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The fragmentation equation with size diffusion: Well posedness and long-term behaviour

Part of: Groups and semigroups of linear operators, their generalizations and applications Special classes of linear operators Partial differential equations Miscellaneous applications of operator theory

Published online by Cambridge University Press:  16 December 2021

PH. LAURENÇOT Open the ORCID record for PH. LAURENÇOT [Opens in a new window]  and
CH. WALKER
PH. LAURENÇOT
Affiliation:
Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, F–31062 Toulouse Cedex 9, France email: [email protected]
CH. WALKER
Affiliation:
Leibniz Universität Hannover, Institut für Angewandte Mathematik, Welfengarten 1, D-30167 Hannover, Germany email: [email protected]
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Abstract

The dynamics of the fragmentation equation with size diffusion is investigated when the size ranges in $(0,\infty)$ . The associated linear operator involves three terms and can be seen as a nonlocal perturbation of a Schrödinger operator. A Miyadera perturbation argument is used to prove that it is the generator of a positive, analytic semigroup on a weighted $L_1$ -space. Moreover, if the overall fragmentation rate does not vanish at infinity, then there is a unique stationary solution with given mass. Assuming further that the overall fragmentation rate diverges to infinity for large sizes implies the immediate compactness of the semigroup and that it eventually stabilizes at an exponential rate to a one-dimensional projection carrying the information of the mass of the initial value.

Keywords

Fragmentationsize diffusionwell posednessconvergencesemigroupperturbation

MSC classification

Primary: 45K05: Integro-partial differential equations 47D06: One-parameter semigroups and linear evolution equations
Secondary: 47B65: Positive operators and order-bounded operators 47N50: Applications in the physical sciences 35B40: Asymptotic behavior of solutions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 6 , December 2022 , pp. 1083 - 1116
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Aliprantis, C. D. & Burkinshaw, O. (2006) Positive Operators, Springer, Dordrecht. Reprint of the 1985 original.Google Scholar
Amann, H. (1990) Ordinary Differential Equations, de Gruyter Studies in Mathematics, Vol. 13, Walter de Gruyter & Co., Berlin. An introduction to nonlinear analysis, Translated from the German by Gerhard Metzen.Google Scholar
Arendt, W. & Batty, C. J. K. (1993) Absorption semigroups and Dirichlet boundary conditions. Math. Ann. 295, 427–448.CrossRefGoogle Scholar
Banasiak, J. (2004) Conservative and shattering solutions for some classes of fragmentation models. Math. Models Methods Appl. Sci. 14, 483–501.CrossRefGoogle Scholar
Banasiak, J. (2006) Shattering and non-uniqueness in fragmentation models—an analytic approach. Phys. D 222, 63–72.CrossRefGoogle Scholar
Banasiak, J. (2020) Global solutions of continuous coagulation-fragmentation equations with unbounded coefficients. Discrete Contin. Dyn. Syst. Ser. S 13, 3319–3334.Google Scholar
Banasiak, J. & Arlotti, L. (2006) Perturbations of Positive Semigroups with Applications , Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London.Google Scholar
Banasiak, J., Lamb, W. & Laurençot, Ph. (2020) Analytic Methods for Coagulation-Fragmentation Models , Monographs and Research Notes in Mathematics, Vol. I, CRC Press, Boca Raton, FL.Google Scholar
Banasiak, J., Lamb, W. & Laurençot, Ph. (2020) Analytic Methods for Coagulation-Fragmentation Models , Monographs and Research Notes in Mathematics, Vol. II, CRC Press, Boca Raton, FL.Google Scholar
Bertoin, J. (2006) Random Fragmentation and Coagulation Processes , Cambridge Studies in Advanced Mathematics, Vol. 102, Cambridge University Press, Cambridge.Google Scholar
Cheng, Z. & Redner, S. (1990) Kinetics of fragmentation. J. Phys. A 23, 1233–1258.CrossRefGoogle Scholar
Clément, P., Heijmans, H. J. A. M., Angenent, S., van Duijn, C. J. & de Pagter, B. (1987) One-Parameter Semigroups , CWI Monographs, Vol. 5, North-Holland Publishing Co., Amsterdam.Google Scholar
Craig, W. (2018) A Course on Partial Differential Equations , Graduate Studies in Mathematics, Vol. 197, American Mathematical Society, Providence, RI.Google Scholar
Engel, K.-J. & Nagel, R. (2000) One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, Vol. 194, Springer-Verlag, New York. With contributions by S. Brendle, M. Campiti, T. Hahn, G. Metafune, G. Nickel, D. Pallara, C. Perazzoli, A. Rhandi, S. Romanelli and R. Schnaubelt.CrossRefGoogle Scholar
Escobedo, M., Mischler, S. & Rodriguez Ricard, M. (2005) On self-similarity and stationary problem for fragmentation and coagulation models. Ann. Inst. H. Poincaré Anal. Non Linéaire 22, 99–125.CrossRefGoogle Scholar
Ferkinghoff-Borg, J., Jensen, M. H., Mathiesen, J., Olesen, P. & Sneppen, K. (2003) Competition between diffusion and fragmentation: an important evolutionary process of nature. Phys. Rev. Lett. 91, 266103.CrossRefGoogle ScholarPubMed
Filippov, A. F. (1961) On the distribution of the sizes of particles which undergo splitting. Theory Probab. Appl. 6, 275–294.CrossRefGoogle Scholar
Flyvbjerg, H., Holy, T. E. & Leibler, S. (1994) Stochastic dynamics of microtubules: a model for caps and catastrophes. Phys. Rev. Lett. 73, 2372–2375.CrossRefGoogle Scholar
Fonseca, I. & Leoni, G. (2007) Modern Methods in the Calculus of Variations: $L^p$ Spaces, Springer Monographs in Mathematics, Springer, New York.Google Scholar
Gamba, I. M., Panferov, V. & Villani, C. (2004) On the Boltzmann equation for diffusively excited granular media. Comm. Math. Phys. 246, 503–541.Google Scholar
Haas, B. (2003) Loss of mass in deterministic and random fragmentations. Stochastic Process. Appl. 106, 245–277.CrossRefGoogle Scholar
Jeon, I. (2002) Stochastic fragmentation and some sufficient conditions for shattering transition. J. Korean Math. Soc. 39, 543–558.CrossRefGoogle Scholar
Kato, T. (1972) Schrödinger operators with singular potentials. Israel J. Math. 13, 135–148.CrossRefGoogle Scholar
Kinderlehrer, D. & Stampacchia, G. (2000) An Introduction to Variational Inequalities and their Applications, Classics in Applied Mathematics, Vol. 31, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. Reprint of the 1980 original.Google Scholar
Laurençot, Ph. (2004) Steady states for a fragmentation equation with size diffusion. In: Nonlocal Elliptic and Parabolic Problems, Banach Center Publications, Vol. 66, Mathematical Institute of the Polish Academy of Sciences, Warsaw, pp. 211–219.CrossRefGoogle Scholar
Laurençot, Ph. & Walker, Ch. (2021) The fragmentation equation with size diffusion: small and large size behavior of stationary solutions. Kinet. Relat. Models.CrossRefGoogle Scholar
Mathiesen, J., Ferkinghoff-Borg, J., Jensen, M. H., Levinsen, M., Olesen, P., Dahl-Jensen, D. & Svenson, A. (2004) Dynamics of crystal formation in the greenland NorthGRIP ice core. J. Glaciol. 50, 325–328.CrossRefGoogle Scholar
McGrady, E. D. & Ziff, R. M. (1987) “Shattering” transition in fragmentation. Phys. Rev. Lett. 58, 892–895.CrossRefGoogle ScholarPubMed
Michel, P., Mischler, S. & Perthame, B. (2005) General relative entropy inequality: an illustration on growth models. J. Math. Pures Appl. 84(9), 1235–1260.CrossRefGoogle Scholar
Pazy, A. (1983) Semigroups of Linear Operators and Applications to Partial Differential Equations , Applied Mathematical Sciences, Vol. 44, Springer-Verlag, New York.Google Scholar
Voigt, J. (1987) On substochastic $C_0$ -semigroups and their generators. Trans. Theory Statist. Phys. 16, 453–466.Google Scholar
Ziff, R. M. & McGrady, E. D. (1985) The kinetics of cluster fragmentation and depolymerisation. J. Phys. A 18, 3027–3037.CrossRefGoogle Scholar