June  2010, 3(2): 299-309. doi: 10.3934/krm.2010.3.299

A mean-field toy model for resonant transport

1. 

Université Paris-Sud XI, Laboratoire de Mathématiques, 91405 Orsay Cedex, France, France

Received  September 2009 Revised  November 2009 Published  May 2010

We consider a simple one dimensional mean-field equation modeling a resonant setting in a coupled wave + transport system. Using elementary methods, we obtain sufficient conditions on the initial data to ensure global existence or blow-up in finite time.
Citation: Patrick Gerard, Christophe Pallard. A mean-field toy model for resonant transport. Kinetic and Related Models, 2010, 3 (2) : 299-309. doi: 10.3934/krm.2010.3.299
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[6]

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

[7]

Yu-Zhu Wang, Weibing Zuo. On the blow-up criterion of smooth solutions for Hall-magnetohydrodynamics system with partial viscosity. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1327-1336. doi: 10.3934/cpaa.2014.13.1327

[8]

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

[9]

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

[10]

Yemin Chen. Smoothness of classical solutions to the Vlasov-Maxwell-Landau system near Maxwellians. Discrete and Continuous Dynamical Systems, 2008, 20 (4) : 889-910. doi: 10.3934/dcds.2008.20.889

[11]

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

[12]

Yan Jia, Xingwei Zhang, Bo-Qing Dong. Remarks on the blow-up criterion for smooth solutions of the Boussinesq equations with zero diffusion. Communications on Pure and Applied Analysis, 2013, 12 (2) : 923-937. doi: 10.3934/cpaa.2013.12.923

[13]

Baoquan Yuan, Xiao Li. Blow-up criteria of smooth solutions to the three-dimensional micropolar fluid equations in Besov space. Discrete and Continuous Dynamical Systems - S, 2016, 9 (6) : 2167-2179. doi: 10.3934/dcdss.2016090

[14]

Qiong Chen, Chunlai Mu, Zhaoyin Xiang. Blow-up and asymptotic behavior of solutions to a semilinear integrodifferential system. Communications on Pure and Applied Analysis, 2006, 5 (3) : 435-446. doi: 10.3934/cpaa.2006.5.435

[15]

Huiling Li, Mingxin Wang. Properties of blow-up solutions to a parabolic system with nonlinear localized terms. Discrete and Continuous Dynamical Systems, 2005, 13 (3) : 683-700. doi: 10.3934/dcds.2005.13.683

[16]

Hongwei Chen. Blow-up estimates of positive solutions of a reaction-diffusion system. Conference Publications, 2003, 2003 (Special) : 182-188. doi: 10.3934/proc.2003.2003.182

[17]

Stephen Pankavich, Nicholas Michalowski. Global classical solutions for the "One and one-half'' dimensional relativistic Vlasov-Maxwell-Fokker-Planck system. Kinetic and Related Models, 2015, 8 (1) : 169-199. doi: 10.3934/krm.2015.8.169

[18]

Jörg Weber. Confined steady states of the relativistic Vlasov–Maxwell system in an infinitely long cylinder. Kinetic and Related Models, 2020, 13 (6) : 1135-1161. doi: 10.3934/krm.2020040

[19]

Shuangqian Liu, Qinghua Xiao. The relativistic Vlasov-Maxwell-Boltzmann system for short range interaction. Kinetic and Related Models, 2016, 9 (3) : 515-550. doi: 10.3934/krm.2016005

[20]

Yūki Naito, Takasi Senba. Blow-up behavior of solutions to a parabolic-elliptic system on higher dimensional domains. Discrete and Continuous Dynamical Systems, 2012, 32 (10) : 3691-3713. doi: 10.3934/dcds.2012.32.3691

2021 Impact Factor: 1.398

Metrics

  • PDF downloads (83)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]