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, in preparation.

[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.

show all references

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, in preparation.

[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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