August  2022, 15(4): 663-679. doi: 10.3934/krm.2021048

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 Published  August 2022 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 and Related Models, 2022, 15 (4) : 663-679. 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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[7]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[1]

Yunbai Cao, Chanwoo Kim. Glassey-Strauss representation of Vlasov-Maxwell systems in a Half Space. Kinetic and Related Models, 2022, 15 (3) : 385-401. doi: 10.3934/krm.2021034

[2]

Sergiu Klainerman, Gigliola Staffilani. A new approach to study the Vlasov-Maxwell system. Communications on Pure and Applied Analysis, 2002, 1 (1) : 103-125. doi: 10.3934/cpaa.2002.1.103

[3]

Jonathan Ben-Artzi, Stephen Pankavich, Junyong Zhang. A toy model for the relativistic Vlasov-Maxwell system. Kinetic and Related Models, 2022, 15 (3) : 341-354. doi: 10.3934/krm.2021053

[4]

Adina Luminiţa Sasu, Bogdan Sasu. Discrete admissibility and exponential trichotomy of dynamical systems. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 2929-2962. doi: 10.3934/dcds.2014.34.2929

[5]

Toan T. Nguyen, Truyen V. Nguyen, Walter A. Strauss. Global magnetic confinement for the 1.5D Vlasov-Maxwell system. Kinetic and Related Models, 2015, 8 (1) : 153-168. doi: 10.3934/krm.2015.8.153

[6]

Adina Luminiţa Sasu, Bogdan Sasu. Exponential trichotomy and $(r, p)$-admissibility for discrete dynamical systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (8) : 3199-3220. doi: 10.3934/dcdsb.2017170

[7]

Mohammad Asadzadeh, Piotr Kowalczyk, Christoffer Standar. On hp-streamline diffusion and Nitsche schemes for the relativistic Vlasov-Maxwell system. Kinetic and Related Models, 2019, 12 (1) : 105-131. doi: 10.3934/krm.2019005

[8]

Toan T. Nguyen, Truyen V. Nguyen, Walter A. Strauss. Erratum to: Global magnetic confinement for the 1.5D Vlasov-Maxwell system. Kinetic and Related Models, 2015, 8 (3) : 615-616. doi: 10.3934/krm.2015.8.615

[9]

Jin Woo Jang, Robert M. Strain, Tak Kwong Wong. Magnetic confinement for the 2D axisymmetric relativistic Vlasov-Maxwell system in an annulus. Kinetic and Related Models, 2022, 15 (4) : 569-604. doi: 10.3934/krm.2021039

[10]

Abed Bounemoura, Edouard Pennamen. Instability for a priori unstable Hamiltonian systems: A dynamical approach. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 753-793. doi: 10.3934/dcds.2012.32.753

[11]

Marian Gidea, Rafael De La Llave. Topological methods in the instability problem of Hamiltonian systems. Discrete and Continuous Dynamical Systems, 2006, 14 (2) : 295-328. doi: 10.3934/dcds.2006.14.295

[12]

Rod Cross, Victor Kozyakin. Double exponential instability of triangular arbitrage systems. Discrete and Continuous Dynamical Systems - B, 2013, 18 (2) : 349-376. doi: 10.3934/dcdsb.2013.18.349

[13]

Frédérique Charles, Bruno Després, Benoît Perthame, Rémis Sentis. Nonlinear stability of a Vlasov equation for magnetic plasmas. Kinetic and Related Models, 2013, 6 (2) : 269-290. doi: 10.3934/krm.2013.6.269

[14]

Pietro-Luciano Buono, Daniel C. Offin. Instability criterion for periodic solutions with spatio-temporal symmetries in Hamiltonian systems. Journal of Geometric Mechanics, 2017, 9 (4) : 439-457. doi: 10.3934/jgm.2017017

[15]

Daniel Franco, J. R. L. Webb. Collisionless orbits of singular and nonsingular dynamical systems. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 747-757. doi: 10.3934/dcds.2006.15.747

[16]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete and Continuous Dynamical Systems, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[17]

Daniela Cárcamo-Díaz, Jesús F. Palacián, Claudio Vidal, Patricia Yanguas. Nonlinear stability of elliptic equilibria in hamiltonian systems with exponential time estimates. Discrete and Continuous Dynamical Systems, 2021, 41 (11) : 5183-5208. doi: 10.3934/dcds.2021073

[18]

Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete and Continuous Dynamical Systems, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166

[19]

Gunther Dirr, Hiroshi Ito, Anders Rantzer, Björn S. Rüffer. Separable Lyapunov functions for monotone systems: Constructions and limitations. Discrete and Continuous Dynamical Systems - B, 2015, 20 (8) : 2497-2526. doi: 10.3934/dcdsb.2015.20.2497

[20]

Reinhard Racke. Instability of coupled systems with delay. Communications on Pure and Applied Analysis, 2012, 11 (5) : 1753-1773. doi: 10.3934/cpaa.2012.11.1753

2020 Impact Factor: 1.432

Metrics

  • PDF downloads (183)
  • HTML views (128)
  • Cited by (0)

Other articles
by authors

[Back to Top]