American Institute of Mathematical Sciences

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

A mean-field toy model for resonant transport

Patrick Gerard 1, and  Christophe Pallard 1,

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.
Keywords: blow-up., smooth solutions, Vlasov-Maxwell system, resonant coupling.
Mathematics Subject Classification: Primary: 35Q83, 35B40; Secondary: 82D1.
Citation: Patrick Gerard, Christophe Pallard. A mean-field toy model for resonant transport. Kinetic & 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 & Applied Analysis, 2002, 1 (1) : 103-125. doi: 10.3934/cpaa.2002.1.103
[2]

Toan T. Nguyen, Truyen V. Nguyen, Walter A. Strauss. Global magnetic confinement for the 1.5D Vlasov-Maxwell system. Kinetic & Related Models, 2015, 8 (1) : 153-168. doi: 10.3934/krm.2015.8.153
[3]

Mohammad Asadzadeh, Piotr Kowalczyk, Christoffer Standar. On hp-streamline diffusion and Nitsche schemes for the relativistic Vlasov-Maxwell system. Kinetic & Related Models, 2019, 12 (1) : 105-131. doi: 10.3934/krm.2019005
[4]

Toan T. Nguyen, Truyen V. Nguyen, Walter A. Strauss. Erratum to: Global magnetic confinement for the 1.5D Vlasov-Maxwell system. Kinetic & Related Models, 2015, 8 (3) : 615-616. doi: 10.3934/krm.2015.8.615
[5]

Yu-Zhu Wang, Weibing Zuo. On the blow-up criterion of smooth solutions for Hall-magnetohydrodynamics system with partial viscosity. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1327-1336. doi: 10.3934/cpaa.2014.13.1327
[6]

Yemin Chen. Smoothness of classical solutions to the Vlasov-Maxwell-Landau system near Maxwellians. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 889-910. doi: 10.3934/dcds.2008.20.889
[7]

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

Qiong Chen, Chunlai Mu, Zhaoyin Xiang. Blow-up and asymptotic behavior of solutions to a semilinear integrodifferential system. Communications on Pure & Applied Analysis, 2006, 5 (3) : 435-446. doi: 10.3934/cpaa.2006.5.435
[9]

Huiling Li, Mingxin Wang. Properties of blow-up solutions to a parabolic system with nonlinear localized terms. Discrete & Continuous Dynamical Systems - A, 2005, 13 (3) : 683-700. doi: 10.3934/dcds.2005.13.683
[10]

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
[11]

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 & Applied Analysis, 2013, 12 (2) : 923-937. doi: 10.3934/cpaa.2013.12.923
[12]

Baoquan Yuan, Xiao Li. Blow-up criteria of smooth solutions to the three-dimensional micropolar fluid equations in Besov space. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2167-2179. doi: 10.3934/dcdss.2016090
[13]

Shuangqian Liu, Qinghua Xiao. The relativistic Vlasov-Maxwell-Boltzmann system for short range interaction. Kinetic & Related Models, 2016, 9 (3) : 515-550. doi: 10.3934/krm.2016005
[14]

Yūki Naito, Takasi Senba. Blow-up behavior of solutions to a parabolic-elliptic system on higher dimensional domains. Discrete & Continuous Dynamical Systems - A, 2012, 32 (10) : 3691-3713. doi: 10.3934/dcds.2012.32.3691
[15]

Shu-Xiang Huang, Fu-Cai Li, Chun-Hong Xie. Global existence and blow-up of solutions to a nonlocal reaction-diffusion system. Discrete & Continuous Dynamical Systems - A, 2003, 9 (6) : 1519-1532. doi: 10.3934/dcds.2003.9.1519
[16]

Yongsheng Mi, Boling Guo, Chunlai Mu. Well-posedness and blow-up scenario for a new integrable four-component system with peakon solutions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (4) : 2171-2191. doi: 10.3934/dcds.2016.36.2171
[17]

Johannes Lankeit. Infinite time blow-up of many solutions to a general quasilinear parabolic-elliptic Keller-Segel system. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 233-255. doi: 10.3934/dcdss.2020013
[18]

Vo Anh Khoa, Le Thi Phuong Ngoc, Nguyen Thanh Long. Existence, blow-up and exponential decay of solutions for a porous-elastic system with damping and source terms. Evolution Equations & Control Theory, 2019, 8 (2) : 359-395. doi: 10.3934/eect.2019019
[19]

C. Y. Chan. Recent advances in quenching and blow-up of solutions. Conference Publications, 2001, 2001 (Special) : 88-95. doi: 10.3934/proc.2001.2001.88
[20]

Yong Zhou, Zhengguang Guo. Blow up and propagation speed of solutions to the DGH equation. Discrete & Continuous Dynamical Systems - B, 2009, 12 (3) : 657-670. doi: 10.3934/dcdsb.2009.12.657

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences