Sur quelques limites de la physique des particules chargées vers la (magnéto)hydrodynamique

Brenier, Yann; Mauser, Norbert J.; Puel, Marjolaine

Comptes Rendus. Mathématique, Tome 334 (2002) no. 3, pp. 239-244.

Nous discutons les connexions entre les modèles obtenus par différents « scalings » à partir du système de Dirac–Maxwell quantique-relativiste. En particulier, nous examinons des limites quasi-neutres/non-relativistes du système de Vlasov–Maxwell. Dans le cas d'un scaling où les effets relativistes sont partiellement conservés, on obtient un modèle du type magnéto-hydrodynamique (MHD), sinon on obtient les équations d'Euler des fluides incompressibles. Un point clef de notre analyse asymptotique rigoureuse est la méthode d'énergie modulée.

We discuss the connection between different scalings limits of the quantum-relativistic Dirac–Maxwell system. In particular we give rigorous results for the quasi-neutral/non-relativistic limit of the Vlasov–Maxwell system: we obtain a magneto-hydro-dynamic system when we consider the magnetic field as a non-relativistic effect and we obtain the Euler equation when we see it as a relativistic effect. A mathematical key is the modulated energy method.

[1] Asano, K.; Ukai, S. On the Vlasov–Poisson limit of the Vlasov Maxwell equation, Pattern and Waves (1986), pp. 369-383

[2] Bechouche, P.; Mauser, N.J.; Poupaud, F. (Semi)-nonrelativistic limits of the Dirac equation with external time-dependent electromagnetic field, Comm. Math. Phys., Volume 197 (1998), pp. 405-425

[3] Brenier, Y. Convergence of the Vlasov–Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, Volume 25 (2000), pp. 737-754

[4] Y. Brenier, N.J. Mauser, M. Puel, Incompressible Euler and e-MHD as scaling limits of the Vlasov–Maxwell system, to be submitted to Comm. Math. Phys. (2001)

[5] Degond, P. Local existence of solutions of the Vlasov–Maxwell equations and convergence to the Vlasov–Poisson equations for infinite light velocity, Math. Methods Appl. Sci., Volume 8 (1986), pp. 533-558

[6] Gérard, P.; Markowich, P.A.; Mauser, N.J.; Poupaud, F. Homogenization limits and wigner transforms, Comm. Pure Appl. Math., Volume 50 (1997), pp. 321-377

[7] Guo, Y.; Nakamitsu, K.; Strauss, W. Global finite-energy solutions of the Maxwell–Schrödinger system, Comm. Math. Phys., Volume 170 (1995), pp. 181-196

[8] D.D. Holm, Notes on electron MHD vs Euler-alpha, Private communication

[9] Lions, P.-L. Mathematical Topics in Fluid Mechanics. Vol. 1. Incompressible Models, Oxford Lecture in Math. Appl., Oxford University Press, 1996

[10] Lions, P.-L.; Paul, T. Sur les mesures de Wigner, Rev. Mat. Iberoamericana, Volume 9 (1993), pp. 553-618

[11] Markowich, P.A.; Mauser, N.J. The classical limit of a self-consistent quantum-Vlasov equation in 3-d, Math. Models Methods Appl. Sci., Volume 9 (1993), pp. 109-124

[12] Masmoudi, N.; Mauser, N.J. The self-consistent Pauli equation, Monatsh. Math., Volume 132 (2001), pp. 19-24

[13] Mauser, N.J. Nonrelativistic limits of the Dirac equation with non-static field: first and second order corrections, Trans. Theory Statist. Phys., Volume 29 (2000), pp. 122-137

[14] M. Puel, Convergence of the Schrödinger–Poisson system to the incompressible Euler equations (2001) (submitted)

[15] Schaeffer, J. The classical limit of the relativistic Vlasov–Maxwell system, Comm. Math. Phys., Volume 104 (1986), pp. 403-421

[16] P. Zhang, Y. Zheng, N.J. Mauser, Classical limit from Schrödinger–Poisson to Vlasov–Poisson equations for general initial data including the pure state case, in 1-d, Comm. Pure Appl. Math. (2001) (to appear)