Euler des fluides parfaits incompressibles, Inria report no 1070 INRIA-Rocquencourt (1989). 
  8. Y.Brenier. Convergence of the Vlasov-Poisson system to the incompressible Euler equations. Comm. Partial Differential Equations 25 (2000), 737–754. Zbl0970.35110MR1748352
  9. Y. Brenier, Homogeneous hydrostatic flows with convex velocity profiles, Nonlinearity 12, (1999), 495-512,. Zbl0984.35131MR1690189
  10. J. Chazarain, Problèmes de Cauchy abstraits et applications à quelques problèmes mixtes, (French) J. Funct. Anal. 7 (1971), 386–446. Zbl0211.12902MR276830
  11. C.Q. Chen, P.G. Lefloch, Existence theory for the isentropic Euler equations, Arch. Rational Mech. Anal. 166 (2003), 81–98. Zbl1027.76043MR1952080
  12. M.G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc., 78 (1980), 385-390 . Zbl0449.47059MR553381
  13. C. Dafermos, “Hyperbolic conservation Laws in continuum Physics”, Springer-Verlag, New York, 2000. Zbl1078.35001MR1763936
  14. M.R. Feix, F. Hohl, L.D. Staton, Nonlinear effects in plasmas, (eds. Kalman and Feix), Gordon and Breach, (1969), 3–21. 
  15. P. Gérard, Remarques sur l’analyse semi-classique de l’équation de Schrödinger non linéaire, Séminaire sur les Equations aux Dérivées Partielles, 1992-1993, Exp. No. XIII, 13 pp., Ecole Polytech., Palaiseau, 1993. Zbl0874.35111
  16. P. Gérard, P. Markowich, N. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math. 50 (1997), no 4, 323–379. Zbl0881.35099MR1438151
  17. E. Grenier, Limite semi-classique de l’équation de Schrödinger non linéaire en temps petit, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no 6, 691–694. Zbl0835.35132MR1323939
  18. E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time, Proc. Amer. Math. Soc. 126 (1998), no 2, 523–530. Zbl0910.35115MR1425123
  19. E. Grenier, Limite quasineutre en dimension 1, Journées “Equations aux Dérivées Partielles” (Saint-Jean-de-Monts, 1999), Exp. No II, 8 pp., Univ. Nantes, Nantes, 1999. Zbl1008.35054MR1718954
  20. Y. Guo, I. Tice, Compressible, inviscid Rayleigh-Taylor instability, Indiana Univ. Math. J. 60 (2011), no. 2, 677–711. Zbl1248.35153MR2963789
  21. D. Han-Kwan, Quasineutral limit of the Vlasov-Poisson system with massless electrons Comm. Partial Differential Equations 36 (2011), no. 8, 1385–1425. Zbl1228.35251MR2825596
  22. P. E. Jabin, A. Nouri, Analytic solutions to a strongly nonlinear Vlasov, C. R. Math. Acad. Sci. Paris 349 (2011), no. 9-10, 541–546. Zbl1219.35046MR2802921
  23. S. Jin, C. D., Levermore, D.W. McLaughlin, The semiclassical limit of the defocusing NLS hierarchy, Comm. Pure Appl. Math. 52 (1999), no. 5, 613–654. Zbl0935.35148MR1670048
  24. T. Kato, “Perturbation Theory for Linear Operators”, Springer-Verlag, New York, 1966. Zbl0435.47001MR203473
  25. N. Krall and A. Trivelpeice, “Principles of Plasma Physics”, International Series in Pure and Apllied Physics MacGraw-Hill Book Company New York 1973. 
  26. J.-L. Lions, Les semi groupes distributions, (French) Port. Math. 19 (1960), 141–164. Zbl0103.09001MR143045
  27. P.-L. Lions, T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana 9 (1993), no. 3, 553–618. Zbl0801.35117MR1251718
  28. G. Loeper, Uniqueness of the solution to the Vlasov-Poisson system with bounded density, J. Math. Pures Appl. (9) 86 (2006), no. 1, 68Ð79. Zbl1111.35045MR2246357
  29. A. Majda, “Compressible fluid flow and systems of conservation laws in several space variables”, Applied Mathematical Sciences, 53. Springer-Verlag, New York, 1984. Zbl0537.76001MR748308
  30. R.C. Paley, N. Wiener, “Fourier transforms in the complex plane”, AMS Vol 19. (1934). Zbl0011.01601
  31. O. Penrose, Electronic Instabilities of a non Uniform Plasma Phys. of Fluids 3 (1960), no2, 258–265. Zbl0090.22801
  32. V. M. Teshukov, On hyperbolicity of long-wave equations, Soviet Math. Dokl. 32 (1985), 469–473. Zbl0615.35051MR805829
  33. V.M.Teshukov, On Cauchy problem for long-wave equations In: Numerical Methods for Free Boundary Problems, ISMN 92, vol. 106, pp. 333–338. Birkhäuser, Boston, 1992. Zbl0817.35082MR1229551
  34. V.M. Teshukov, Long waves in an eddying barotropic liquid, J. Appl. Mech. Tech. Phys. 35 (1994), 823–831. Zbl0999.76520MR1363888
  35. E.C. Titchmarsh, “Introduction to the theory of Fourier integrals”, Oxford, Clarendon press (1948). Zbl0017.40404
  36. K. Yosida, “Functional Analysis”, Springer -Verlag Berlin Heidelberg New York 1968. Zbl0435.46002MR239384
  37. V. E Zakharov, Benney equations and quasiclassical approximation in the inverse problem method, (Russian) Funktsional. Anal. i Prilozhen. 14 (1980), no. 2, 15–24. Zbl0453.35086MR575201

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