doi: 10.3934/krm.2021048
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Linear instability of Vlasov-Maxwell systems revisited-A Hamiltonian approach

School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA

* Corresponding author: Zhiwu Lin

This paper is dedicated to the memory of Robert Glassey

Received  July 2021 Revised  November 2021 Early access January 2022

We consider linear stability of steady states of 1$ \frac{1}{2} $ and 3DVlasov-Maxwell systems for collisionless plasmas. The linearized systems canbe written as separable Hamiltonian systems with constraints. By using ageneral theory for separable Hamiltonian systems, we recover the sharp linearstability criteria obtained previously by different approaches. Moreover, weobtain the exponential trichotomy estimates for the linearized Vlasov-Maxwellsystems in both relativistic and nonrelativistic cases.

Citation: Zhiwu Lin. Linear instability of Vlasov-Maxwell systems revisited-A Hamiltonian approach. Kinetic & Related Models, doi: 10.3934/krm.2021048
References:
[1]

J. Ben-Artzi, Instabilities in kinetic theory and their relationship to the ergodic theorem, Complex Analysis and Dynamical Systems Ⅵ. Part 1, 25–39, Contemp. Math., 653, Israel Math. Conf. Proc., Amer. Math. Soc., Providence, RI, 2015. doi: 10.1090/conm/653/13176.  Google Scholar

[2]

J. Ben-Artzi, Instability of nonmonotone magnetic equilibria of the relativistic Vlasov-Maxwell system, Nonlinearity, 24 (2011), 3353-3389.  doi: 10.1088/0951-7715/24/12/004.  Google Scholar

[3]

J. Ben-Artzi, Instability of nonsymmetric nonmonotone equilibria of the Vlasov-Maxwell system, J. Math. Phys., 52 (2011), 123703, 21 pp. doi: 10.1063/1.3670874.  Google Scholar

[4]

J. Ben-Artzi and T. Holding, Instabilities of the relativistic Vlasov-Maxwell system on unbounded domains, SIAM J. Math. Anal., 49 (2017), 4024-4063.  doi: 10.1137/15M1025396.  Google Scholar

[5]

Y. Guo and Z. Lin, Unstable and stable galaxy models, Comm. Math. Phys., 279 (2008), 789-813.  doi: 10.1007/s00220-008-0439-z.  Google Scholar

[6]

Y. Guo and W. A. Strauss, Magnetically created instability in a collisionless plasma, J. Math. Pures. Appl., 79 (2000), 975-1009.  doi: 10.1016/S0021-7824(00)01186-7.  Google Scholar

[7]

Z. Lin, Instability of periodic BGK waves, Math. Res. Lett., 8 (2001), 521-534.  doi: 10.4310/MRL.2001.v8.n4.a11.  Google Scholar

[8]

Z. Lin and W. A. Strauss, Linear stability and instability of relativistic Vlasov-Maxwell systems, Comm. Pure Appl. Math., 60 (2007), 724-787.  doi: 10.1002/cpa.20158.  Google Scholar

[9]

Z. Lin and W. Strauss, Nonlinear stability and instability of relativistic Vlasov-Maxwell systems, Comm. Pure. Appl. Math., 60 (2007), 789-837.  doi: 10.1002/cpa.20161.  Google Scholar

[10]

Z. Lin and W. A. Strauss, A sharp stability criterion for Vlasov-Maxwell systems, Invent. Math., 173 (2008), 497-546.  doi: 10.1007/s00222-008-0122-1.  Google Scholar

[11]

Z. Lin and C. Zeng, Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs, Mem. Amer. Math. Soc., 275 (2022), 1347.  doi: 10.1090/memo/1347.  Google Scholar

[12]

Z. Lin and C. Zeng, Separable Hamiltonian PDEs and Turning point principle for stability of gaseous stars, arXiv: 2005.00973, accepted by Comm. Pure. Appl. Math. doi: 10.1002/cpa.22027.  Google Scholar

[13]

T. T. Nguyen and W. A. Strauss, Linear stability analysis of a hot plasma in a solid torus, Arch. Ration. Mech. Anal., 211 (2014), 619-672.  doi: 10.1007/s00205-013-0680-2.  Google Scholar

[14]

T. T. Nguyen and W. A. Strauss, Stability analysis of collisionless plasmas with specularly reflecting boundary, SIAM J. Math. Anal., 45 (2013), 777-808.  doi: 10.1137/110859695.  Google Scholar

[15]

K. Z. Zhang, Linear stability analysis of the relativistic Vlasov-Maxwell system in an axisymmetric domain, SIAM J. Math. Anal., 51 (2019), 4683-4723.  doi: 10.1137/18M1206825.  Google Scholar

show all references

References:
[1]

J. Ben-Artzi, Instabilities in kinetic theory and their relationship to the ergodic theorem, Complex Analysis and Dynamical Systems Ⅵ. Part 1, 25–39, Contemp. Math., 653, Israel Math. Conf. Proc., Amer. Math. Soc., Providence, RI, 2015. doi: 10.1090/conm/653/13176.  Google Scholar

[2]

J. Ben-Artzi, Instability of nonmonotone magnetic equilibria of the relativistic Vlasov-Maxwell system, Nonlinearity, 24 (2011), 3353-3389.  doi: 10.1088/0951-7715/24/12/004.  Google Scholar

[3]

J. Ben-Artzi, Instability of nonsymmetric nonmonotone equilibria of the Vlasov-Maxwell system, J. Math. Phys., 52 (2011), 123703, 21 pp. doi: 10.1063/1.3670874.  Google Scholar

[4]

J. Ben-Artzi and T. Holding, Instabilities of the relativistic Vlasov-Maxwell system on unbounded domains, SIAM J. Math. Anal., 49 (2017), 4024-4063.  doi: 10.1137/15M1025396.  Google Scholar

[5]

Y. Guo and Z. Lin, Unstable and stable galaxy models, Comm. Math. Phys., 279 (2008), 789-813.  doi: 10.1007/s00220-008-0439-z.  Google Scholar

[6]

Y. Guo and W. A. Strauss, Magnetically created instability in a collisionless plasma, J. Math. Pures. Appl., 79 (2000), 975-1009.  doi: 10.1016/S0021-7824(00)01186-7.  Google Scholar

[7]

Z. Lin, Instability of periodic BGK waves, Math. Res. Lett., 8 (2001), 521-534.  doi: 10.4310/MRL.2001.v8.n4.a11.  Google Scholar

[8]

Z. Lin and W. A. Strauss, Linear stability and instability of relativistic Vlasov-Maxwell systems, Comm. Pure Appl. Math., 60 (2007), 724-787.  doi: 10.1002/cpa.20158.  Google Scholar

[9]

Z. Lin and W. Strauss, Nonlinear stability and instability of relativistic Vlasov-Maxwell systems, Comm. Pure. Appl. Math., 60 (2007), 789-837.  doi: 10.1002/cpa.20161.  Google Scholar

[10]

Z. Lin and W. A. Strauss, A sharp stability criterion for Vlasov-Maxwell systems, Invent. Math., 173 (2008), 497-546.  doi: 10.1007/s00222-008-0122-1.  Google Scholar

[11]

Z. Lin and C. Zeng, Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs, Mem. Amer. Math. Soc., 275 (2022), 1347.  doi: 10.1090/memo/1347.  Google Scholar

[12]

Z. Lin and C. Zeng, Separable Hamiltonian PDEs and Turning point principle for stability of gaseous stars, arXiv: 2005.00973, accepted by Comm. Pure. Appl. Math. doi: 10.1002/cpa.22027.  Google Scholar

[13]

T. T. Nguyen and W. A. Strauss, Linear stability analysis of a hot plasma in a solid torus, Arch. Ration. Mech. Anal., 211 (2014), 619-672.  doi: 10.1007/s00205-013-0680-2.  Google Scholar

[14]

T. T. Nguyen and W. A. Strauss, Stability analysis of collisionless plasmas with specularly reflecting boundary, SIAM J. Math. Anal., 45 (2013), 777-808.  doi: 10.1137/110859695.  Google Scholar

[15]

K. Z. Zhang, Linear stability analysis of the relativistic Vlasov-Maxwell system in an axisymmetric domain, SIAM J. Math. Anal., 51 (2019), 4683-4723.  doi: 10.1137/18M1206825.  Google Scholar

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