# American Institute of Mathematical Sciences

October  2016, 9(5): 1447-1473. doi: 10.3934/dcdss.2016058

## Expansion of a singularly perturbed equation with a two-scale converging convection term

 1 Laboratoire Paul Painlevé, CNRS & Universit é de Sciences et Technologies Lille 1, Cit é Scientifique, F-59655 Villeneuve d'Ascq, France

Received  February 2015 Revised  September 2015 Published  October 2016

In many physical contexts, evolution convection equations may present some very large amplitude convective terms. As an example, in the context of magnetic confinement fusion, the distribution function that describes the plasma satisfies the Vlasov equation in which some terms are of the same order as $\epsilon^{-1}$, $\epsilon \ll 1$ being the characteristic gyrokinetic period of the particles around the magnetic lines. In this paper, we aim to present a model hierarchy for modeling the distribution function for any value of $\epsilon$ by using some two-scale convergence tools. Following Frénod & Sonnendrücker's recent work, we choose the framework of a singularly perturbed convection equation where the convective terms admit either a high amplitude part or a an oscillating part with high frequency $\epsilon^{-1} \gg 1$. In this abstract framework, we derive an expansion with respect to the small parameter $\epsilon$ and we recursively identify each term of this expansion. Finally, we apply this new model hierarchy to the context of a linear Vlasov equation in three physical contexts linked to the magnetic confinement fusion and the evolution of charged particle beams.
Citation: Alexandre Mouton. Expansion of a singularly perturbed equation with a two-scale converging convection term. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1447-1473. doi: 10.3934/dcdss.2016058
##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. [2] M. Bostan, The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime, Asymptot. Anal., 61 (2009), 91-123. [3] M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, J. Differential Equations, 249 (2010), 1620-1663. doi: 10.1016/j.jde.2010.07.010. [4] F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183, Springer, New-York, 2013. doi: 10.1007/978-1-4614-5975-0. [5] A. Brizard, Nonlinear Gyrokinetic Tokamak Physics, Ph.D thesis, Princeton University, 1990. [6] A. Brizard and T.-S. Hahm, Foundations of nonlinear gyrokinetic theory, Rev. Mod. Phys., 79 (2007), 421-468. doi: 10.1103/RevModPhys.79.421. [7] P. Degond and P.-A. Raviart, On the paraxial approximation of the stationary Vlasov-Maxwell system, Math. Model. Meth. Appl. Sci., 3 (1993), 513-562. doi: 10.1142/S0218202593000278. [8] D.-H. Dubin, J.-A. Krommes, C. Oberman and W.-W. Lee, Nonlinear gyrokinetic equations, Phys. Fluids, 26 (1983), 3524-3535. doi: 10.1063/1.864113. [9] F. Filbet and É. Sonnendrücker, Modeling and numerical simulation of space charge dominated beams in the paraxial approximation, Math. Model. Meth. Appl. Sci., 16 (2006), 763-791. doi: 10.1142/S0218202506001340. [10] E. Frénod, M. Gutnic and S. Hirstoaga, First order two-scale particle-in-cell numerical method for the Vlasov equation, ESAIM Proc., 38 (2012), 348-360. doi: 10.1051/proc/201238019. [11] E. Frnod and A. Mouton, Two-dimensional Finite Larmor Radius approximation in canonical gyrokinetic coordinates, J. Pure Appl. Math. Adv. Appl., 4 (2010), 135-169. [12] E. Frénod, P.-A. Raviart and E. Sonnendrücker, Two-scale expansion of a singularly perturbed convection equation, J. Math. Pures Appl., 80 (2001), 815-843. doi: 10.1016/S0021-7824(01)01215-6. [13] E. Frénod, F. Salvarani and E. Sonnendrücker, Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method, Math. Models Methods Appl. Sci., 19 (2009), 175-197. doi: 10.1142/S0218202509003395. [14] E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field, Asymptot. Anal., 18 (1998), 193-213. [15] E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247. doi: 10.1137/S0036141099364243. [16] F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures Appl., 78 (1999), 791-817. doi: 10.1016/S0021-7824(99)00021-5. [17] F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field in quasineutral regime, Math. Models Methods Appl. Sci., 13 (2003), 661-714. doi: 10.1142/S0218202503002647. [18] D. Han-Kwan, Effect of the polarization drift in a strongly magnetized plasma, ESAIM Math. Model. Numer. Anal., 46 (2012), 929-947. doi: 10.1051/m2an/2011068. [19] D. Han-Kwan, On the confinement of a tokamak plasma, SIAM J. Math. Anal., 42 (2010), 2337-2367. doi: 10.1137/090774574. [20] D. Han-Kwan, The three-dimensional finite Larmor radius approximation, Asymptot. Anal., 66 (2010), 9-33. [21] E. Kamke, Zue Theorie der Systeme gewühnlicher Differentialgleichungen, Acta Math., 58 (1932), 57-85. doi: 10.1007/BF02547774. [22] H. Knobloch, An existence theorem for periodic solutions of nonlinear ordinary differential equations, Michigan Math. J., 9 (1962), 303-309. [23] W.-W. Lee, Gyrokinetic approach in particle simulation, Phys. Fluids, 26 (1983), 555-562. [24] W.-W. Lee, Gyrokinetic particle simulation model, J. Comp. Phys., 72 (1987), 243-269. [25] R.-G. Littlejohn, A guiding center Hamiltonian: A new approach, J. Math. Phys., 20 (1979), 2445-2458. doi: 10.1063/1.524053. [26] A. Mouton, Approximation Multi-échelles de L'équation de Vlasov, Ph.D thesis, Universitéde Strasbourg, 2009. [27] A. Mouton, Two-scale semi-lagrangian simulation of a charged particle beam in a periodic focusing channel, Kinet. Relat. Models, 2 (2009), 251-274. doi: 10.3934/krm.2009.2.251. [28] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043. [29] K. Schmitt, Periodic Solutions of Nonlinear Differential Systems, J. Math. Anal. Appl., 40 (1972), 174-182.

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##### References:
 [1] G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. [2] M. Bostan, The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime, Asymptot. Anal., 61 (2009), 91-123. [3] M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, J. Differential Equations, 249 (2010), 1620-1663. doi: 10.1016/j.jde.2010.07.010. [4] F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models, Applied Mathematical Sciences, 183, Springer, New-York, 2013. doi: 10.1007/978-1-4614-5975-0. [5] A. Brizard, Nonlinear Gyrokinetic Tokamak Physics, Ph.D thesis, Princeton University, 1990. [6] A. Brizard and T.-S. Hahm, Foundations of nonlinear gyrokinetic theory, Rev. Mod. Phys., 79 (2007), 421-468. doi: 10.1103/RevModPhys.79.421. [7] P. Degond and P.-A. Raviart, On the paraxial approximation of the stationary Vlasov-Maxwell system, Math. Model. Meth. Appl. Sci., 3 (1993), 513-562. doi: 10.1142/S0218202593000278. [8] D.-H. Dubin, J.-A. Krommes, C. Oberman and W.-W. Lee, Nonlinear gyrokinetic equations, Phys. Fluids, 26 (1983), 3524-3535. doi: 10.1063/1.864113. [9] F. Filbet and É. Sonnendrücker, Modeling and numerical simulation of space charge dominated beams in the paraxial approximation, Math. Model. Meth. Appl. Sci., 16 (2006), 763-791. doi: 10.1142/S0218202506001340. [10] E. Frénod, M. Gutnic and S. Hirstoaga, First order two-scale particle-in-cell numerical method for the Vlasov equation, ESAIM Proc., 38 (2012), 348-360. doi: 10.1051/proc/201238019. [11] E. Frnod and A. Mouton, Two-dimensional Finite Larmor Radius approximation in canonical gyrokinetic coordinates, J. Pure Appl. Math. Adv. Appl., 4 (2010), 135-169. [12] E. Frénod, P.-A. Raviart and E. Sonnendrücker, Two-scale expansion of a singularly perturbed convection equation, J. Math. Pures Appl., 80 (2001), 815-843. doi: 10.1016/S0021-7824(01)01215-6. [13] E. Frénod, F. Salvarani and E. Sonnendrücker, Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method, Math. Models Methods Appl. Sci., 19 (2009), 175-197. doi: 10.1142/S0218202509003395. [14] E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field, Asymptot. Anal., 18 (1998), 193-213. [15] E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247. doi: 10.1137/S0036141099364243. [16] F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures Appl., 78 (1999), 791-817. doi: 10.1016/S0021-7824(99)00021-5. [17] F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field in quasineutral regime, Math. Models Methods Appl. Sci., 13 (2003), 661-714. doi: 10.1142/S0218202503002647. [18] D. Han-Kwan, Effect of the polarization drift in a strongly magnetized plasma, ESAIM Math. Model. Numer. Anal., 46 (2012), 929-947. doi: 10.1051/m2an/2011068. [19] D. Han-Kwan, On the confinement of a tokamak plasma, SIAM J. Math. Anal., 42 (2010), 2337-2367. doi: 10.1137/090774574. [20] D. Han-Kwan, The three-dimensional finite Larmor radius approximation, Asymptot. Anal., 66 (2010), 9-33. [21] E. Kamke, Zue Theorie der Systeme gewühnlicher Differentialgleichungen, Acta Math., 58 (1932), 57-85. doi: 10.1007/BF02547774. [22] H. Knobloch, An existence theorem for periodic solutions of nonlinear ordinary differential equations, Michigan Math. J., 9 (1962), 303-309. [23] W.-W. Lee, Gyrokinetic approach in particle simulation, Phys. Fluids, 26 (1983), 555-562. [24] W.-W. Lee, Gyrokinetic particle simulation model, J. Comp. Phys., 72 (1987), 243-269. [25] R.-G. Littlejohn, A guiding center Hamiltonian: A new approach, J. Math. Phys., 20 (1979), 2445-2458. doi: 10.1063/1.524053. [26] A. Mouton, Approximation Multi-échelles de L'équation de Vlasov, Ph.D thesis, Universitéde Strasbourg, 2009. [27] A. Mouton, Two-scale semi-lagrangian simulation of a charged particle beam in a periodic focusing channel, Kinet. Relat. Models, 2 (2009), 251-274. doi: 10.3934/krm.2009.2.251. [28] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623. doi: 10.1137/0520043. [29] K. Schmitt, Periodic Solutions of Nonlinear Differential Systems, J. Math. Anal. Appl., 40 (1972), 174-182.
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