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

Claude Bardos; Nicolas Besse

doi:10.3934/krm.2013.6.893

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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

Abstract
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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. Google Scholar
[2]	C. Bardos and A. Nouri, A Vlasov equation with Dirac potential used in fusion plasmas,, J. Math. Phys., 53 (2012), 115621. doi: 10.1063/1.4765338. Google Scholar
[3]	D. J. Benney, Instabilities associated with forced nonlinear waves,, Stud. Appl. Math., 60 (1979), 27. Google Scholar
[4]	N. Besse, On the waterbag continuum,, Arch. Rat. Mech. Anal., 199* (2011), 453. 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. 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., (1070). Google Scholar
[7]	Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations,, Comm. Partial Differential Equations, 25 (2000), 737. doi: 10.1080/03605300008821529. Google Scholar
[8]	Y. Brenier, Homogeneous hydrostatic flows with convex velocity profiles,, Nonlinearity, 12 (1999), 495. doi: 10.1088/0951-7715/12/3/004. Google Scholar
[9]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext, (2011). Google Scholar
[10]	J. Chazarain, Problèmes de Cauchy abstraits et applications à quelques problèmes mixtes, (French), J. Funct. Anal., 7 (1971), 386. 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. 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. 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. 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], (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, (2000). Google Scholar
[16]	M. R. Feix, F. Hohl and L. D. Staton, Nonlinear effects in plasmas,, (eds. Kalman and Feix), (1969), 3. 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, (1967). 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, (1993), 1992. Google Scholar
[19]	P. Gérard, P. Markowich, N. Mauser and F. Poupaud, Homogenization limits and Wigner transforms,, Comm. Pure Appl. Math., 50 (1997), 323. 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. Google Scholar*
[21]	E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time,, Proc. Amer. Math. Soc., 126* (1998), 523. doi: 10.1090/S0002-9939-98-04164-1. Google Scholar*
[22]	E. Grenier, Limite Quasineutre En Dimension 1,, Journées Equations aux Dérivées Partielles, (1999). Google Scholar
[23]	Y. Guo and I. Tice, Compressible, inviscid Rayleigh-Taylor instability,, Indiana Univ. Math. J., 60 (2011), 677. 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. 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. 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. 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, (1966). 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. Google Scholar
[30]	P.-L. Lions and T. Paul, Sur les mesures de Wigner,, Rev. Mat. Iberoamericana, 9 (1993), 553. 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. 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, 53 (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, (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, 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, (1996). Google Scholar
[36]	O. Penrose, Electronic instabilities of a non uniform plasma,, Phys. of Fluids, 3 (1960), 258. Google Scholar
[37]	M. Riesz, Sur les fonctions conjuguées,, Math. Zeit., 27 (1928), 218. doi: 10.1007/BF01171098. Google Scholar
[38]	E. M. Stein, Harmonic Analysis,, Monographs in Harmonic Analysis, (1993). Google Scholar
[39]	V. M. Teshukov, On hyperbolicity of long-wave equations,, Soviet Math. Dokl., 32 (1985), 469. Google Scholar
[40]	V. M. Teshukov, On Cauchy problem for long-wave equations,, In Numerical Methods for Free Boundary Problems, 106* (1992), 331. Google Scholar*
[41]	V. M. Teshukov, Long waves in an eddying barotropic liquid,, J. Appl. Mech. Tech. Phys., 35 (1994), 823. doi: 10.1007/BF02369574. Google Scholar
[42]	E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals,, Oxford, (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. 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. Google Scholar
[2]	C. Bardos and A. Nouri, A Vlasov equation with Dirac potential used in fusion plasmas,, J. Math. Phys., 53 (2012), 115621. doi: 10.1063/1.4765338. Google Scholar
[3]	D. J. Benney, Instabilities associated with forced nonlinear waves,, Stud. Appl. Math., 60 (1979), 27. Google Scholar
[4]	N. Besse, On the waterbag continuum,, Arch. Rat. Mech. Anal., 199* (2011), 453. 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. 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., (1070). Google Scholar
[7]	Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations,, Comm. Partial Differential Equations, 25 (2000), 737. doi: 10.1080/03605300008821529. Google Scholar
[8]	Y. Brenier, Homogeneous hydrostatic flows with convex velocity profiles,, Nonlinearity, 12 (1999), 495. doi: 10.1088/0951-7715/12/3/004. Google Scholar
[9]	H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations,, Universitext, (2011). Google Scholar
[10]	J. Chazarain, Problèmes de Cauchy abstraits et applications à quelques problèmes mixtes, (French), J. Funct. Anal., 7 (1971), 386. 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. 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. 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. 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], (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, (2000). Google Scholar
[16]	M. R. Feix, F. Hohl and L. D. Staton, Nonlinear effects in plasmas,, (eds. Kalman and Feix), (1969), 3. 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, (1967). 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, (1993), 1992. Google Scholar
[19]	P. Gérard, P. Markowich, N. Mauser and F. Poupaud, Homogenization limits and Wigner transforms,, Comm. Pure Appl. Math., 50 (1997), 323. 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. Google Scholar*
[21]	E. Grenier, Semiclassical limit of the nonlinear Schrödinger equation in small time,, Proc. Amer. Math. Soc., 126* (1998), 523. doi: 10.1090/S0002-9939-98-04164-1. Google Scholar*
[22]	E. Grenier, Limite Quasineutre En Dimension 1,, Journées Equations aux Dérivées Partielles, (1999). Google Scholar
[23]	Y. Guo and I. Tice, Compressible, inviscid Rayleigh-Taylor instability,, Indiana Univ. Math. J., 60 (2011), 677. 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. 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. 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. 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, (1966). 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. Google Scholar
[30]	P.-L. Lions and T. Paul, Sur les mesures de Wigner,, Rev. Mat. Iberoamericana, 9 (1993), 553. 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. 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, 53 (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, (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, 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, (1996). Google Scholar
[36]	O. Penrose, Electronic instabilities of a non uniform plasma,, Phys. of Fluids, 3 (1960), 258. Google Scholar
[37]	M. Riesz, Sur les fonctions conjuguées,, Math. Zeit., 27 (1928), 218. doi: 10.1007/BF01171098. Google Scholar
[38]	E. M. Stein, Harmonic Analysis,, Monographs in Harmonic Analysis, (1993). Google Scholar
[39]	V. M. Teshukov, On hyperbolicity of long-wave equations,, Soviet Math. Dokl., 32 (1985), 469. Google Scholar
[40]	V. M. Teshukov, On Cauchy problem for long-wave equations,, In Numerical Methods for Free Boundary Problems, 106* (1992), 331. Google Scholar*
[41]	V. M. Teshukov, Long waves in an eddying barotropic liquid,, J. Appl. Mech. Tech. Phys., 35 (1994), 823. doi: 10.1007/BF02369574. Google Scholar
[42]	E. C. Titchmarsh, Introduction to the Theory of Fourier Integrals,, Oxford, (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. Google Scholar

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

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