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December  2013, 6(4): 893-917. doi: 10.3934/krm.2013.6.893

The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits

Claude Bardos 1, and  Nicolas Besse 2,

1. 

Laboratoire Jacques Louis Lions UMR CNRS 7598, Université Denis Diderot et Université Pierre et, Marie Curie 2 Place Jussieu 75005 Paris, France
2. 

Institut Jean Lamour UMR CNRS 7198, Université de Lorraine BP 70239 54506, Vandoeuvre-lès-Nancy Cedex, France

Received  August 2013 Revised  September 2013 Published  November 2013

This contribution concerns a one-dimensional version of the Vlasov equation dubbed the Vlasov$-$Dirac$-$Benney equation (in short V$-$D$-$B) where the self interacting potential is replaced by a Dirac mass. Emphasis is put on the relations between the linearized version, the full nonlinear problem and equations of fluids. In particular the connection with the so-called Benney equation leads to new stability results. Eventually the V$-$D$-$B appears to be at the ``cross road" of several problems of mathematical physics which have as far as stability is concerned very similar properties.
Keywords: Collisionless kinetic equations with Dirac potential, ill-posed and well-posed Cauchy problems., nonlinear equations, hyperbolic systems of conservation laws.
Mathematics Subject Classification: Primary: 3Q83, 75X05; Secondary: 82D1.
Citation: Claude Bardos, Nicolas Besse. The Cauchy problem for the Vlasov-Dirac-Benney equation and related issues in fluid mechanics and semi-classical limits. Kinetic & Related Models, 2013, 6 (4) : 893-917. doi: 10.3934/krm.2013.6.893
References:
[1]

V. I. Arnold, An a priori estimate in the theory of hydrodynamical stability, Izv. Vyss. Ucebn. Zaved. Matematika, 54 (1966), 3-5.  Google Scholar

[2]

C. Bardos and A. Nouri, A Vlasov equation with Dirac potential used in fusion plasmas, J. Math. Phys., 53 (2012), 115621-115637. doi: 10.1063/1.4765338.  Google Scholar

[3]

D. J. Benney, Instabilities associated with forced nonlinear waves, Stud. Appl. Math., 60 (1979), 27-41.  Google Scholar

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N. Besse, On the waterbag continuum, Arch. Rat. Mech. Anal., 199 (2011), 453-491. doi: 10.1007/s00205-010-0392-9.  Google Scholar

[5]

N. Besse, F. Berthelin, Y. Brenier and P. Bertrand, The multi-water-bag equations for collision less kinetic modelization, Kin. Relat. Models, 2 (2009), 39-80. doi: 10.3934/krm.2009.2.39.  Google Scholar

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[7]

Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 25 (2000), 737-754. doi: 10.1080/03605300008821529.  Google Scholar

[8]

Y. Brenier, Homogeneous hydrostatic flows with convex velocity profiles, Nonlinearity, 12 (1999), 495-512. doi: 10.1088/0951-7715/12/3/004.  Google Scholar

[9]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, 2011.  Google Scholar

[10]

J. Chazarain, Problèmes de Cauchy abstraits et applications à quelques problèmes mixtes, (French) J. Funct. Anal., 7 (1971), 386-446. doi: 10.1016/0022-1236(71)90027-9.  Google Scholar

[11]

C. Q. Chen and P. G. Lefloch, Compressible Euler equations with general pressure law, Arch. Rational Mech. Anal., 153 (2000), 221-259. doi: 10.1007/s002050000091.  Google Scholar

[12]

C. Q. Chen and P. G. Lefloch, Existence theory for the isentropic Euler equations, Arch. Rational Mech. Anal., 166 (2003), 81-98. doi: 10.1007/s00205-002-0229-2.  Google Scholar

[13]

M. G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc., 78 (1980), 385-390. doi: 10.1090/S0002-9939-1980-0553381-X.  Google Scholar

[14]

C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 325. Springer-Verlag, Berlin, 2000. doi: 10.1007/3-540-29089-3_14.  Google Scholar

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K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer, 2000.  Google Scholar

[16]

M. R. Feix, F. Hohl and L. D. Staton, Nonlinear effects in plasmas, (eds. Kalman and Feix), Gordon and Breach (1969), 3-21. Google Scholar

[17]

I. M. Gelfand and G. E. Shilov, Generalized Functions, Vol. 3: Theory of differential equations. Translated from the Russian by Meinhard E. Mayer Academic Press, New York-London 1967 x+222 pp.  Google Scholar

[18]

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P. Gérard, P. Markowich, N. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C.  Google Scholar

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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), 691-694.  Google Scholar

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E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time, Proc. Amer. Math. Soc., 126 (1998), 523-530. doi: 10.1090/S0002-9939-98-04164-1.  Google Scholar

[22]

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.  Google Scholar

[23]

Y. Guo and I. Tice, Compressible, inviscid Rayleigh-Taylor instability, Indiana Univ. Math. J., 60 (2011), 677-711. doi: 10.1512/iumj.2011.60.4193.  Google Scholar

[24]

D. Han-Kwan, Quasineutral limit of the Vlasov-Poisson system with massless electrons, Comm. Partial Differential Equations, 36 (2011), 1385-1425. doi: 10.1080/03605302.2011.555804.  Google Scholar

[25]

P. E. Jabin and A. Nouri, Analytic solutions to a strongly nonlinear Vlasov, C. R. Math. Acad. Sci. Paris, 349 (2011), 541-546. doi: 10.1016/j.crma.2011.03.024.  Google Scholar

[26]

S. Jin, C. D. Levermore and D. W. McLaughlin, The semiclassical limit of the defocusing NLS hierarchy, Comm. Pure Appl. Math., 52 (1999), 613-654. doi: 10.1002/(SICI)1097-0312(199905)52:5<613::AID-CPA2>3.0.CO;2-L.  Google Scholar

[27]

T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132 Springer-Verlag New York, Inc., New York 1966 xix+592 pp.  Google Scholar

[28]

N. Krall and A. Trivelpiece, Principles of Plasma Physics, International Series in Pure and Apllied Physics MacGraw-Hill Book Company, 1973. Google Scholar

[29]

J.-L. Lions, Les semi groupes distributions, (French) Port. Math., 19 (1960), 141-164.  Google Scholar

[30]

P.-L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618. doi: 10.4171/RMI/143.  Google Scholar

[31]

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

[32]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, Vol 53 Springer-Verlag, 1984. doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[33]

S. G Mikhlin and S. Prössdorf, Singular Integral Operators, Translated from the German by Albrecht Böttcher and Reinhard Lehmann. Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61631-0.  Google Scholar

[34]

R. C. Paley and N. Wiener, Fourier Transforms in the Complex Plane, AMS Vol 19, 1934. Google Scholar

[35]

J. N. Pandey, The Hilbert Transform of Schwartz Distributions and Applications, Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1996.  Google Scholar

[36]

O. Penrose, Electronic instabilities of a non uniform plasma, Phys. of Fluids, 3 (1960), 258-265. Google Scholar

[37]

M. Riesz, Sur les fonctions conjuguées, Math. Zeit., 27 (1928), 218-244. doi: 10.1007/BF01171098.  Google Scholar

[38]

E. M. Stein, Harmonic Analysis, Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993.  Google Scholar

[39]

V. M. Teshukov, On hyperbolicity of long-wave equations, Soviet Math. Dokl., 32 (1985), 469-473. Google Scholar

[40]

V. M. Teshukov, On Cauchy problem for long-wave equations, In Numerical Methods for Free Boundary Problems, Birkhäuser, Boston, 106 (1992), 331-338.  Google Scholar

[41]

V. M. Teshukov, Long waves in an eddying barotropic liquid, J. Appl. Mech. Tech. Phys., 35 (1994), 823-831. doi: 10.1007/BF02369574.  Google Scholar

[42]

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Oxford, Clarendon press, 1948. Google Scholar

[43]

K. Yosida, Functional Analysis, Springer-Verlag, 1968. Google Scholar

[44]

V. E. Zakharov, Benney equations and quasiclassical approximation in the inverse problem method, (Russian) Funktsional. Anal. i Prilozhen, 14 (1980), 15-24.  Google Scholar

show all references

References:
[1]

V. I. Arnold, An a priori estimate in the theory of hydrodynamical stability, Izv. Vyss. Ucebn. Zaved. Matematika, 54 (1966), 3-5.  Google Scholar

[2]

C. Bardos and A. Nouri, A Vlasov equation with Dirac potential used in fusion plasmas, J. Math. Phys., 53 (2012), 115621-115637. doi: 10.1063/1.4765338.  Google Scholar

[3]

D. J. Benney, Instabilities associated with forced nonlinear waves, Stud. Appl. Math., 60 (1979), 27-41.  Google Scholar

[4]

N. Besse, On the waterbag continuum, Arch. Rat. Mech. Anal., 199 (2011), 453-491. doi: 10.1007/s00205-010-0392-9.  Google Scholar

[5]

N. Besse, F. Berthelin, Y. Brenier and P. Bertrand, The multi-water-bag equations for collision less kinetic modelization, Kin. Relat. Models, 2 (2009), 39-80. doi: 10.3934/krm.2009.2.39.  Google Scholar

[6]

Y. Brenier, Une Formulation De Type Vlasov-Poisson Pour Les Équations D'Euler Des Fluides Parfaits, Incompressibles, Inria report No 1070 INRIA-Rocquencourt 1989. Google Scholar

[7]

Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 25 (2000), 737-754. doi: 10.1080/03605300008821529.  Google Scholar

[8]

Y. Brenier, Homogeneous hydrostatic flows with convex velocity profiles, Nonlinearity, 12 (1999), 495-512. doi: 10.1088/0951-7715/12/3/004.  Google Scholar

[9]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, 2011.  Google Scholar

[10]

J. Chazarain, Problèmes de Cauchy abstraits et applications à quelques problèmes mixtes, (French) J. Funct. Anal., 7 (1971), 386-446. doi: 10.1016/0022-1236(71)90027-9.  Google Scholar

[11]

C. Q. Chen and P. G. Lefloch, Compressible Euler equations with general pressure law, Arch. Rational Mech. Anal., 153 (2000), 221-259. doi: 10.1007/s002050000091.  Google Scholar

[12]

C. Q. Chen and P. G. Lefloch, Existence theory for the isentropic Euler equations, Arch. Rational Mech. Anal., 166 (2003), 81-98. doi: 10.1007/s00205-002-0229-2.  Google Scholar

[13]

M. G. Crandall and L. Tartar, Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc., 78 (1980), 385-390. doi: 10.1090/S0002-9939-1980-0553381-X.  Google Scholar

[14]

C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 325. Springer-Verlag, Berlin, 2000. doi: 10.1007/3-540-29089-3_14.  Google Scholar

[15]

K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Springer, 2000.  Google Scholar

[16]

M. R. Feix, F. Hohl and L. D. Staton, Nonlinear effects in plasmas, (eds. Kalman and Feix), Gordon and Breach (1969), 3-21. Google Scholar

[17]

I. M. Gelfand and G. E. Shilov, Generalized Functions, Vol. 3: Theory of differential equations. Translated from the Russian by Meinhard E. Mayer Academic Press, New York-London 1967 x+222 pp.  Google Scholar

[18]

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.  Google Scholar

[19]

P. Gérard, P. Markowich, N. Mauser and F. Poupaud, Homogenization limits and Wigner transforms, Comm. Pure Appl. Math., 50 (1997), 323-379. doi: 10.1002/(SICI)1097-0312(199704)50:4<323::AID-CPA4>3.0.CO;2-C.  Google Scholar

[20]

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), 691-694.  Google Scholar

[21]

E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time, Proc. Amer. Math. Soc., 126 (1998), 523-530. doi: 10.1090/S0002-9939-98-04164-1.  Google Scholar

[22]

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.  Google Scholar

[23]

Y. Guo and I. Tice, Compressible, inviscid Rayleigh-Taylor instability, Indiana Univ. Math. J., 60 (2011), 677-711. doi: 10.1512/iumj.2011.60.4193.  Google Scholar

[24]

D. Han-Kwan, Quasineutral limit of the Vlasov-Poisson system with massless electrons, Comm. Partial Differential Equations, 36 (2011), 1385-1425. doi: 10.1080/03605302.2011.555804.  Google Scholar

[25]

P. E. Jabin and A. Nouri, Analytic solutions to a strongly nonlinear Vlasov, C. R. Math. Acad. Sci. Paris, 349 (2011), 541-546. doi: 10.1016/j.crma.2011.03.024.  Google Scholar

[26]

S. Jin, C. D. Levermore and D. W. McLaughlin, The semiclassical limit of the defocusing NLS hierarchy, Comm. Pure Appl. Math., 52 (1999), 613-654. doi: 10.1002/(SICI)1097-0312(199905)52:5<613::AID-CPA2>3.0.CO;2-L.  Google Scholar

[27]

T. Kato, Perturbation Theory for Linear Operators, Die Grundlehren der mathematischen Wissenschaften, Band 132 Springer-Verlag New York, Inc., New York 1966 xix+592 pp.  Google Scholar

[28]

N. Krall and A. Trivelpiece, Principles of Plasma Physics, International Series in Pure and Apllied Physics MacGraw-Hill Book Company, 1973. Google Scholar

[29]

J.-L. Lions, Les semi groupes distributions, (French) Port. Math., 19 (1960), 141-164.  Google Scholar

[30]

P.-L. Lions and T. Paul, Sur les mesures de Wigner, Rev. Mat. Iberoamericana, 9 (1993), 553-618. doi: 10.4171/RMI/143.  Google Scholar

[31]

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

[32]

A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Applied Mathematical Sciences, Vol 53 Springer-Verlag, 1984. doi: 10.1007/978-1-4612-1116-7.  Google Scholar

[33]

S. G Mikhlin and S. Prössdorf, Singular Integral Operators, Translated from the German by Albrecht Böttcher and Reinhard Lehmann. Springer-Verlag, Berlin, 1986. doi: 10.1007/978-3-642-61631-0.  Google Scholar

[34]

R. C. Paley and N. Wiener, Fourier Transforms in the Complex Plane, AMS Vol 19, 1934. Google Scholar

[35]

J. N. Pandey, The Hilbert Transform of Schwartz Distributions and Applications, Pure and Applied Mathematics (New York). A Wiley-Interscience Publication. John Wiley & Sons, Inc., New York, 1996.  Google Scholar

[36]

O. Penrose, Electronic instabilities of a non uniform plasma, Phys. of Fluids, 3 (1960), 258-265. Google Scholar

[37]

M. Riesz, Sur les fonctions conjuguées, Math. Zeit., 27 (1928), 218-244. doi: 10.1007/BF01171098.  Google Scholar

[38]

E. M. Stein, Harmonic Analysis, Monographs in Harmonic Analysis, III. Princeton University Press, Princeton, NJ, 1993.  Google Scholar

[39]

V. M. Teshukov, On hyperbolicity of long-wave equations, Soviet Math. Dokl., 32 (1985), 469-473. Google Scholar

[40]

V. M. Teshukov, On Cauchy problem for long-wave equations, In Numerical Methods for Free Boundary Problems, Birkhäuser, Boston, 106 (1992), 331-338.  Google Scholar

[41]

V. M. Teshukov, Long waves in an eddying barotropic liquid, J. Appl. Mech. Tech. Phys., 35 (1994), 823-831. doi: 10.1007/BF02369574.  Google Scholar

[42]

E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals, Oxford, Clarendon press, 1948. Google Scholar

[43]

K. Yosida, Functional Analysis, Springer-Verlag, 1968. Google Scholar

[44]

V. E. Zakharov, Benney equations and quasiclassical approximation in the inverse problem method, (Russian) Funktsional. Anal. i Prilozhen, 14 (1980), 15-24.  Google Scholar

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