September  2017, 37(9): 4637-4676. doi: 10.3934/dcds.2017200

Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach

1. 

Aix Marseille Université, CNRS, Centrale Marseille, Institut de Mathématiques de Marseille, UMR 7373, Château Gombert 39 rue F. Joliot Curie, 13453 Marseille Cedex 13, France

2. 

Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France

* Corresponding author: Mihai Bostan

Received  December 2015 Revised  March 2017 Published  June 2017

We perform the asymptotic analysis of parabolic equations with stiff transport terms. This kind of problem occurs, for example, in collisional gyrokinetic theory for tokamak plasmas, where the velocity diffusion of the collision mechanism is dominated by the velocity advection along the Laplace force corresponding to a strong magnetic field. This work appeal to the filtering techniques. Removing the fast oscillations associated to the singular transport operator, leads to a stable family of profiles. The limit profile comes by averaging with respect to the fast time variable, and still satisfies a parabolic model, whose diffusion matrix is completely characterized in terms of the original diffusion matrix and the stiff transport operator. Introducing first order correctors allows us to obtain strong convergence results, for general initial conditions (not necessarily well prepared).

Citation: Thomas Blanc, Mihai Bostan, Franck Boyer. Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4637-4676. doi: 10.3934/dcds.2017200
References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. Google Scholar

[2]

V. I. Arnold, Ordinary Differential Equations 2nd edition, Universitext, Springer-Verlag, Berlin, 2006. Google Scholar

[3]

V. I. Arnold, Mathematical Methods of Classical Mechanics Springer, 1989. doi: 10.1007/978-1-4757-2063-1. Google Scholar

[4]

A. Bensoussan, J. -L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures AMS Chelsea Publishing, Providence, RI, 2011. Google Scholar

[5]

H. BerestyckiF. Hamel and N. Nadirashvili, Elliptic eigenvalue problems with large drift and applications to non linear propagation phenomena, Comm. Math. Phys., 253 (2005), 451-480. doi: 10.1007/s00220-004-1201-9. Google Scholar

[6]

T. Blanc, M. Bostan and F. Boyer, Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach, preprint, arXiv: 1512.04099.Google Scholar

[7]

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. Google Scholar

[8]

M. Bostan and C. Caldini-Queiros, Finite Larmor radius approximation for collisional magnetic confinement. Part Ⅰ: The linear Boltzmann equation, Quart. Appl. Math., 72 (2014), 323-345. doi: 10.1090/S0033-569X-2014-01356-1. Google Scholar

[9]

M. Bostan and C. Caldini-Queiros, Finite Larmor radius approximation for collisional magnetic confinement. Part Ⅱ: The Fokker-Planck-Landau equation, Quart. Appl. Math., 72 (2014), 513-548. doi: 10.1090/S0033-569X-2014-01357-4. Google Scholar

[10]

M. Bostan, Transport of charged particles under fast oscillating magnetic fields, SIAM J. Math. Anal., 44 (2012), 1415-1447. doi: 10.1137/100797400. Google Scholar

[11]

M. Bostan, Strongly anisotropic diffusion problems; asymptotic analysis, J. Differential Equations, 256 (2014), 1043-1092. doi: 10.1016/j.jde.2013.10.008. Google Scholar

[12]

M. Bostan, Multi-scale analysis for linear first order PDEs. The finite Larmor radius regime, SIAM J. Math. Anal., 48 (2016), 2133-2188. doi: 10.1137/15M1033034. Google Scholar

[13]

S. Brahim-OtsmaneG. A. Francfort and F. Murat, Correctors for the homogenization on the wave and heat equations, J. Math. Pures Appl., 71 (1992), 197-231. Google Scholar

[14]

A. J. Brizard, Variational principle for non linear gyrokinetic Vlasov-Maxwell equations, Phys. Plasmas, 7 (2000), 4816-4822. doi: 10.1063/1.1322063. Google Scholar

[15]

A. J. Brizard, A guiding-center Fokker-Planck collision operator for nonuniform magnetic fields, Phys. Plasmas, 11 (2004), 4429-4438. doi: 10.1063/1.1780532. Google Scholar

[16]

A. J. Brizard and T. S. Hahm, Foundations of nonlinear gyrokinetic theory, Rev. Modern Phys., 79 (2007), 421-468. doi: 10.1103/RevModPhys.79.421. Google Scholar

[17]

N. CrouseillesM. Kuhn and G. Latu, Comparison of numerical solvers for anisotropic diffusion equations arising in plasma physics, J. Sc. Comput., 65 (2015), 1091-1128. doi: 10.1007/s10915-015-9999-1. Google Scholar

[18]

A.-L. Dalibard, Homogenization of linear transport equations in a stationary ergodic setting, Comm. Partial Differential Equations, 33 (2008), 881-921. doi: 10.1080/03605300701518216. Google Scholar

[19]

R. Dautray and J. -L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. 5: Evolution problems 1, Springer, 2000. doi: 10.1007/978-3-642-58090-1. Google Scholar

[20]

P. DegondF. Deluzet and C. Negulescu, An asymptotic preserving scheme for strongly anisotropic elliptic problems, Multiscale Model. Simul, 8 (2009/10), 645-666. doi: 10.1137/090754200. Google Scholar

[21]

A. FannjiangA. Kiselev and L. Ryzhik, Quenching of reaction by cellular flows, Geom. Funct. Anal., 16 (2006), 40-69. doi: 10.1007/s00039-006-0554-y. Google Scholar

[22]

A. Fannjiang and G. Papanicolaou, Convection enhanced diffusion for periodic flows, SIAM J. Appl. Math., 54 (1994), 333-408. doi: 10.1137/S0036139992236785. Google Scholar

[23]

F. FilbetC. Negulescu and C. Yang, Numerical study of a nonlinear heat equation for plasma physics, Int. J. Comput. Math., 89 (2012), 1060-1082. doi: 10.1080/00207160.2012.679732. Google Scholar

[24]

E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with strong external magnetic field, Asymptotic Anal., 18 (1998), 193-213. Google Scholar

[25]

E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247. doi: 10.1137/S0036141099364243. Google Scholar

[26]

X. Garbet, G. Dif-Pradalier, C. Nguyen, Y. Sarazin, V. Grandgirard and Ph. Ghendrih, Neoclassical equilibrium in gyrokinetic simulations Phys. Plasmas 16 (2009), 062503. doi: 10.1063/1.3153328. Google Scholar

[27]

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. Google Scholar

[28]

T. HoldingH. Hutridurga and J. Rauch, Convergence along mean flows, SIAM J. Math. Anal., 49 (2017), 222-271, arXiv:1603. doi: 10.1137/16M1068657. Google Scholar

[29]

G. IyerT. KomorowskiA. Novikow and L. Ryzhik, From homogenization to averaging in cellular flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 957-983. doi: 10.1016/j.anihpc.2013.06.003. Google Scholar

[30]

J. -L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications vol. I, Springer Berlin Heidelberg, 1972. doi: 10.1007/978-3-642-65161-8. Google Scholar

[31]

G. N'Guetseng, 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. Google Scholar

[32]

A. NovikovG. Papanicolaou and L. Ryzhik, Boundary layers for cellular flows at high Péclet numbers, Comm. Pure Appl. Math., 58 (2005), 867-922. doi: 10.1002/cpa.20058. Google Scholar

[33]

G. A. Pavliotis and A. M. Stuart, Multiscale Methods: Averaging and Homogenization, Texts in Applied Mathematics vol. 53, Springer, New York, 2008. doi: 10.1007/978-0-387-73829-1. Google Scholar

[34]

M. Reed and B. Simon, Methods of Modern Mathematical Physics vol. Ⅰ, Functional Analysis, Academic Press, 1980. Google Scholar

[35]

X. Q. Xu and M. N. Rosenbluth, Numerical simulation of ion-temperature-gradient-driven modes, Phys. Fluids B, 3 (1991), 627-643. doi: 10.1063/1.859862. Google Scholar

show all references

References:
[1]

G. Allaire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518. doi: 10.1137/0523084. Google Scholar

[2]

V. I. Arnold, Ordinary Differential Equations 2nd edition, Universitext, Springer-Verlag, Berlin, 2006. Google Scholar

[3]

V. I. Arnold, Mathematical Methods of Classical Mechanics Springer, 1989. doi: 10.1007/978-1-4757-2063-1. Google Scholar

[4]

A. Bensoussan, J. -L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures AMS Chelsea Publishing, Providence, RI, 2011. Google Scholar

[5]

H. BerestyckiF. Hamel and N. Nadirashvili, Elliptic eigenvalue problems with large drift and applications to non linear propagation phenomena, Comm. Math. Phys., 253 (2005), 451-480. doi: 10.1007/s00220-004-1201-9. Google Scholar

[6]

T. Blanc, M. Bostan and F. Boyer, Asymptotic analysis of parabolic equations with stiff transport terms by a multi-scale approach, preprint, arXiv: 1512.04099.Google Scholar

[7]

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. Google Scholar

[8]

M. Bostan and C. Caldini-Queiros, Finite Larmor radius approximation for collisional magnetic confinement. Part Ⅰ: The linear Boltzmann equation, Quart. Appl. Math., 72 (2014), 323-345. doi: 10.1090/S0033-569X-2014-01356-1. Google Scholar

[9]

M. Bostan and C. Caldini-Queiros, Finite Larmor radius approximation for collisional magnetic confinement. Part Ⅱ: The Fokker-Planck-Landau equation, Quart. Appl. Math., 72 (2014), 513-548. doi: 10.1090/S0033-569X-2014-01357-4. Google Scholar

[10]

M. Bostan, Transport of charged particles under fast oscillating magnetic fields, SIAM J. Math. Anal., 44 (2012), 1415-1447. doi: 10.1137/100797400. Google Scholar

[11]

M. Bostan, Strongly anisotropic diffusion problems; asymptotic analysis, J. Differential Equations, 256 (2014), 1043-1092. doi: 10.1016/j.jde.2013.10.008. Google Scholar

[12]

M. Bostan, Multi-scale analysis for linear first order PDEs. The finite Larmor radius regime, SIAM J. Math. Anal., 48 (2016), 2133-2188. doi: 10.1137/15M1033034. Google Scholar

[13]

S. Brahim-OtsmaneG. A. Francfort and F. Murat, Correctors for the homogenization on the wave and heat equations, J. Math. Pures Appl., 71 (1992), 197-231. Google Scholar

[14]

A. J. Brizard, Variational principle for non linear gyrokinetic Vlasov-Maxwell equations, Phys. Plasmas, 7 (2000), 4816-4822. doi: 10.1063/1.1322063. Google Scholar

[15]

A. J. Brizard, A guiding-center Fokker-Planck collision operator for nonuniform magnetic fields, Phys. Plasmas, 11 (2004), 4429-4438. doi: 10.1063/1.1780532. Google Scholar

[16]

A. J. Brizard and T. S. Hahm, Foundations of nonlinear gyrokinetic theory, Rev. Modern Phys., 79 (2007), 421-468. doi: 10.1103/RevModPhys.79.421. Google Scholar

[17]

N. CrouseillesM. Kuhn and G. Latu, Comparison of numerical solvers for anisotropic diffusion equations arising in plasma physics, J. Sc. Comput., 65 (2015), 1091-1128. doi: 10.1007/s10915-015-9999-1. Google Scholar

[18]

A.-L. Dalibard, Homogenization of linear transport equations in a stationary ergodic setting, Comm. Partial Differential Equations, 33 (2008), 881-921. doi: 10.1080/03605300701518216. Google Scholar

[19]

R. Dautray and J. -L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, vol. 5: Evolution problems 1, Springer, 2000. doi: 10.1007/978-3-642-58090-1. Google Scholar

[20]

P. DegondF. Deluzet and C. Negulescu, An asymptotic preserving scheme for strongly anisotropic elliptic problems, Multiscale Model. Simul, 8 (2009/10), 645-666. doi: 10.1137/090754200. Google Scholar

[21]

A. FannjiangA. Kiselev and L. Ryzhik, Quenching of reaction by cellular flows, Geom. Funct. Anal., 16 (2006), 40-69. doi: 10.1007/s00039-006-0554-y. Google Scholar

[22]

A. Fannjiang and G. Papanicolaou, Convection enhanced diffusion for periodic flows, SIAM J. Appl. Math., 54 (1994), 333-408. doi: 10.1137/S0036139992236785. Google Scholar

[23]

F. FilbetC. Negulescu and C. Yang, Numerical study of a nonlinear heat equation for plasma physics, Int. J. Comput. Math., 89 (2012), 1060-1082. doi: 10.1080/00207160.2012.679732. Google Scholar

[24]

E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with strong external magnetic field, Asymptotic Anal., 18 (1998), 193-213. Google Scholar

[25]

E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247. doi: 10.1137/S0036141099364243. Google Scholar

[26]

X. Garbet, G. Dif-Pradalier, C. Nguyen, Y. Sarazin, V. Grandgirard and Ph. Ghendrih, Neoclassical equilibrium in gyrokinetic simulations Phys. Plasmas 16 (2009), 062503. doi: 10.1063/1.3153328. Google Scholar

[27]

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. Google Scholar

[28]

T. HoldingH. Hutridurga and J. Rauch, Convergence along mean flows, SIAM J. Math. Anal., 49 (2017), 222-271, arXiv:1603. doi: 10.1137/16M1068657. Google Scholar

[29]

G. IyerT. KomorowskiA. Novikow and L. Ryzhik, From homogenization to averaging in cellular flows, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 957-983. doi: 10.1016/j.anihpc.2013.06.003. Google Scholar

[30]

J. -L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications vol. I, Springer Berlin Heidelberg, 1972. doi: 10.1007/978-3-642-65161-8. Google Scholar

[31]

G. N'Guetseng, 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. Google Scholar

[32]

A. NovikovG. Papanicolaou and L. Ryzhik, Boundary layers for cellular flows at high Péclet numbers, Comm. Pure Appl. Math., 58 (2005), 867-922. doi: 10.1002/cpa.20058. Google Scholar

[33]

G. A. Pavliotis and A. M. Stuart, Multiscale Methods: Averaging and Homogenization, Texts in Applied Mathematics vol. 53, Springer, New York, 2008. doi: 10.1007/978-0-387-73829-1. Google Scholar

[34]

M. Reed and B. Simon, Methods of Modern Mathematical Physics vol. Ⅰ, Functional Analysis, Academic Press, 1980. Google Scholar

[35]

X. Q. Xu and M. N. Rosenbluth, Numerical simulation of ion-temperature-gradient-driven modes, Phys. Fluids B, 3 (1991), 627-643. doi: 10.1063/1.859862. Google Scholar

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