May  2005, 5(2): 299-318. doi: 10.3934/dcdsb.2005.5.299

On the $L^2$-moment closure of transport equations: The general case

1. 

Department of Mathematical and Statistical Sciences, University of Alberta, Edmonton, AB, T6G 2G1

Received  August 2003 Revised  June 2004 Published  February 2005

Transport equations are intensively used in Mathematical Biology. In this article the moment closure for transport equations for an arbitrary finite number of moments is presented. With use of a variational principle the closure can be obtained by minimizing the $L^2(V)$-norm with constraints. An $H$-Theorem for the negative $L^2$-norm is shown and the existence of Lagrange multipliers is proven. The Cattaneo closure is a special case for two moments and was studied in Part I (Hillen 2003). Here the general theory is given and the three moment closure for two space dimensions is calculated explicitly. It turns out that the steady states of the two and three moment systems are determined by the steady states of a corresponding diffusion problem.
Citation: T. Hillen. On the $L^2$-moment closure of transport equations: The general case. Discrete and Continuous Dynamical Systems - B, 2005, 5 (2) : 299-318. doi: 10.3934/dcdsb.2005.5.299
[1]

T. Hillen. On the $L^2$-moment closure of transport equations: The Cattaneo approximation. Discrete and Continuous Dynamical Systems - B, 2004, 4 (4) : 961-982. doi: 10.3934/dcdsb.2004.4.961

[2]

YunKyong Hyon. Hysteretic behavior of a moment-closure approximation for FENE model. Kinetic and Related Models, 2014, 7 (3) : 493-507. doi: 10.3934/krm.2014.7.493

[3]

Martin Frank, Benjamin Seibold. Optimal prediction for radiative transfer: A new perspective on moment closure. Kinetic and Related Models, 2011, 4 (3) : 717-733. doi: 10.3934/krm.2011.4.717

[4]

Teddy Pichard. A moment closure based on a projection on the boundary of the realizability domain: Extension and analysis. Kinetic and Related Models, , () : -. doi: 10.3934/krm.2022014

[5]

Radosław Kurek, Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1955-1981. doi: 10.3934/dcds.2018079

[6]

Teddy Pichard. A moment closure based on a projection on the boundary of the realizability domain: 1D case. Kinetic and Related Models, 2020, 13 (6) : 1243-1280. doi: 10.3934/krm.2020045

[7]

Zhenning Cai, Yuwei Fan, Ruo Li. On hyperbolicity of 13-moment system. Kinetic and Related Models, 2014, 7 (3) : 415-432. doi: 10.3934/krm.2014.7.415

[8]

Yinsong Bai, Lin He, Huijiang Zhao. Nonlinear stability of rarefaction waves for a hyperbolic system with Cattaneo's law. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2441-2474. doi: 10.3934/cpaa.2021049

[9]

Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.. Evolution Equations and Control Theory, 2014, 3 (1) : 83-118. doi: 10.3934/eect.2014.3.83

[10]

Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part I: The linearized system.. Evolution Equations and Control Theory, 2014, 3 (1) : 59-82. doi: 10.3934/eect.2014.3.59

[11]

Léo Bois, Emmanuel Franck, Laurent Navoret, Vincent Vigon. A neural network closure for the Euler-Poisson system based on kinetic simulations. Kinetic and Related Models, 2022, 15 (1) : 49-89. doi: 10.3934/krm.2021044

[12]

Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. I. Invariant torus and its normal hyperbolicity. Journal of Geometric Mechanics, 2012, 4 (4) : 443-467. doi: 10.3934/jgm.2012.4.443

[13]

Darryl D. Holm, Cesare Tronci. Geodesic Vlasov equations and their integrable moment closures. Journal of Geometric Mechanics, 2009, 1 (2) : 181-208. doi: 10.3934/jgm.2009.1.181

[14]

Miroslava Růžičková, Irada Dzhalladova, Jitka Laitochová, Josef Diblík. Solution to a stochastic pursuit model using moment equations. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 473-485. doi: 10.3934/dcdsb.2018032

[15]

Julian Koellermeier, Giovanni Samaey. Projective integration schemes for hyperbolic moment equations. Kinetic and Related Models, 2021, 14 (2) : 353-387. doi: 10.3934/krm.2021008

[16]

Guillaume Bal, Alexandre Jollivet. Boundary control for transport equations. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022014

[17]

Alain Miranville. Asymptotic behavior of the conserved Caginalp phase-field system based on the Maxwell-Cattaneo law. Communications on Pure and Applied Analysis, 2014, 13 (5) : 1971-1987. doi: 10.3934/cpaa.2014.13.1971

[18]

Ahmad Makki, Alain Miranville, Georges Sadaka. On the nonconserved Caginalp phase-field system based on the Maxwell-Cattaneo law with two temperatures and logarithmic potentials. Discrete and Continuous Dynamical Systems - B, 2019, 24 (3) : 1341-1365. doi: 10.3934/dcdsb.2019019

[19]

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

[20]

Jessy Mallet, Stéphane Brull, Bruno Dubroca. General moment system for plasma physics based on minimum entropy principle. Kinetic and Related Models, 2015, 8 (3) : 533-558. doi: 10.3934/krm.2015.8.533

2020 Impact Factor: 1.327

Metrics

  • PDF downloads (44)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]