# American Institute of Mathematical Sciences

2015, 12(2): 279-290. doi: 10.3934/mbe.2015.12.279

## A note on modelling with measures: Two-features balance equations

 1 Center for Industrial Mathematics, University of Bremen, Bibliothekstrasse 1, D-28359 Bremen, Germany, Germany

Received  April 2014 Revised  October 2014 Published  December 2014

In this note we explain by an example what we understand by a balance situation and by a balance equation in terms of measures.
The latter ones are an attempt to start modelling of (not only) diffusion-reaction or mass-conservation scenarios in terms of measures rather than by derivatives and other rates.
By means of three examples this concept is extended to two-features (= two-traits-) balance situations, which, e.g., combine features like aging and physical motion in populations or physical motion and formation of polymers by means of a single model equation.
Citation: Michael Böhm, Martin Höpker. A note on modelling with measures: Two-features balance equations. Mathematical Biosciences & Engineering, 2015, 12 (2) : 279-290. doi: 10.3934/mbe.2015.12.279
##### References:
 [1] V. Agoshkov, Boundary Value Problems for Transport Equations, BirkhäuserBoston, Inc., Boston, MA, 1998. doi: 10.1007/978-1-4612-1994-1. [2] M. Böhm, Mathematical Modelling,, Lecture Notes, (). [3] P. R. Halmos, Measure Theory, Springer, 1974. [4] P. M. Gschwend and M. D. Reynolds, Monodisperse ferrous phosphate colloids in an anoxic groundwater plume, Journal of Contaminant Hydrology, 1 (1987), 309-327. doi: 10.1016/0169-7722(87)90011-8. [5] O. Krehel, A. Muntean and P. Knabner, On Modeling and Simulation of Flocculation in Porous Media, XIX International Conference on Water Resources, CMWR, 2012. [6] A. Marzocchi and A. Musesti, Decomposition and integral representation of Cauchy interactions associated with measures, Continuum Mech. Thermodyn., 13 (2001), 149-169. doi: 10.1007/s001610100046. [7] A. Muntean, E. N. M. Cirillo, O. Krehel and M. Böhm, Pedestrians moving in the dark: Balancing measures and playing games on lattices, Collective Dynamics from Bacteria to Crowds, CISM International Centre for Mechanical Sciences, 553 (2014), 75-103. doi: 10.1007/978-3-7091-1785-9_3. [8] F. Schuricht, A new mathematical foundation for contact interactions in continuum physics, Arch. Ration. Mech. Anal., 1984 (2007), 169-196. [9] R. Segev, The geometry of Cauchy fluxes, Arch. Rational Mech. Anal., 154 (2000), 183-198. doi: 10.1007/s002050000089. [10] M. Šilhavý, The Mechanics and Thermodynamics of Continuous Media, Springer, Berlin, 1997. doi: 10.1007/978-3-662-03389-0. [11] S. Slomkowski, J. Alemán, R. G. Gilbert, M. Hess, K. Horie, R. G. Jones, P. Kubisa, I. Meisel, W. Mormann, S. Penczek and R. F. T. Stepto, Terminology of polymers and polymerization processes in dispersed systems (IUPAC Recommendations 2011), Pure and Applied Chemistry, 83 (2011), 2229-2259. doi: 10.1351/PAC-REC-10-06-03. [12] R. Temam and A. Miranville, Mathematical Modeling in Continuum Mechanics, $2^{nd}$ edition, Cambridge University Press, 2001. [13] C. Truesdell, A First Course in Rational Continuum Mechanics, $1^{st}$ edition, AcademicPress, Boston, 1991.

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##### References:
 [1] V. Agoshkov, Boundary Value Problems for Transport Equations, BirkhäuserBoston, Inc., Boston, MA, 1998. doi: 10.1007/978-1-4612-1994-1. [2] M. Böhm, Mathematical Modelling,, Lecture Notes, (). [3] P. R. Halmos, Measure Theory, Springer, 1974. [4] P. M. Gschwend and M. D. Reynolds, Monodisperse ferrous phosphate colloids in an anoxic groundwater plume, Journal of Contaminant Hydrology, 1 (1987), 309-327. doi: 10.1016/0169-7722(87)90011-8. [5] O. Krehel, A. Muntean and P. Knabner, On Modeling and Simulation of Flocculation in Porous Media, XIX International Conference on Water Resources, CMWR, 2012. [6] A. Marzocchi and A. Musesti, Decomposition and integral representation of Cauchy interactions associated with measures, Continuum Mech. Thermodyn., 13 (2001), 149-169. doi: 10.1007/s001610100046. [7] A. Muntean, E. N. M. Cirillo, O. Krehel and M. Böhm, Pedestrians moving in the dark: Balancing measures and playing games on lattices, Collective Dynamics from Bacteria to Crowds, CISM International Centre for Mechanical Sciences, 553 (2014), 75-103. doi: 10.1007/978-3-7091-1785-9_3. [8] F. Schuricht, A new mathematical foundation for contact interactions in continuum physics, Arch. Ration. Mech. Anal., 1984 (2007), 169-196. [9] R. Segev, The geometry of Cauchy fluxes, Arch. Rational Mech. Anal., 154 (2000), 183-198. doi: 10.1007/s002050000089. [10] M. Šilhavý, The Mechanics and Thermodynamics of Continuous Media, Springer, Berlin, 1997. doi: 10.1007/978-3-662-03389-0. [11] S. Slomkowski, J. Alemán, R. G. Gilbert, M. Hess, K. Horie, R. G. Jones, P. Kubisa, I. Meisel, W. Mormann, S. Penczek and R. F. T. Stepto, Terminology of polymers and polymerization processes in dispersed systems (IUPAC Recommendations 2011), Pure and Applied Chemistry, 83 (2011), 2229-2259. doi: 10.1351/PAC-REC-10-06-03. [12] R. Temam and A. Miranville, Mathematical Modeling in Continuum Mechanics, $2^{nd}$ edition, Cambridge University Press, 2001. [13] C. Truesdell, A First Course in Rational Continuum Mechanics, $1^{st}$ edition, AcademicPress, Boston, 1991.
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