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.

[2]

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

[3]

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

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[34]

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

[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.

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[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.

[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.

[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.

[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.

[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.

[10]

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

[11]

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

[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.

[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.

[14]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[22]

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

[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.

[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.

[25]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[34]

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

[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.

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