December  2021, 14(6): 1035-1079. doi: 10.3934/krm.2021042

Uniform lifetime for classical solutions to the Hot, Magnetized, Relativistic Vlasov Maxwell system

1. 

Univ Rennes, CNRS, IRMAR - UMR 6625, F-35000 Rennes, France

2. 

Department of Mathematics and Statistics, University of Victoria, British Columbia, Canada

*Corresponding author: Dayton Preissl

Received  March 2021 Revised  August 2021 Published  December 2021 Early access  December 2021

Fund Project: Dayton Preissl and Slim Ibrahim were supported by NSERC grant (371637-2019)

This article is devoted to the kinetic description in phase space of magnetically confined plasmas. It addresses the problem of stability near equilibria of the Relativistic Vlasov Maxwell system. We work under the Glassey-Strauss compactly supported momentum assumption on the density function $ f(t,\cdot) $. Magnetically confined plasmas are characterized by the presence of a strong external magnetic field $ x \mapsto \epsilon^{-1} \mathbf{B}_e(x) $, where $ \epsilon $ is a small parameter related to the inverse gyrofrequency of electrons. In comparison, the self consistent internal electromagnetic fields $ (E,B) $ are supposed to be small. In the non-magnetized setting, local $ C^1 $-solutions do exist but do not exclude the possibility of blow up in finite time for large data. Consequently, in the strongly magnetized case, since $ \epsilon^{-1} $ is large, standard results predict that the lifetime $ T_\epsilon $ of solutions may shrink to zero when $ \epsilon $ goes to $ 0 $. In this article, through field straightening, and a time averaging procedure we show a uniform lower bound ($ 0<T<T_\epsilon $) on the lifetime of solutions and uniform Sup-Norm estimates. Furthermore, a bootstrap argument shows $ f $ remains at a distance $ \epsilon $ from the linearized system, while the internal fields can differ by order 1 for well prepared initial data.

Citation: Dayton Preissl, Christophe Cheverry, Slim Ibrahim. Uniform lifetime for classical solutions to the Hot, Magnetized, Relativistic Vlasov Maxwell system. Kinetic & Related Models, 2021, 14 (6) : 1035-1079. doi: 10.3934/krm.2021042
References:
[1]

I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hungar., 7 (1956), 81-94.  doi: 10.1007/BF02022967.  Google Scholar

[2]

M. Bostan, The Vlasov-Maxwell system with strong initial magnetic field: Guiding-center approximation, Multiscale Model. Simul., 6 (2007), 1026-1058.  doi: 10.1137/070689383.  Google Scholar

[3]

F. BouchutF. Golse and C. Pallard, Classical solutions and the Glassey-Strauss theorem for the 3D Vlasov-Maxwell system, Arch. Ration. Mech. Anal., 170 (2003), 1-15.  doi: 10.1007/s00205-003-0265-6.  Google Scholar

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C. Cheverry, Anomalous transport, J. Differential Equations, 262 (2017), 2987-3033.  doi: 10.1016/j.jde.2016.11.012.  Google Scholar

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C. Cheverry, Can one hear whistler waves?, Comm. Math. Phys., 338 (2015), 641–-703. doi: 10.1007/s00220-015-2389-6.  Google Scholar

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C. Cheverry and S. Ibrahim, The relativistic Vlasov Maxwell equations for strongly magnetized plasmas, Commun. Math. Sci., 18 (2020), 123-162.  doi: 10.4310/CMS.2020.v18.n1.a6.  Google Scholar

[7]

R. J. DiPerna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 42 (1989), 729-757.  doi: 10.1002/cpa.3160420603.  Google Scholar

[8]

L. C. Evans, Partial Differential Equations, 2$^nd$ edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[9]

I. Gallagher and L. Saint Raymond, Asymptotic results for pressureless magnet–Hydrodynamics, arXiv: math/0312021. Google Scholar

[10]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.  Google Scholar

[11]

R. T. Glassey and J. W. Schaeffer, Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data, Comm. Math. Phys., 119 (1988), 353-384.  doi: 10.1007/BF01218078.  Google Scholar

[12]

R. T. Glassey and W. A. Strauss, Absence of shocks in an initially dilute collisionless plasma, Comm. Math. Phys., 113 (1987), 191-208.   Google Scholar

[13]

R. T. Glassey and W. A. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rational Mech. Anal., 92 (1986), 59-90.  doi: 10.1007/BF00250732.  Google Scholar

[14]

F. Golse, Distributions, analyse de Fourier, équations aux dérivées partielles, Cours de l'École Polytechnique, 2012. Available from: http://www.cmls.polytechnique.fr/perso/golse/MAT431-10/POLY431.pdf. Google Scholar

[15]

S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Commun. Pure Appl. Math., 38 (1985), 321-332.  doi: 10.1002/cpa.3160380305.  Google Scholar

[16]

S. Klainerman and G. Staffilani, A new approach to study the Vlasov-Maxwell system, Commun. Pure Appl. Anal., 1 (2002), 103-125.  doi: 10.3934/cpaa.2002.1.103.  Google Scholar

[17]

J. Luk and R. M. Strain, Strichartz estimates and moment bounds for the relativistic Vlasov-Maxwell system, Arch. Ration. Mech. Anal., 219 (2016), 445-552.  doi: 10.1007/s00205-015-0899-1.  Google Scholar

[18]

D. Preissl, The Hot, Magnetized Relativistic Vlasov Maxwell System, MSc thesis, University of Victoria, 2020. Available from: https://dspace.library.uvic.ca:8443/handle/1828/12510. Google Scholar

[19]

X. Wang, Global solution of the 3D relativistic Vlasov-Maxwell system for the large radial data, preprint, arXiv: 2003.14192. Google Scholar

[20]

X. Wang, Propagation of regularity and long time behavior of the 3D massive relativistic transport equation Ⅱ: Vlasov-Maxwell system, preprint, arXiv: 1804.06566. doi: 10.1007/s00220-021-03987-2.  Google Scholar

show all references

References:
[1]

I. Bihari, A generalization of a lemma of Bellman and its application to uniqueness problems of differential equations, Acta Math. Acad. Sci. Hungar., 7 (1956), 81-94.  doi: 10.1007/BF02022967.  Google Scholar

[2]

M. Bostan, The Vlasov-Maxwell system with strong initial magnetic field: Guiding-center approximation, Multiscale Model. Simul., 6 (2007), 1026-1058.  doi: 10.1137/070689383.  Google Scholar

[3]

F. BouchutF. Golse and C. Pallard, Classical solutions and the Glassey-Strauss theorem for the 3D Vlasov-Maxwell system, Arch. Ration. Mech. Anal., 170 (2003), 1-15.  doi: 10.1007/s00205-003-0265-6.  Google Scholar

[4]

C. Cheverry, Anomalous transport, J. Differential Equations, 262 (2017), 2987-3033.  doi: 10.1016/j.jde.2016.11.012.  Google Scholar

[5]

C. Cheverry, Can one hear whistler waves?, Comm. Math. Phys., 338 (2015), 641–-703. doi: 10.1007/s00220-015-2389-6.  Google Scholar

[6]

C. Cheverry and S. Ibrahim, The relativistic Vlasov Maxwell equations for strongly magnetized plasmas, Commun. Math. Sci., 18 (2020), 123-162.  doi: 10.4310/CMS.2020.v18.n1.a6.  Google Scholar

[7]

R. J. DiPerna and P.-L. Lions, Global weak solutions of Vlasov-Maxwell systems, Comm. Pure Appl. Math., 42 (1989), 729-757.  doi: 10.1002/cpa.3160420603.  Google Scholar

[8]

L. C. Evans, Partial Differential Equations, 2$^nd$ edition, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.  Google Scholar

[9]

I. Gallagher and L. Saint Raymond, Asymptotic results for pressureless magnet–Hydrodynamics, arXiv: math/0312021. Google Scholar

[10]

R. T. Glassey, The Cauchy Problem in Kinetic Theory, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1996. doi: 10.1137/1.9781611971477.  Google Scholar

[11]

R. T. Glassey and J. W. Schaeffer, Global existence for the relativistic Vlasov-Maxwell system with nearly neutral initial data, Comm. Math. Phys., 119 (1988), 353-384.  doi: 10.1007/BF01218078.  Google Scholar

[12]

R. T. Glassey and W. A. Strauss, Absence of shocks in an initially dilute collisionless plasma, Comm. Math. Phys., 113 (1987), 191-208.   Google Scholar

[13]

R. T. Glassey and W. A. Strauss, Singularity formation in a collisionless plasma could occur only at high velocities, Arch. Rational Mech. Anal., 92 (1986), 59-90.  doi: 10.1007/BF00250732.  Google Scholar

[14]

F. Golse, Distributions, analyse de Fourier, équations aux dérivées partielles, Cours de l'École Polytechnique, 2012. Available from: http://www.cmls.polytechnique.fr/perso/golse/MAT431-10/POLY431.pdf. Google Scholar

[15]

S. Klainerman, Uniform decay estimates and the Lorentz invariance of the classical wave equation, Commun. Pure Appl. Math., 38 (1985), 321-332.  doi: 10.1002/cpa.3160380305.  Google Scholar

[16]

S. Klainerman and G. Staffilani, A new approach to study the Vlasov-Maxwell system, Commun. Pure Appl. Anal., 1 (2002), 103-125.  doi: 10.3934/cpaa.2002.1.103.  Google Scholar

[17]

J. Luk and R. M. Strain, Strichartz estimates and moment bounds for the relativistic Vlasov-Maxwell system, Arch. Ration. Mech. Anal., 219 (2016), 445-552.  doi: 10.1007/s00205-015-0899-1.  Google Scholar

[18]

D. Preissl, The Hot, Magnetized Relativistic Vlasov Maxwell System, MSc thesis, University of Victoria, 2020. Available from: https://dspace.library.uvic.ca:8443/handle/1828/12510. Google Scholar

[19]

X. Wang, Global solution of the 3D relativistic Vlasov-Maxwell system for the large radial data, preprint, arXiv: 2003.14192. Google Scholar

[20]

X. Wang, Propagation of regularity and long time behavior of the 3D massive relativistic transport equation Ⅱ: Vlasov-Maxwell system, preprint, arXiv: 1804.06566. doi: 10.1007/s00220-021-03987-2.  Google Scholar

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