American Institute of Mathematical Sciences

doi: 10.3934/krm.2021032
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The fragmentation equation with size diffusion: Small and large size behavior of stationary solutions

 1 Institut de Mathématiques de Toulouse, UMR 5219, Université de Toulouse, CNRS, F–31062 Toulouse Cedex 9, France 2 Leibniz Universität Hannover, Institut für Angewandte Mathematik, Welfengarten 1, D–30167 Hannover, Germany

Received  May 2021 Revised  September 2021 Early access November 2021

Fund Project: The first author is partially supported by Deutscher Akademischer Austauschdienst funding programme Research Stays for University Academics and Scientists, 2021 (57552334)

The small and large size behavior of stationary solutions to the fragmentation equation with size diffusion is investigated. It is shown that these solutions behave like stretched exponentials for large sizes, the exponent in the exponential being solely given by the behavior of the overall fragmentation rate at infinity. In contrast, the small size behavior is partially governed by the daughter fragmentation distribution and is at most linear, with possibly non-algebraic behavior. Explicit solutions are also provided for particular fragmentation coefficients.

Citation: Philippe Laurençot, Christoph Walker. The fragmentation equation with size diffusion: Small and large size behavior of stationary solutions. Kinetic & Related Models, doi: 10.3934/krm.2021032
References:
 [1] R. Alonso, J. A. Cañizo, I. Gamba and C. Mouhot, A new approach to the creation and propagation of exponential moments in the Boltzmann equation, Commun. Partial Differ. Equations, 38 (2013), 155-169.  doi: 10.1080/03605302.2012.715707.  Google Scholar [2] D. Balagué, J. A. Cañizo and P. Gabriel, Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates, Kinet. Relat. Models, 6 (2013), 219-243.  doi: 10.3934/krm.2013.6.219.  Google Scholar [3] J. Banasiak, W. Lamb and Ph. Laurençot, Analytic Methods for Coagulation-Fragmentation Models, Vol. Ⅱ, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2020.   Google Scholar [4] W. Biedrzycka and M. Tyran-Kamińska, Self-similar solutions of fragmentation equations revisited, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 13-27.  doi: 10.3934/dcdsb.2018002.  Google Scholar [5] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, vol. 27 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9780511721434.  Google Scholar [6] M. J. Cáceres, J. A. Cañizo and S. Mischler, Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations, J. Math. Pures Appl. (9), 96 (2011), 334-362.  doi: 10.1016/j.matpur.2011.01.003.  Google Scholar [7] \it NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.1.1 of 2021-03-15, F.W.J. Olver, A.B. {Olde Daalhuis}, D.W. Lozier, B.I. Schneider, R.F. Boisvert, C.W. Clark, B.R. Miller, B.V. Saunders, H.S. Cohl, and M.A. McClain, eds. Google Scholar [8] J. Ferkinghoff-Borg, M. H. Jensen, J. Mathiesen, P. Olesen and K. Sneppen, Competition between diffusion and fragmentation: An important evolutionary process of nature, Phys. Rev. Lett., 91 (2003), 266103.  doi: 10.1103/PhysRevLett.91.266103.  Google Scholar [9] A. F. Filippov, On the distribution of the sizes of particles which undergo splitting, Theory Probab. Appl., 6 (1961), 275-294.  doi: 10.1137/1106036.  Google Scholar [10] I. M. Gamba, V. Panferov and C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, Arch. Ration. Mech. Anal., 194 (2009), 253-282.  doi: 10.1007/s00205-009-0250-9.  Google Scholar [11] I. M. Gamba, N. Pavlović and M. Tasković, On pointwise exponentially weighted estimates for the Boltzmann equation, SIAM J. Math. Anal., 51 (2019), 3921-3955.  doi: 10.1137/18M1213191.  Google Scholar [12] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition.  Google Scholar [13] Ph. Laurençot, Steady states for a fragmentation equation with size diffusion, in Nonlocal Elliptic and Parabolic Problems, vol. 66 of Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw, 2004,211–219. doi: 10.4064/bc66-0-14.  Google Scholar [14] Ph. Laurençot and Ch. Walker, The fragmentation equation with size diffusion: Well-posedness and long-term behavior, 2021, arXiv: 2104.14798. Google Scholar [15] J. Mathiesen, J. Ferkinghoff-Borg, M. H. Jensen, M. Levinsen, P. Olesen, D. Dahl-Jensen and A. Svenson, Dynamics of crystal formation in the Greenland NorthGRIP ice core, J. Glaciol., 50 (2004), 325-328.  doi: 10.3189/172756504781829873.  Google Scholar [16] E. D. McGrady and R. M. Ziff, "Shattering" transition in fragmentation, Phys. Rev. Lett., 58 (1987), 892-895.  doi: 10.1103/PhysRevLett.58.892.  Google Scholar [17] M. Pavić-Čolić and M. Tasković, Propagation of stretched exponential moments for the Kac equation and Boltzmann equation with Maxwell molecules, Kinet. Relat. Models, 11 (2018), 597-613.  doi: 10.3934/krm.2018025.  Google Scholar [18] M. Tasković, R. J. Alonso, I. M. Gamba and N. Pavlović, On Mittag-Leffler moments for the Boltzmann equation for hard potentials without cutoff, SIAM J. Math. Anal., 50 (2018), 834-869.  doi: 10.1137/17M1117926.  Google Scholar

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References:
 [1] R. Alonso, J. A. Cañizo, I. Gamba and C. Mouhot, A new approach to the creation and propagation of exponential moments in the Boltzmann equation, Commun. Partial Differ. Equations, 38 (2013), 155-169.  doi: 10.1080/03605302.2012.715707.  Google Scholar [2] D. Balagué, J. A. Cañizo and P. Gabriel, Fine asymptotics of profiles and relaxation to equilibrium for growth-fragmentation equations with variable drift rates, Kinet. Relat. Models, 6 (2013), 219-243.  doi: 10.3934/krm.2013.6.219.  Google Scholar [3] J. Banasiak, W. Lamb and Ph. Laurençot, Analytic Methods for Coagulation-Fragmentation Models, Vol. Ⅱ, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2020.   Google Scholar [4] W. Biedrzycka and M. Tyran-Kamińska, Self-similar solutions of fragmentation equations revisited, Discrete Contin. Dyn. Syst. Ser. B, 23 (2018), 13-27.  doi: 10.3934/dcdsb.2018002.  Google Scholar [5] N. H. Bingham, C. M. Goldie and J. L. Teugels, Regular Variation, vol. 27 of Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9780511721434.  Google Scholar [6] M. J. Cáceres, J. A. Cañizo and S. Mischler, Rate of convergence to an asymptotic profile for the self-similar fragmentation and growth-fragmentation equations, J. Math. Pures Appl. (9), 96 (2011), 334-362.  doi: 10.1016/j.matpur.2011.01.003.  Google Scholar [7] \it NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, Release 1.1.1 of 2021-03-15, F.W.J. Olver, A.B. {Olde Daalhuis}, D.W. Lozier, B.I. Schneider, R.F. Boisvert, C.W. Clark, B.R. Miller, B.V. Saunders, H.S. Cohl, and M.A. McClain, eds. Google Scholar [8] J. Ferkinghoff-Borg, M. H. Jensen, J. Mathiesen, P. Olesen and K. Sneppen, Competition between diffusion and fragmentation: An important evolutionary process of nature, Phys. Rev. Lett., 91 (2003), 266103.  doi: 10.1103/PhysRevLett.91.266103.  Google Scholar [9] A. F. Filippov, On the distribution of the sizes of particles which undergo splitting, Theory Probab. Appl., 6 (1961), 275-294.  doi: 10.1137/1106036.  Google Scholar [10] I. M. Gamba, V. Panferov and C. Villani, Upper Maxwellian bounds for the spatially homogeneous Boltzmann equation, Arch. Ration. Mech. Anal., 194 (2009), 253-282.  doi: 10.1007/s00205-009-0250-9.  Google Scholar [11] I. M. Gamba, N. Pavlović and M. Tasković, On pointwise exponentially weighted estimates for the Boltzmann equation, SIAM J. Math. Anal., 51 (2019), 3921-3955.  doi: 10.1137/18M1213191.  Google Scholar [12] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Classics in Mathematics, Springer-Verlag, Berlin, 2001, Reprint of the 1998 edition.  Google Scholar [13] Ph. Laurençot, Steady states for a fragmentation equation with size diffusion, in Nonlocal Elliptic and Parabolic Problems, vol. 66 of Banach Center Publ., Polish Acad. Sci. Inst. Math., Warsaw, 2004,211–219. doi: 10.4064/bc66-0-14.  Google Scholar [14] Ph. Laurençot and Ch. Walker, The fragmentation equation with size diffusion: Well-posedness and long-term behavior, 2021, arXiv: 2104.14798. Google Scholar [15] J. Mathiesen, J. Ferkinghoff-Borg, M. H. Jensen, M. Levinsen, P. Olesen, D. Dahl-Jensen and A. Svenson, Dynamics of crystal formation in the Greenland NorthGRIP ice core, J. Glaciol., 50 (2004), 325-328.  doi: 10.3189/172756504781829873.  Google Scholar [16] E. D. McGrady and R. M. Ziff, "Shattering" transition in fragmentation, Phys. Rev. Lett., 58 (1987), 892-895.  doi: 10.1103/PhysRevLett.58.892.  Google Scholar [17] M. Pavić-Čolić and M. Tasković, Propagation of stretched exponential moments for the Kac equation and Boltzmann equation with Maxwell molecules, Kinet. Relat. Models, 11 (2018), 597-613.  doi: 10.3934/krm.2018025.  Google Scholar [18] M. Tasković, R. J. Alonso, I. M. Gamba and N. Pavlović, On Mittag-Leffler moments for the Boltzmann equation for hard potentials without cutoff, SIAM J. Math. Anal., 50 (2018), 834-869.  doi: 10.1137/17M1117926.  Google Scholar
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