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.
@article{SLSEDP_2012-2013____A15_0, author = {Claude Bardos}, title = {About a {Variant} of the $1d$ {Vlasov} equation, dubbed {{\textquotedblleft}Vlasov-Dirac-Benney} {Equation"}}, journal = {S\'eminaire Laurent Schwartz {\textemdash} EDP et applications}, note = {talk:15}, pages = {1--21}, publisher = {Institut des hautes \'etudes scientifiques & Centre de math\'ematiques Laurent Schwartz, \'Ecole polytechnique}, year = {2012-2013}, doi = {10.5802/slsedp.42}, language = {en}, url = {https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.42/} }
TY - JOUR AU - Claude Bardos TI - About a Variant of the $1d$ Vlasov equation, dubbed “Vlasov-Dirac-Benney Equation" JO - Séminaire Laurent Schwartz — EDP et applications N1 - talk:15 PY - 2012-2013 SP - 1 EP - 21 PB - Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique UR - https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.42/ DO - 10.5802/slsedp.42 LA - en ID - SLSEDP_2012-2013____A15_0 ER -
%0 Journal Article %A Claude Bardos %T About a Variant of the $1d$ Vlasov equation, dubbed “Vlasov-Dirac-Benney Equation" %J Séminaire Laurent Schwartz — EDP et applications %Z talk:15 %D 2012-2013 %P 1-21 %I Institut des hautes études scientifiques & Centre de mathématiques Laurent Schwartz, École polytechnique %U https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.42/ %R 10.5802/slsedp.42 %G en %F SLSEDP_2012-2013____A15_0
Claude Bardos. About a Variant of the $1d$ Vlasov equation, dubbed “Vlasov-Dirac-Benney Equation". Séminaire Laurent Schwartz — EDP et applications (2012-2013), Talk no. 15, 21 p. doi : 10.5802/slsedp.42. https://proceedings.centre-mersenne.org/articles/10.5802/slsedp.42/
[1] V. I. Arnold, On an a priori estimate in the theory of hydrodynamical stability;, Amer. Math. Soc. Transl. 79 (1969) 267–269. | Zbl
[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.
[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. | MR
[4] D. J. Benney, Some properties of long nonlinear waves, Stud. Appl. Math. 53 (1973), 45–50. | Zbl
[5] N. Besse, On the Waterbag Continuum, Arch. Rat. Mech. Anal. 199 (2011), no. 2, 453–491. | MR | Zbl
[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. | MR | Zbl
[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. | MR | Zbl
[9] Y. Brenier, Homogeneous hydrostatic flows with convex velocity profiles, Nonlinearity 12, (1999), 495-512,. | MR | Zbl
[10] J. Chazarain, Problèmes de Cauchy abstraits et applications à quelques problèmes mixtes, (French) J. Funct. Anal. 7 (1971), 386–446. | MR | Zbl
[11] C.Q. Chen, P.G. Lefloch, Existence theory for the isentropic Euler equations, Arch. Rational Mech. Anal. 166 (2003), 81–98. | MR | Zbl
[12] M.G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc., 78 (1980), 385-390 . | MR | Zbl
[13] C. Dafermos, “Hyperbolic conservation Laws in continuum Physics”, Springer-Verlag, New York, 2000. | MR | Zbl
[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. | Zbl
[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. | MR | Zbl
[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. | MR | Zbl
[18] E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time, Proc. Amer. Math. Soc. 126 (1998), no 2, 523–530. | MR | Zbl
[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. | MR | Zbl
[20] Y. Guo, I. Tice, Compressible, inviscid Rayleigh-Taylor instability, Indiana Univ. Math. J. 60 (2011), no. 2, 677–711. | MR | Zbl
[21] D. Han-Kwan, Quasineutral limit of the Vlasov-Poisson system with massless electrons Comm. Partial Differential Equations 36 (2011), no. 8, 1385–1425. | MR | Zbl
[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. | MR | Zbl
[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. | MR | Zbl
[24] T. Kato, “Perturbation Theory for Linear Operators”, Springer-Verlag, New York, 1966. | MR | Zbl
[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. | MR | Zbl
[27] P.-L. Lions, T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana 9 (1993), no. 3, 553–618. | MR | Zbl
[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. | MR | Zbl
[29] A. Majda, “Compressible fluid flow and systems of conservation laws in several space variables”, Applied Mathematical Sciences, 53. Springer-Verlag, New York, 1984. | MR | Zbl
[30] R.C. Paley, N. Wiener, “Fourier transforms in the complex plane”, AMS Vol 19. (1934). | Zbl
[31] O. Penrose, Electronic Instabilities of a non Uniform Plasma Phys. of Fluids 3 (1960), no2, 258–265. | Zbl
[32] V. M. Teshukov, On hyperbolicity of long-wave equations, Soviet Math. Dokl. 32 (1985), 469–473. | MR | Zbl
[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. | MR | Zbl
[34] V.M. Teshukov, Long waves in an eddying barotropic liquid, J. Appl. Mech. Tech. Phys. 35 (1994), 823–831. | MR | Zbl
[35] E.C. Titchmarsh, “Introduction to the theory of Fourier integrals”, Oxford, Clarendon press (1948). | Zbl
[36] K. Yosida, “Functional Analysis”, Springer -Verlag Berlin Heidelberg New York 1968. | MR | Zbl
[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. | MR | Zbl
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