April  2020, 13(2): 309-344. doi: 10.3934/krm.2020011

Trend to the equilibrium for the Fokker-Planck system with an external magnetic field

Université de Nantes, Laboratoire de Mathematiques Jean Leray, 2, rue de la Houssinière, BP 92208 F-44322 Nantes Cedex 3, France

Received  January 2019 Revised  September 2019 Published  January 2020

We consider the Fokker-Planck equation with an external magnetic field. Global-in-time solutions are built near the Maxwellian, the global equilibrium state for the system. Moreover, we prove the convergence to equilibrium at exponential rate. The results are first obtained on spaces with an exponential weight. Then they are extended to larger functional spaces, like certain Lebesgue spaces with polynomial weights and modified weighted Sobolev spaces, by the method of factorization and enlargement of the functional space developed in [Gualdani, Mischler, Mouhot, 2017].

Citation: Zeinab Karaki. Trend to the equilibrium for the Fokker-Planck system with an external magnetic field. Kinetic and Related Models, 2020, 13 (2) : 309-344. doi: 10.3934/krm.2020011
References:
[1]

C. Ané, D. Bakry and M. Ledoux., Sur les Inégalités de Sobolev Logarithmiques, volume 10., Société mathématique de France Paris, 2000.

[2]

D. BakryF. BartheP. CattiauxA. Guillin and et al., A simple proof of the Poincaré inequality for a large class of probability measures, Electronic Communications in Probability, 13 (2008), 60-66.  doi: 10.1214/ECP.v13-1352.

[3]

J. Bedrossian and F. Wang, The linearized Vlasov and Vlasov-Fokker-Planck equations in a uniform magnetic field, Journal of Statistical Physics, (2019), arXiv: 1805.10756. doi: 10.1007/s10955-019-02441-x.

[4]

E. Bouin, J. Dolbeault, S. Mischler, C. Mouhot and C. Schmeiser, Hypocoercivity without confinement, preprint, arXiv: 1708.06180, 2017.

[5]

D. Chafaï, Entropies, convexity, and functional inequalities: On $\phi $-entropies and $\phi $-Sobolev inequalities, Journal of Mathematics of Kyoto University, 44 (2004), 325-363.  doi: 10.1215/kjm/1250283556.

[6]

Z. Chen and X. Zhang, Cometary flow equation with a self-consistent electromagnetic field, Journal of Mathematical Physics, 53 (2012), 053301, 21pp. doi: 10.1063/1.4708622.

[7]

L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear fokker-planck equation, Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, 54 (2001), 1-42.  doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q.

[8]

J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Am. Math. Soc., 367 (2015), 3807-3828.  doi: 10.1090/S0002-9947-2015-06012-7.

[9]

G. Dujardin, F. Hérau and P. Lafitte-Godillon, Coercivity, hypocoercivity, exponential time decay and simulations for discrete Fokker- Planck equations, Numerische Mathematik, (2019). doi: 10.1007/s00211-019-01094-y.

[10]

M. P. Gualdani, S. Mischler and C. Mouhot, Factorization of non-symmetric operators and exponential $H$-theorem, Mém. Soc. Math. Fr. (N.S.), 153 (2017), 137pp.

[11]

Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Communications on Pure and Applied Mathematics, 55 (2002), 1104-1135.  doi: 10.1002/cpa.10040.

[12]

Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Inventiones Mathematicae, 153 (2003), 593-630.  doi: 10.1007/s00222-003-0301-z.

[13]

F. Hérau, Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptotic Analysis, 46 (2006), 349-359. 

[14]

F. Hérau, Introduction to hypocoercive methods and applications for simple linear inhomogeneous kinetic models, Lectures on the Analysis of Nonlinear Partial Differential Equations. Part 5, 119–147, Morningside Lect. Math., 5, Int. Press, Somerville, MA, 2018, arXiv: 1710.05588.

[15]

F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Archive for Rational Mechanics and Analysis, 171 (2004), 151-218.  doi: 10.1007/s00205-003-0276-3.

[16]

F. Hérau and L. Thomann, On global existence and trend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system with exterior confining potential, Journal of Functional Analysis, 271 (2016), 1301-1340.  doi: 10.1016/j.jfa.2016.04.030.

[17]

F. Hérau, D. Tonon and I. Tristani, Regularization estimates and cauchy theory for inhomogeneous Boltzmann equation for hard potential without cut-off, preprint, arXiv: 1710.01098, 2017.

[18]

M. Herda, On massless electron limit for a multispecies kinetic system with external magnetic field, Journal of Differential Equations, 260 (2016), 7861-7891.  doi: 10.1016/j.jde.2016.02.005.

[19]

M. Herda and L. M. Rodrigues, Anisotropic Boltzmann-Gibbs dynamics of strongly magnetized Vlasov-Fokker-Planck equations, Kinetic and Related Models, 12 (2018), 593-636.  doi: 10.3934/krm.2019024.

[20]

L. Hörmander, Hypoelliptic second order differential equations, Acta Mathematica, 119 (1967), 147-171.  doi: 10.1007/BF02392081.

[21]

L. Hormander, The analysis of linear partial differential operators Ⅲ, Springer-Verlag, Berlin, 1985.

[22]

S. Mischler and C. Mouhot, Exponential stability of slowly decaying solutions to the Kinetic-Fokker-Planck equation, Archive for Rational Mechanics and Analysis, 221 (2016), 677-723.  doi: 10.1007/s00205-016-0972-4.

[23]

C. Mouhot, Rate of convergence to equilibrium for the spatially homogeneous boltzmann equation with hard potentials, Communications in Mathematical Physics, 261 (2006), 629-672.  doi: 10.1007/s00220-005-1455-x.

[24]

C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998.  doi: 10.1088/0951-7715/19/4/011.

[25]

J. Nash, Continuity of solutions of parabolic and elliptic equations, American Journal of Mathematics, 80 (1958), 931-954.  doi: 10.2307/2372841.

[26]

F. Nier and B. Helffer, Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians, Springer-Verlag, Berlin, 2005. doi: 10.1007/b104762.

[27]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[28]

C. Villani, Hypocoercive diffusion operators, International Congress of Mathematicians, 3 (2006), 473-498. 

[29]

C. Villani, Hypocoercivity, Mem.Amer.Math.Soc, 202 (2009), ⅳ+141 pp. doi: 10.1090/S0065-9266-09-00567-5.

[30]

T. Yang and H. Yu, Global classical solutions for the Vlasov-Maxwell-Fokker-Planck system, SIAM Journal on Mathematical Analysis, 42 (2010), 459-488.  doi: 10.1137/090755796.

[31]

X. Zhang and X. Yin, Existence theorems for the initial value problem of the cometary flow equation with an external force, Journal of Mathematical Analysis and Applications, 405 (2013), 574-594.  doi: 10.1016/j.jmaa.2013.04.042.

show all references

References:
[1]

C. Ané, D. Bakry and M. Ledoux., Sur les Inégalités de Sobolev Logarithmiques, volume 10., Société mathématique de France Paris, 2000.

[2]

D. BakryF. BartheP. CattiauxA. Guillin and et al., A simple proof of the Poincaré inequality for a large class of probability measures, Electronic Communications in Probability, 13 (2008), 60-66.  doi: 10.1214/ECP.v13-1352.

[3]

J. Bedrossian and F. Wang, The linearized Vlasov and Vlasov-Fokker-Planck equations in a uniform magnetic field, Journal of Statistical Physics, (2019), arXiv: 1805.10756. doi: 10.1007/s10955-019-02441-x.

[4]

E. Bouin, J. Dolbeault, S. Mischler, C. Mouhot and C. Schmeiser, Hypocoercivity without confinement, preprint, arXiv: 1708.06180, 2017.

[5]

D. Chafaï, Entropies, convexity, and functional inequalities: On $\phi $-entropies and $\phi $-Sobolev inequalities, Journal of Mathematics of Kyoto University, 44 (2004), 325-363.  doi: 10.1215/kjm/1250283556.

[6]

Z. Chen and X. Zhang, Cometary flow equation with a self-consistent electromagnetic field, Journal of Mathematical Physics, 53 (2012), 053301, 21pp. doi: 10.1063/1.4708622.

[7]

L. Desvillettes and C. Villani, On the trend to global equilibrium in spatially inhomogeneous entropy-dissipating systems: The linear fokker-planck equation, Communications on Pure and Applied Mathematics: A Journal Issued by the Courant Institute of Mathematical Sciences, 54 (2001), 1-42.  doi: 10.1002/1097-0312(200101)54:1<1::AID-CPA1>3.0.CO;2-Q.

[8]

J. DolbeaultC. Mouhot and C. Schmeiser, Hypocoercivity for linear kinetic equations conserving mass, Am. Math. Soc., 367 (2015), 3807-3828.  doi: 10.1090/S0002-9947-2015-06012-7.

[9]

G. Dujardin, F. Hérau and P. Lafitte-Godillon, Coercivity, hypocoercivity, exponential time decay and simulations for discrete Fokker- Planck equations, Numerische Mathematik, (2019). doi: 10.1007/s00211-019-01094-y.

[10]

M. P. Gualdani, S. Mischler and C. Mouhot, Factorization of non-symmetric operators and exponential $H$-theorem, Mém. Soc. Math. Fr. (N.S.), 153 (2017), 137pp.

[11]

Y. Guo, The Vlasov-Poisson-Boltzmann system near Maxwellians, Communications on Pure and Applied Mathematics, 55 (2002), 1104-1135.  doi: 10.1002/cpa.10040.

[12]

Y. Guo, The Vlasov-Maxwell-Boltzmann system near Maxwellians, Inventiones Mathematicae, 153 (2003), 593-630.  doi: 10.1007/s00222-003-0301-z.

[13]

F. Hérau, Hypocoercivity and exponential time decay for the linear inhomogeneous relaxation Boltzmann equation, Asymptotic Analysis, 46 (2006), 349-359. 

[14]

F. Hérau, Introduction to hypocoercive methods and applications for simple linear inhomogeneous kinetic models, Lectures on the Analysis of Nonlinear Partial Differential Equations. Part 5, 119–147, Morningside Lect. Math., 5, Int. Press, Somerville, MA, 2018, arXiv: 1710.05588.

[15]

F. Hérau and F. Nier, Isotropic hypoellipticity and trend to equilibrium for the Fokker-Planck equation with a high-degree potential, Archive for Rational Mechanics and Analysis, 171 (2004), 151-218.  doi: 10.1007/s00205-003-0276-3.

[16]

F. Hérau and L. Thomann, On global existence and trend to the equilibrium for the Vlasov-Poisson-Fokker-Planck system with exterior confining potential, Journal of Functional Analysis, 271 (2016), 1301-1340.  doi: 10.1016/j.jfa.2016.04.030.

[17]

F. Hérau, D. Tonon and I. Tristani, Regularization estimates and cauchy theory for inhomogeneous Boltzmann equation for hard potential without cut-off, preprint, arXiv: 1710.01098, 2017.

[18]

M. Herda, On massless electron limit for a multispecies kinetic system with external magnetic field, Journal of Differential Equations, 260 (2016), 7861-7891.  doi: 10.1016/j.jde.2016.02.005.

[19]

M. Herda and L. M. Rodrigues, Anisotropic Boltzmann-Gibbs dynamics of strongly magnetized Vlasov-Fokker-Planck equations, Kinetic and Related Models, 12 (2018), 593-636.  doi: 10.3934/krm.2019024.

[20]

L. Hörmander, Hypoelliptic second order differential equations, Acta Mathematica, 119 (1967), 147-171.  doi: 10.1007/BF02392081.

[21]

L. Hormander, The analysis of linear partial differential operators Ⅲ, Springer-Verlag, Berlin, 1985.

[22]

S. Mischler and C. Mouhot, Exponential stability of slowly decaying solutions to the Kinetic-Fokker-Planck equation, Archive for Rational Mechanics and Analysis, 221 (2016), 677-723.  doi: 10.1007/s00205-016-0972-4.

[23]

C. Mouhot, Rate of convergence to equilibrium for the spatially homogeneous boltzmann equation with hard potentials, Communications in Mathematical Physics, 261 (2006), 629-672.  doi: 10.1007/s00220-005-1455-x.

[24]

C. Mouhot and L. Neumann, Quantitative perturbative study of convergence to equilibrium for collisional kinetic models in the torus, Nonlinearity, 19 (2006), 969-998.  doi: 10.1088/0951-7715/19/4/011.

[25]

J. Nash, Continuity of solutions of parabolic and elliptic equations, American Journal of Mathematics, 80 (1958), 931-954.  doi: 10.2307/2372841.

[26]

F. Nier and B. Helffer, Hypoelliptic Estimates and Spectral Theory for Fokker-Planck Operators and Witten Laplacians, Springer-Verlag, Berlin, 2005. doi: 10.1007/b104762.

[27]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[28]

C. Villani, Hypocoercive diffusion operators, International Congress of Mathematicians, 3 (2006), 473-498. 

[29]

C. Villani, Hypocoercivity, Mem.Amer.Math.Soc, 202 (2009), ⅳ+141 pp. doi: 10.1090/S0065-9266-09-00567-5.

[30]

T. Yang and H. Yu, Global classical solutions for the Vlasov-Maxwell-Fokker-Planck system, SIAM Journal on Mathematical Analysis, 42 (2010), 459-488.  doi: 10.1137/090755796.

[31]

X. Zhang and X. Yin, Existence theorems for the initial value problem of the cometary flow equation with an external force, Journal of Mathematical Analysis and Applications, 405 (2013), 574-594.  doi: 10.1016/j.jmaa.2013.04.042.

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