The Vlasov equation with strong magnetic field and oscillating electric field as a model of isotope resonant separation

We study qualitative behavior of the Vlasov equation with strong external magnetic field and oscillating electric field. This model is relevant in order to understand isotop resonant separation. We show that the effective equation is a kinetic equation with a memory term. This memory term involves a pseudo-differential operator whose kernel is characterized by an integral equation involving Bessel functions. In some particular cases, the kernel is explicitly given.

[1]  J. Krommes,et al.  Nonlinear gyrokinetic equations , 1983 .

[2]  E. Grenier.  Oscillatory perturbations of the Navier Stokes equations , 1997 .

[3]  R. A. Silverman,et al.  Special functions and their applications , 1966 .

[4]  Emmanuel Frénod,et al.  The Finite Larmor Radius Approximation , 2001, SIAM J. Math. Anal..

[5]  L. Saint-Raymond.  THE GYROKINETIC APPROXIMATION FOR THE VLASOV–POISSON SYSTEM , 2000 .

[6]  P. Omnes,et al.  Self-consistent numerical simulation of isotope separation by selective ion cyclotron resonance heating in a magnetically confined plasma , 2001 .

[7]  Bruce I. Cohen,et al.  Orbit Averaging and Subcycling in Particle Simulation of Plasmas , 1985 .

[8]  L. Heflinger,et al.  Isotope Separation in Plasmas by Use of Ion Cyclotron Resonance , 1976 .

[9]  L. Milne‐Thomson.  A Treatise on the Theory of Bessel Functions , 1945, Nature.

[10]  Y. Amirat,et al.  Homogenisation of parametrised families of hyperbolic problems , 1992, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[11]  François Golse,et al.  THE VLASOV-POISSON SYSTEM WITH STRONG MAGNETIC FIELD , 1999 .

[12]  E. Grenier.  Pseudo-differential energy estimates of singular perturbations , 1997 .

[13]  J. Joly,et al.  Generic rigorous asymptotic expansions for weakly nonlinear multidimensional oscillatory waves , 1993 .

[14]  C. Meunier,et al.  Multiphase Averaging for Classical Systems: With Applications To Adiabatic Theorems , 1988 .

[15]  D. Serre.  Oscillations non linéaires des systèmes hyperboliques: méthodes et résultats qualitatifs , 1991 .

[16]  J. Joly,et al.  Global Solutions to Maxwell Equations in a Ferromagnetic Medium , 2000 .

[17]  G. Allaire.  Homogenization and two-scale convergence , 1992 .

[18]  Olivier Guès,et al.  Développement asymptotique de solutions exactes de systèmes hyperboliques quasilinéaires , 1993 .

[19]  Homogénéisation d’équations hyperboliques du premier ordre et application aux écoulements miscibles en milieu poreux , 1989 .

[20]  F. Golse,et al.  L'approximation centre-guide pour l'équation de Vlasov-Poisson 2D , 1998 .

[21]  P. Jabin,et al.  Large time concentrations for solutions to kinetic equations with energy dissipation , 2000 .

[22]  W. W. Lee,et al.  Gyrokinetic approach in particle simulation , 1981 .

[23]  Emmanuel Frénod.  Homogénéisation d'équations cinétiques avec potentiels oscillants , 1994 .

[24]  G. Nguetseng.  A general convergence result for a functional related to the theory of homogenization , 1989 .

[25]  A. C. Sim,et al.  Distribution Theory and Transform Analysis , 1966 .

[26]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[27]  J. Joly,et al.  Nonlinear oscillations beyond caustics , 1996 .

[28]  M. Czubak,et al.  PSEUDODIFFERENTIAL OPERATORS , 2020, Introduction to Partial Differential Equations.

[29]  D. Stern,et al.  Hamiltonian formulation of guiding center motion , 1971 .

[30]  Some results in homogenization tackling memory effects , 1997 .

[31]  Y. Brenier,et al.  convergence of the vlasov-poisson system to the incompressible euler equations , 2000 .