December  2019, 12(6): 1313-1327. doi: 10.3934/krm.2019051

Focusing solutions of the Vlasov-Poisson system

Department of Mathematics, Brown University, 151 Thayer Street, Providence, RI 02912, USA

Received  February 2019 Revised  May 2019 Published  September 2019

We study smooth, spherically-symmetric solutions to the Vlasov-Poisson system and relativistic Vlasov-Poisson system in the plasma physical case. We construct solutions that initially possess arbitrarily small $ C^k $ norms ($ k \geq 1 $) for the charge densities and the electric fields, but attain arbitrarily large $ L^\infty $ norms of them at some later time.

Citation: Katherine Zhiyuan Zhang. Focusing solutions of the Vlasov-Poisson system. Kinetic & Related Models, 2019, 12 (6) : 1313-1327. doi: 10.3934/krm.2019051
References:
[1]

J. Ben-ArtziS. Calogero and S. Pankavich, Arbitrarily large solutions of the Vlasov-Poisson system, SIAM Journal on Mathematical Analysis, 50 (2018), 4311-4326.  doi: 10.1137/17M1142715.  Google Scholar

[2]

J. Ben-Artzi, S. Calogero and S. Pankavich, Concentrating solutions of the relativistic Vlasov-Maxwell system, Commun. Math. Sci., 17 (2019), 377–392, arXiv: 1807.02801. doi: 10.4310/CMS.2019.v17.n2.a4.  Google Scholar

[3] J. P. Friedberg, Ideal Magnetohydrodynamics, Plenum Press, New York, 1987.   Google Scholar
[4]

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

[5]

R. GlasseyS. Pankavich and J. Schaeffer, Decay in time for a one-dimensional two-component plasma, Math. Methods Appl. Sci., 31 (2008), 2115-2132.  doi: 10.1002/mma.1015.  Google Scholar

[6]

R. GlasseyS. Pankavich and J. Schaeffer, On long-time behavior of monocharged and neutral plasma in one and one-half dimensions, Kinetic and Related Models, 2 (2009), 465-488.  doi: 10.3934/krm.2009.2.465.  Google Scholar

[7]

R. GlasseyS. Pankavich and J. Schaeffer, Large time behavior of the relativistic Vlasov-Maxwell system in low space dimension, Differential and Integral Equations, 23 (2010), 61-77.   Google Scholar

[8]

R. GlasseyS. Pankavich and J. Schaeffer, Time decay for solutions to one-dimensional two component plasma equations, Quarterly of Applied Mathematics, 68 (2010), 135-141.  doi: 10.1090/S0033-569X-09-01143-4.  Google Scholar

[9]

R. T. Glassey and J. Schaeffer, On symmetric solutions of the relativistic Vlasov-Poisson system, Comm. Math. Phys., 101 (1985), 459-473.  doi: 10.1007/BF01210740.  Google Scholar

[10]

E. Horst, Symmetric plasmas and their decay, Comm. Math. Phys., 126 (1990), 613-633.  doi: 10.1007/BF02125703.  Google Scholar

[11]

R. Illner and G. Rein, Time decay of the solutions of the Vlasov-Poisson system in the plasma physical case, Math. Methods Appl. Sci., 19 (1996), 1409-1413.  doi: 10.1002/(SICI)1099-1476(19961125)19:17<1409::AID-MMA836>3.0.CO;2-2.  Google Scholar

[12]

P.-L. Lions and B. Perthame, Propogation of moments and regularity for the three dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430.  doi: 10.1007/BF01232273.  Google Scholar

[13]

D. R. Nicholson, Introduction to Plasma Theory, Wiley, New York, 1983. Google Scholar

[14]

K. Pfaffelmoser, Global classical solution of the Vlasov-Poisson system in three dimensions for general initial data, J. Diff. Eq., 95 (1992), 281-303.  doi: 10.1016/0022-0396(92)90033-J.  Google Scholar

[15]

G. Rein and L. Taegert, Gravitational collapse and the Vlasov-Poisson system, Annales Henri Poincaré, 17 (2016), 1415-1427.  doi: 10.1007/s00023-015-0424-y.  Google Scholar

show all references

References:
[1]

J. Ben-ArtziS. Calogero and S. Pankavich, Arbitrarily large solutions of the Vlasov-Poisson system, SIAM Journal on Mathematical Analysis, 50 (2018), 4311-4326.  doi: 10.1137/17M1142715.  Google Scholar

[2]

J. Ben-Artzi, S. Calogero and S. Pankavich, Concentrating solutions of the relativistic Vlasov-Maxwell system, Commun. Math. Sci., 17 (2019), 377–392, arXiv: 1807.02801. doi: 10.4310/CMS.2019.v17.n2.a4.  Google Scholar

[3] J. P. Friedberg, Ideal Magnetohydrodynamics, Plenum Press, New York, 1987.   Google Scholar
[4]

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

[5]

R. GlasseyS. Pankavich and J. Schaeffer, Decay in time for a one-dimensional two-component plasma, Math. Methods Appl. Sci., 31 (2008), 2115-2132.  doi: 10.1002/mma.1015.  Google Scholar

[6]

R. GlasseyS. Pankavich and J. Schaeffer, On long-time behavior of monocharged and neutral plasma in one and one-half dimensions, Kinetic and Related Models, 2 (2009), 465-488.  doi: 10.3934/krm.2009.2.465.  Google Scholar

[7]

R. GlasseyS. Pankavich and J. Schaeffer, Large time behavior of the relativistic Vlasov-Maxwell system in low space dimension, Differential and Integral Equations, 23 (2010), 61-77.   Google Scholar

[8]

R. GlasseyS. Pankavich and J. Schaeffer, Time decay for solutions to one-dimensional two component plasma equations, Quarterly of Applied Mathematics, 68 (2010), 135-141.  doi: 10.1090/S0033-569X-09-01143-4.  Google Scholar

[9]

R. T. Glassey and J. Schaeffer, On symmetric solutions of the relativistic Vlasov-Poisson system, Comm. Math. Phys., 101 (1985), 459-473.  doi: 10.1007/BF01210740.  Google Scholar

[10]

E. Horst, Symmetric plasmas and their decay, Comm. Math. Phys., 126 (1990), 613-633.  doi: 10.1007/BF02125703.  Google Scholar

[11]

R. Illner and G. Rein, Time decay of the solutions of the Vlasov-Poisson system in the plasma physical case, Math. Methods Appl. Sci., 19 (1996), 1409-1413.  doi: 10.1002/(SICI)1099-1476(19961125)19:17<1409::AID-MMA836>3.0.CO;2-2.  Google Scholar

[12]

P.-L. Lions and B. Perthame, Propogation of moments and regularity for the three dimensional Vlasov-Poisson system, Invent. Math., 105 (1991), 415-430.  doi: 10.1007/BF01232273.  Google Scholar

[13]

D. R. Nicholson, Introduction to Plasma Theory, Wiley, New York, 1983. Google Scholar

[14]

K. Pfaffelmoser, Global classical solution of the Vlasov-Poisson system in three dimensions for general initial data, J. Diff. Eq., 95 (1992), 281-303.  doi: 10.1016/0022-0396(92)90033-J.  Google Scholar

[15]

G. Rein and L. Taegert, Gravitational collapse and the Vlasov-Poisson system, Annales Henri Poincaré, 17 (2016), 1415-1427.  doi: 10.1007/s00023-015-0424-y.  Google Scholar

[1]

Justin Holmer, Chang Liu. Blow-up for the 1D nonlinear Schrödinger equation with point nonlinearity II: Supercritical blow-up profiles. Communications on Pure & Applied Analysis, 2021, 20 (1) : 215-242. doi: 10.3934/cpaa.2020264

[2]

Alex H. Ardila, Mykael Cardoso. Blow-up solutions and strong instability of ground states for the inhomogeneous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2021, 20 (1) : 101-119. doi: 10.3934/cpaa.2020259

[3]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[4]

Juan Pablo Pinasco, Mauro Rodriguez Cartabia, Nicolas Saintier. Evolutionary game theory in mixed strategies: From microscopic interactions to kinetic equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020051

[5]

Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052

[6]

Teresa D'Aprile. Bubbling solutions for the Liouville equation around a quantized singularity in symmetric domains. Communications on Pure & Applied Analysis, 2021, 20 (1) : 159-191. doi: 10.3934/cpaa.2020262

[7]

Sihem Guerarra. Maximum and minimum ranks and inertias of the Hermitian parts of the least rank solution of the matrix equation AXB = C. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 75-86. doi: 10.3934/naco.2020016

[8]

Yongxiu Shi, Haitao Wan. Refined asymptotic behavior and uniqueness of large solutions to a quasilinear elliptic equation in a borderline case. Electronic Research Archive, , () : -. doi: 10.3934/era.2020119

[9]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276

[10]

Kai Yang. Scattering of the focusing energy-critical NLS with inverse square potential in the radial case. Communications on Pure & Applied Analysis, 2021, 20 (1) : 77-99. doi: 10.3934/cpaa.2020258

[11]

Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020451

[12]

Jie Zhang, Yuping Duan, Yue Lu, Michael K. Ng, Huibin Chang. Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020071

[13]

Adrian Constantin, Darren G. Crowdy, Vikas S. Krishnamurthy, Miles H. Wheeler. Stuart-type polar vortices on a rotating sphere. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 201-215. doi: 10.3934/dcds.2020263

[14]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256

[15]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458

[16]

Sergey Rashkovskiy. Hamilton-Jacobi theory for Hamiltonian and non-Hamiltonian systems. Journal of Geometric Mechanics, 2020, 12 (4) : 563-583. doi: 10.3934/jgm.2020024

[17]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

[18]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[19]

Meng Chen, Yong Hu, Matteo Penegini. On projective threefolds of general type with small positive geometric genus. Electronic Research Archive, , () : -. doi: 10.3934/era.2020117

[20]

Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171

2019 Impact Factor: 1.311

Metrics

  • PDF downloads (59)
  • HTML views (112)
  • Cited by (0)

Other articles
by authors

[Back to Top]