• Previous Article
    Low field regime for the relativistic Vlasov-Maxwell-Fokker-Planck system; the one and one half dimensional case
  • KRM Home
  • This Issue
  • Next Article
    Discrete transparent boundary conditions for the Schrodinger equation -- a compact higher order scheme
March  2008, 1(1): 127-137. doi: 10.3934/krm.2008.1.127

Fast-reaction limit for the inhomogeneous Aizenman-Bak model

1. 

ICREA-Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain

2. 

ENS Cachan, CMLA, IUF & CNRS, PRES UniverSud, 61, Av. du Pdt Wilson, 94235 Cachan Cedex

3. 

Faculty of Mathematics, University of Vienna, Nordbergstr. 15, 1090 Wien

Received  October 2007 Revised  October 2007 Published  February 2008

Solutions of the spatially inhomogeneous diffusive\linebreak Aizenmann-Bak model for clustering within a bounded domain with homogeneous Neumann boundary conditions are shown to stabilize, in the fast reaction limit, towards local equilibria determined by their monomer density. Moreover, the sequence of monomer densities converges to the solution of a nonlinear diffusion equation whose nonlinearity depends on the size-dependent diffusion coefficient. Initial data are assumed to be integrable, bounded and with a certain number of moments in size. The number density of clusters for the solutions is assumed to verify uniform bounds away from zero and infinity independently of the scale parameter.
Citation: José A. Carrillo, Laurent Desvillettes, Klemens Fellner. Fast-reaction limit for the inhomogeneous Aizenman-Bak model. Kinetic & Related Models, 2008, 1 (1) : 127-137. doi: 10.3934/krm.2008.1.127
[1]

Maxime Breden. Applications of improved duality lemmas to the discrete coagulation-fragmentation equations with diffusion. Kinetic & Related Models, 2018, 11 (2) : 279-301. doi: 10.3934/krm.2018014

[2]

Jacek Banasiak, Luke O. Joel, Sergey Shindin. The discrete unbounded coagulation-fragmentation equation with growth, decay and sedimentation. Kinetic & Related Models, 2019, 12 (5) : 1069-1092. doi: 10.3934/krm.2019040

[3]

Prasanta Kumar Barik, Ankik Kumar Giri. A note on mass-conserving solutions to the coagulation-fragmentation equation by using non-conservative approximation. Kinetic & Related Models, 2018, 11 (5) : 1125-1138. doi: 10.3934/krm.2018043

[4]

Pierre Degond, Maximilian Engel. Numerical approximation of a coagulation-fragmentation model for animal group size statistics. Networks & Heterogeneous Media, 2017, 12 (2) : 217-243. doi: 10.3934/nhm.2017009

[5]

Miguel A. Herrero, Marianito R. Rodrigo. Remarks on accessible steady states for some coagulation-fragmentation systems. Discrete & Continuous Dynamical Systems - A, 2007, 17 (3) : 541-552. doi: 10.3934/dcds.2007.17.541

[6]

Ankik Kumar Giri. On the uniqueness for coagulation and multiple fragmentation equation. Kinetic & Related Models, 2013, 6 (3) : 589-599. doi: 10.3934/krm.2013.6.589

[7]

Martin Frank, Cory D. Hauck, Edgar Olbrant. Perturbed, entropy-based closure for radiative transfer. Kinetic & Related Models, 2013, 6 (3) : 557-587. doi: 10.3934/krm.2013.6.557

[8]

Dieter Bothe, Michel Pierre. The instantaneous limit for reaction-diffusion systems with a fast irreversible reaction. Discrete & Continuous Dynamical Systems - S, 2012, 5 (1) : 49-59. doi: 10.3934/dcdss.2012.5.49

[9]

Marcel Freitag. The fast signal diffusion limit in nonlinear chemotaxis systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019211

[10]

Annegret Glitzky. Energy estimates for electro-reaction-diffusion systems with partly fast kinetics. Discrete & Continuous Dynamical Systems - A, 2009, 25 (1) : 159-174. doi: 10.3934/dcds.2009.25.159

[11]

Jacek Banasiak. Blow-up of solutions to some coagulation and fragmentation equations with growth. Conference Publications, 2011, 2011 (Special) : 126-134. doi: 10.3934/proc.2011.2011.126

[12]

Jean Dolbeault, Giuseppe Toscani. Fast diffusion equations: Matching large time asymptotics by relative entropy methods. Kinetic & Related Models, 2011, 4 (3) : 701-716. doi: 10.3934/krm.2011.4.701

[13]

Lu Tan, Ling Li, Senjian An, Zhenkuan Pan. Nonlinear diffusion based image segmentation using two fast algorithms. Mathematical Foundations of Computing, 2019, 2 (2) : 149-168. doi: 10.3934/mfc.2019011

[14]

Petr Knobloch. Error estimates for a nonlinear local projection stabilization of transient convection--diffusion--reaction equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 901-911. doi: 10.3934/dcdss.2015.8.901

[15]

Florian Schneider, Jochen Kall, Graham Alldredge. A realizability-preserving high-order kinetic scheme using WENO reconstruction for entropy-based moment closures of linear kinetic equations in slab geometry. Kinetic & Related Models, 2016, 9 (1) : 193-215. doi: 10.3934/krm.2016.9.193

[16]

E. C.M. Crooks, E. N. Dancer, Danielle Hilhorst. Fast reaction limit and long time behavior for a competition-diffusion system with Dirichlet boundary conditions. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 39-44. doi: 10.3934/dcdsb.2007.8.39

[17]

Jacek Banasiak. Transport processes with coagulation and strong fragmentation. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 445-472. doi: 10.3934/dcdsb.2012.17.445

[18]

Zbigniew Banach, Wieslaw Larecki. Entropy-based mixed three-moment description of fermionic radiation transport in slab and spherical geometries. Kinetic & Related Models, 2017, 10 (4) : 879-900. doi: 10.3934/krm.2017035

[19]

Prasanta Kumar Barik. Existence of mass-conserving weak solutions to the singular coagulation equation with multiple fragmentation. Evolution Equations & Control Theory, 2019, 0 (0) : 0-0. doi: 10.3934/eect.2020012

[20]

Alessandro Audrito. Bistable reaction equations with doubly nonlinear diffusion. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 2977-3015. doi: 10.3934/dcds.2019124

2018 Impact Factor: 1.38

Metrics

  • PDF downloads (9)
  • HTML views (0)
  • Cited by (6)

[Back to Top]