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

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

Alexandre Mouton 1,

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.
Keywords: homogenization., Vlasov equation, convection equation, two-scale convergence, gyrokinetic approximations.
Mathematics Subject Classification: Primary: 35Q83, 76M40, 78A35, 82D10; Secondary: 76M4.
Citation: Alexandre Mouton. Expansion of a singularly perturbed equation with a two-scale converging convection term. Discrete & 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.  doi: 10.1137/0523084.  Google Scholar

[2]

M. Bostan, The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime,, Asymptot. Anal., 61 (2009), 91.   Google Scholar

[3]

M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics,, J. Differential Equations, 249 (2010), 1620.  doi: 10.1016/j.jde.2010.07.010.  Google Scholar

[4]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models,, Applied Mathematical Sciences, 183 (2013).  doi: 10.1007/978-1-4614-5975-0.  Google Scholar

[5]

A. Brizard, Nonlinear Gyrokinetic Tokamak Physics,, Ph.D thesis, (1990).   Google Scholar

[6]

A. Brizard and T.-S. Hahm, Foundations of nonlinear gyrokinetic theory,, Rev. Mod. Phys., 79 (2007), 421.  doi: 10.1103/RevModPhys.79.421.  Google Scholar

[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.  doi: 10.1142/S0218202593000278.  Google Scholar

[8]

D.-H. Dubin, J.-A. Krommes, C. Oberman and W.-W. Lee, Nonlinear gyrokinetic equations,, Phys. Fluids, 26 (1983), 3524.  doi: 10.1063/1.864113.  Google Scholar

[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.  doi: 10.1142/S0218202506001340.  Google Scholar

[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.  doi: 10.1051/proc/201238019.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/S0021-7824(01)01215-6.  Google Scholar

[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.  doi: 10.1142/S0218202509003395.  Google Scholar

[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.   Google Scholar

[15]

E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation,, SIAM J. Math. Anal., 32 (2001), 1227.  doi: 10.1137/S0036141099364243.  Google Scholar

[16]

F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field,, J. Math. Pures Appl., 78 (1999), 791.  doi: 10.1016/S0021-7824(99)00021-5.  Google Scholar

[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.  doi: 10.1142/S0218202503002647.  Google Scholar

[18]

D. Han-Kwan, Effect of the polarization drift in a strongly magnetized plasma,, ESAIM Math. Model. Numer. Anal., 46 (2012), 929.  doi: 10.1051/m2an/2011068.  Google Scholar

[19]

D. Han-Kwan, On the confinement of a tokamak plasma,, SIAM J. Math. Anal., 42 (2010), 2337.  doi: 10.1137/090774574.  Google Scholar

[20]

D. Han-Kwan, The three-dimensional finite Larmor radius approximation,, Asymptot. Anal., 66 (2010), 9.   Google Scholar

[21]

E. Kamke, Zue Theorie der Systeme gewühnlicher Differentialgleichungen,, Acta Math., 58 (1932), 57.  doi: 10.1007/BF02547774.  Google Scholar

[22]

H. Knobloch, An existence theorem for periodic solutions of nonlinear ordinary differential equations,, Michigan Math. J., 9 (1962), 303.   Google Scholar

[23]

W.-W. Lee, Gyrokinetic approach in particle simulation,, Phys. Fluids, 26 (1983), 555.   Google Scholar

[24]

W.-W. Lee, Gyrokinetic particle simulation model,, J. Comp. Phys., 72 (1987), 243.   Google Scholar

[25]

R.-G. Littlejohn, A guiding center Hamiltonian: A new approach,, J. Math. Phys., 20 (1979), 2445.  doi: 10.1063/1.524053.  Google Scholar

[26]

A. Mouton, Approximation Multi-échelles de L'équation de Vlasov,, Ph.D thesis, (2009).   Google Scholar

[27]

A. Mouton, Two-scale semi-lagrangian simulation of a charged particle beam in a periodic focusing channel,, Kinet. Relat. Models, 2 (2009), 251.  doi: 10.3934/krm.2009.2.251.  Google Scholar

[28]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM J. Math. Anal., 20 (1989), 608.  doi: 10.1137/0520043.  Google Scholar

[29]

K. Schmitt, Periodic Solutions of Nonlinear Differential Systems,, J. Math. Anal. Appl., 40 (1972), 174.   Google Scholar

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482.  doi: 10.1137/0523084.  Google Scholar

[2]

M. Bostan, The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime,, Asymptot. Anal., 61 (2009), 91.   Google Scholar

[3]

M. Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics,, J. Differential Equations, 249 (2010), 1620.  doi: 10.1016/j.jde.2010.07.010.  Google Scholar

[4]

F. Boyer and P. Fabrie, Mathematical Tools for the Study of the Incompressible Navier-Stokes Equations and Related Models,, Applied Mathematical Sciences, 183 (2013).  doi: 10.1007/978-1-4614-5975-0.  Google Scholar

[5]

A. Brizard, Nonlinear Gyrokinetic Tokamak Physics,, Ph.D thesis, (1990).   Google Scholar

[6]

A. Brizard and T.-S. Hahm, Foundations of nonlinear gyrokinetic theory,, Rev. Mod. Phys., 79 (2007), 421.  doi: 10.1103/RevModPhys.79.421.  Google Scholar

[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.  doi: 10.1142/S0218202593000278.  Google Scholar

[8]

D.-H. Dubin, J.-A. Krommes, C. Oberman and W.-W. Lee, Nonlinear gyrokinetic equations,, Phys. Fluids, 26 (1983), 3524.  doi: 10.1063/1.864113.  Google Scholar

[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.  doi: 10.1142/S0218202506001340.  Google Scholar

[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.  doi: 10.1051/proc/201238019.  Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/S0021-7824(01)01215-6.  Google Scholar

[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.  doi: 10.1142/S0218202509003395.  Google Scholar

[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.   Google Scholar

[15]

E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation,, SIAM J. Math. Anal., 32 (2001), 1227.  doi: 10.1137/S0036141099364243.  Google Scholar

[16]

F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field,, J. Math. Pures Appl., 78 (1999), 791.  doi: 10.1016/S0021-7824(99)00021-5.  Google Scholar

[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.  doi: 10.1142/S0218202503002647.  Google Scholar

[18]

D. Han-Kwan, Effect of the polarization drift in a strongly magnetized plasma,, ESAIM Math. Model. Numer. Anal., 46 (2012), 929.  doi: 10.1051/m2an/2011068.  Google Scholar

[19]

D. Han-Kwan, On the confinement of a tokamak plasma,, SIAM J. Math. Anal., 42 (2010), 2337.  doi: 10.1137/090774574.  Google Scholar

[20]

D. Han-Kwan, The three-dimensional finite Larmor radius approximation,, Asymptot. Anal., 66 (2010), 9.   Google Scholar

[21]

E. Kamke, Zue Theorie der Systeme gewühnlicher Differentialgleichungen,, Acta Math., 58 (1932), 57.  doi: 10.1007/BF02547774.  Google Scholar

[22]

H. Knobloch, An existence theorem for periodic solutions of nonlinear ordinary differential equations,, Michigan Math. J., 9 (1962), 303.   Google Scholar

[23]

W.-W. Lee, Gyrokinetic approach in particle simulation,, Phys. Fluids, 26 (1983), 555.   Google Scholar

[24]

W.-W. Lee, Gyrokinetic particle simulation model,, J. Comp. Phys., 72 (1987), 243.   Google Scholar

[25]

R.-G. Littlejohn, A guiding center Hamiltonian: A new approach,, J. Math. Phys., 20 (1979), 2445.  doi: 10.1063/1.524053.  Google Scholar

[26]

A. Mouton, Approximation Multi-échelles de L'équation de Vlasov,, Ph.D thesis, (2009).   Google Scholar

[27]

A. Mouton, Two-scale semi-lagrangian simulation of a charged particle beam in a periodic focusing channel,, Kinet. Relat. Models, 2 (2009), 251.  doi: 10.3934/krm.2009.2.251.  Google Scholar

[28]

G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM J. Math. Anal., 20 (1989), 608.  doi: 10.1137/0520043.  Google Scholar

[29]

K. Schmitt, Periodic Solutions of Nonlinear Differential Systems,, J. Math. Anal. Appl., 40 (1972), 174.   Google Scholar

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