Abstract
Mathematical models arising in the natural sciences often involve equations which describe how the phenomena under investigation evolve in time. In these notes, some mathematical techniques will be presented for analysing a range of evolution equations that can arise in a number of applied disciplines such as biomathematics and population dynamics. Different types of equations will be examined, but a unifying theme will be provided by developing methods from a dynamical systems point of view and using some elegant results from functional analysis. To fix ideas, we will begin with some simple finite-dimensional models from population dynamics which are expressed in terms of ordinary differential equations. We then go on to consider dynamical systems in an infinite-dimensional setting, and provide a gentle introduction to the theory of strongly continuous semigroups of operators. This theory is applied to an infinite system of nonlinear ordinary differential equations that models the time-evolution of the size distribution of a collection of particles that can coagulate to form larger particles or fragment into smaller particles.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
J.M. Ball, J. Carr, The discrete coagulation–fragmentation equations: existence, uniqueness and density conservation. J. Stat. Phys. 61, 203–234 (1990)
J. Banasiak, On an extension of the Kato–Voigt perturbation theorem for substochastic semigroups and its application. Taiwanese J. Math. 5, 169–191 (2001)
J. Banasiak, Kinetic models in natural sciences, in Evolutionary Equations with Applications to Natural Sciences, ed. by J. Banasiak, M. Mokhtar-Kharroubi. Lecture Notes in Mathematics (Springer, Berlin, 2014)
J. Banasiak, L. Arlotti, Perturbations of Positive Semigroups with Applications (Springer, New York, 2006)
J. Banasiak, W. Lamb, The discrete fragmentation equation: semigroups, compactness and asynchronous exponential growth. Kinetic Relat. Model 5, 223–236 (2012)
A. Belleni-Morante, Applied Semigroups and Evolution Equations (Clarendon Press, Oxford, 1979)
A. Belleni-Morante, A.C. McBride, Applied Nonlinear Semigroups (Wiley, Chichester, 1998)
A. Bobrowski, Boundary conditions in evolutionary equations in biology, in Evolutionary Equations with Applications to Natural Sciences, ed. by J. Banasiak, M. Mokhtar-Kharroubi. Lecture Notes in Mathematics (Springer, Berlin, 2014)
N.F. Britton, Essential Mathematical Biology (Springer, London, 2003)
J.F. Collet, Some modelling issues in the theory of fragmentation–coagulation systems. Commun. Math. Sci. 1, 35–54 (2004)
F.P. da Costa, Existence and uniqueness of density conserving solutions to the coagulation–fragmentation equations with strong fragmentation. J. Math. Anal. Appl. 192, 892–914 (1995)
S.P. Ellner, J. Guckenheimer, Dynamic Models in Biology (Princeton University Press, Princeton, 2006)
K.-J. Engel, R. Nagel, One-Parameter Semigroups for Linear Evolution Equations. Graduate Texts in Mathematics (Springer, New York, 2000)
M. Farkas, Dynamical Models in Biology (Academic, San Diego, 2001)
M.W. Hirsch, S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra (Academic, Orlando, 1974)
E. Kreyszig, Introductory Functional Analysis with Applications (Wiley, New York, 1978)
P. Laurençot, The discrete coagulation equations with multiple fragmentation. Proc. Edinburgh Math. Soc. 45, 67–82 (2002)
P. Laurençot, Weak compactness techniques and coagulation equations, in Evolutionary Equations with Applications to Natural Sciences, ed. by J. Banasiak, M. Mokhtar-Kharroubi. Lecture Notes in Mathematics (Springer, Berlin, 2014)
A.C. McBride, Semigroups of Linear Operators: An Introduction. Research Notes in Mathematics (Pitman, Harlow, 1987)
A.C. McBride, A.L. Smith, W. Lamb, Strongly differentiable solutions of the discrete coagulation–fragmentation equation. Physica D 239, 1436–1445 (2010)
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations (Springer, Berlin, 1983)
L. Smith, W. Lamb, M. Langer, A.C. McBride, Discrete fragmentation with mass loss. J. Evol. Equ. 12, 181–201 (2012)
J.A.D. Wattis, An introduction to mathematical models of coagulation–fragmentation processes; a discrete deterministic mean-field approach. Physica D 222, 1–20 (2006)
S. Wiggins, Introduction to Applied Nonlinear Dynamical Systems and Chaos (Springer, New York, 1990)
R.M. Ziff, An explicit solution to a discrete fragmentation model. J. Phys. A Math. Gen. 25, 2569–2576 (1992)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Lamb, W. (2015). Applying Functional Analytic Techniques to Evolution Equations. In: Banasiak, J., Mokhtar-Kharroubi, M. (eds) Evolutionary Equations with Applications in Natural Sciences. Lecture Notes in Mathematics, vol 2126. Springer, Cham. https://doi.org/10.1007/978-3-319-11322-7_1
Download citation
DOI: https://doi.org/10.1007/978-3-319-11322-7_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-11321-0
Online ISBN: 978-3-319-11322-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)