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Fully conservative spectral Galerkin–Petrov method for the inhomogeneous Boltzmann equation
Cyclic asymptotic behaviour of a population reproducing by fission into two equal parts
1. | Laboratoire de Géodésie, IGN-LAREG, Bâtiment Lamarck A et B, 35 rue Hélène Brion, 75013 Paris, France |
2. | Sorbonne Universités, Inria, UPMC Univ Paris 06, Mamba project-team, Laboratoire Jacques-Louis Lions, Paris, France |
3. | Wolfgang Pauli Institute, c/o Faculty of Mathematics of the University of Vienna, Vienna, Austria |
4. | Laboratoire de Mathématiques de Versailles, UVSQ, CNRS, Université Paris-Saclay, 45 Avenue des États-Unis, 78035 Versailles cedex, France |
$ \frac{\partial}{\partial t} u(t,x) + \dfrac{\partial}{ \partial x} \big(x u(t,x)\big) + B(x) u(t,x) = 4 B(2x)u(t,2x), $ |
$ B(x), $ |
$ L^2 $ |
References:
[1] |
W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One-parameter Semigroups of Positive Operators, vol. 1184 of Lecture Notes in Mathematics, Berlin, 1986.
doi: 10.1007/BFb0074922. |
[2] |
O. Arino,
Some spectral properties for the asymptotic behavior of semigroups connected to population dynamics, SIAM Rev., 34 (1992), 445-476.
doi: 10.1137/1034086. |
[3] |
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. |
[4] |
J. Banasiak,
On a non-uniqueness in fragmentation models, Math. Methods Appl. Sci., 25 (2002), 541-556.
doi: 10.1002/mma.301. |
[5] |
J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics, Springer-Verlag, London, 2006. |
[6] |
J. Banasiak and W. Lamb,
The discrete fragmentation equation: Semigroups, compactness and asynchronous exponential growth, Kinet. Relat. Models, 5 (2012), 223-236.
doi: 10.3934/krm.2012.5.223. |
[7] |
J. Banasiak, K. Pichór and R. Rudnicki,
Asynchronous exponential growth of a general structured population model, Acta Appl. Math., 119 (2012), 149-166.
doi: 10.1007/s10440-011-9666-y. |
[8] |
G. I. Bell,
Cell growth and division: Ⅲ. conditions for balanced exponential growth in a mathematical model, Biophys. J., 8 (1968), 431-444.
doi: 10.1016/S0006-3495(68)86498-7. |
[9] |
G. I. Bell and E. C. Anderson,
Cell growth and division: I. a mathematical model with applications to cell volume distributions in mammalian suspension cultures, Biophys. J., 7 (1967), 329-351.
doi: 10.1016/S0006-3495(67)86592-5. |
[10] |
E. Bernard and P. Gabriel, Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate, preprint, arXiv: 1809.10974. |
[11] |
J. Bertoin,
The asymptotic behavior of fragmentation processes, J. Eur. Math. Soc., 5 (2003), 395-416.
doi: 10.1007/s10097-003-0055-3. |
[12] |
J. Bertoin and A. R. Watson,
Probabilistic aspects of critical growth-fragmentation equations, Adv. in Appl. Probab., 48 (2016), 37-61.
doi: 10.1017/apr.2016.41. |
[13] |
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., 96 (2011), 334-362.
doi: 10.1016/j.matpur.2011.01.003. |
[14] |
B. Cloez,
Limit theorems for some branching measure-valued processes, Adv. in Appl. Probab., 49 (2017), 549-580.
doi: 10.1017/apr.2017.12. |
[15] |
O. Diekmann, H. J. A. M. Heijmans and H. R. Thieme,
On the stability of the cell size distribution, J. Math. Biol., 19 (1984), 227-248.
doi: 10.1007/BF00277748. |
[16] |
M. Doumic and M. Escobedo,
Time asymptotics for a critical case in fragmentation and growth-fragmentation equations, Kinet. Relat. Models, 9 (2016), 251-297.
doi: 10.3934/krm.2016.9.251. |
[17] |
M. Doumic and P. Gabriel,
Eigenelements of a general aggregation-fragmentation model, Math. Models Methods Appl. Sci., 20 (2010), 757-783.
doi: 10.1142/S021820251000443X. |
[18] |
M. Doumic, M. Hoffmann, N. Krell and L. Robert,
Statistical estimation of a growth-fragmentation model observed on a genealogical tree, Bernoulli, 21 (2015), 1760-1799.
doi: 10.3150/14-BEJ623. |
[19] |
K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. |
[20] |
M. Escobedo, S. Mischler and M. Rodriguez Ricard,
On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 99-125.
doi: 10.1016/j.anihpc.2004.06.001. |
[21] |
P. Gabriel and F. Salvarani,
Exponential relaxation to self-similarity for the superquadratic fragmentation equation, Appl. Math. Lett., 27 (2014), 74-78.
doi: 10.1016/j.aml.2013.08.001. |
[22] |
G. Greiner and R. Nagel, Growth of cell populations via one-parameter semigroups of positive
operators, in Mathematics Applied to Science, Academic Press, Boston, MA, 1988, 79–105. |
[23] |
P. Gwiazda and E. Wiedemann,
Generalized entropy method for the renewal equation with measure data, Commun. Math. Sci., 15 (2017), 577-586.
doi: 10.4310/CMS.2017.v15.n2.a13. |
[24] |
B. Haas,
Asymptotic behavior of solutions of the fragmentation equation with shattering: an approach via self-similar Markov processes, Ann. Appl. Probab., 20 (2010), 382-429.
doi: 10.1214/09-AAP622. |
[25] |
A. J. Hall and G. C. Wake,
Functional-differential equations determining steady size distributions for populations of cells growing exponentially, J. Austral. Math. Soc. Ser. B, 31 (1990), 434-453.
doi: 10.1017/S0334270000006779. |
[26] |
H. J. A. M. Heijmans,
An eigenvalue problem related to cell growth, J. Math. Anal. Appl., 111 (1985), 253-280.
doi: 10.1016/0022-247X(85)90215-X. |
[27] |
P. Laurençot, B. Niethammer and J. J. L. Velázquez,
Oscillatory dynamics in Smoluchowski's coagulation equation with diagonal kernel, Kinet. Relat. Models, 11 (2018), 933-952.
doi: 10.3934/krm.2018037. |
[28] |
P. Laurençot and B. Perthame,
Exponential decay for the growth-fragmentation/cell-division equation, Commun. Math. Sci., 7 (2009), 503-510.
doi: 10.4310/CMS.2009.v7.n2.a12. |
[29] |
P. Michel, S. Mischler and B. Perthame,
General entropy equations for structured population models and scattering, C. R. Math. Acad. Sci. Paris, 338 (2004), 697-702.
doi: 10.1016/j.crma.2004.03.006. |
[30] |
P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration
on growth models, J. Math. Pures Appl. (9), 84 (2005), 1235–1260.
doi: 10.1016/j.matpur.2005.04.001. |
[31] |
S. Mischler and J. Scher,
Spectral analysis of semigroups and growth-fragmentation equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 849-898.
doi: 10.1016/j.anihpc.2015.01.007. |
[32] |
K. Pakdaman, B. Perthame and D. Salort, Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation, J. Math. Neurosci., 4 (2014), Art. 14, 26 pp.
doi: 10.1186/2190-8567-4-14. |
[33] |
B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. |
[34] |
B. Perthame and L. Ryzhik,
Exponential decay for the fragmentation or cell-division equation, J. Differential Equations, 210 (2005), 155-177.
doi: 10.1016/j.jde.2004.10.018. |
[35] |
J. Sinko and W. Streifer,
A model for populations reproducing by fission, Ecology, 52 (1971), 330-335.
doi: 10.2307/1934592. |
[36] |
C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), ⅳ+141pp.
doi: 10.1090/S0065-9266-09-00567-5. |
[37] |
A. A. Zaidi, B. Van Brunt and G. C. Wake, Solutions to an advanced functional partial differential equation of the pantograph type, Proc. A., 471 (2015), 20140947, 15pp.
doi: 10.1098/rspa.2014.0947. |
[38] |
A. A. Zaidi, B. van Brunt and G. C. Wake,
A model for asymmetrical cell division, Math. Biosc. Eng., 12 (2015), 491-501.
doi: 10.3934/mbe.2015.12.491. |
show all references
References:
[1] |
W. Arendt, A. Grabosch, G. Greiner, U. Groh, H. P. Lotz, U. Moustakas, R. Nagel, F. Neubrander and U. Schlotterbeck, One-parameter Semigroups of Positive Operators, vol. 1184 of Lecture Notes in Mathematics, Berlin, 1986.
doi: 10.1007/BFb0074922. |
[2] |
O. Arino,
Some spectral properties for the asymptotic behavior of semigroups connected to population dynamics, SIAM Rev., 34 (1992), 445-476.
doi: 10.1137/1034086. |
[3] |
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. |
[4] |
J. Banasiak,
On a non-uniqueness in fragmentation models, Math. Methods Appl. Sci., 25 (2002), 541-556.
doi: 10.1002/mma.301. |
[5] |
J. Banasiak and L. Arlotti, Perturbations of Positive Semigroups with Applications, Springer Monographs in Mathematics, Springer-Verlag, London, 2006. |
[6] |
J. Banasiak and W. Lamb,
The discrete fragmentation equation: Semigroups, compactness and asynchronous exponential growth, Kinet. Relat. Models, 5 (2012), 223-236.
doi: 10.3934/krm.2012.5.223. |
[7] |
J. Banasiak, K. Pichór and R. Rudnicki,
Asynchronous exponential growth of a general structured population model, Acta Appl. Math., 119 (2012), 149-166.
doi: 10.1007/s10440-011-9666-y. |
[8] |
G. I. Bell,
Cell growth and division: Ⅲ. conditions for balanced exponential growth in a mathematical model, Biophys. J., 8 (1968), 431-444.
doi: 10.1016/S0006-3495(68)86498-7. |
[9] |
G. I. Bell and E. C. Anderson,
Cell growth and division: I. a mathematical model with applications to cell volume distributions in mammalian suspension cultures, Biophys. J., 7 (1967), 329-351.
doi: 10.1016/S0006-3495(67)86592-5. |
[10] |
E. Bernard and P. Gabriel, Asynchronous exponential growth of the growth-fragmentation equation with unbounded fragmentation rate, preprint, arXiv: 1809.10974. |
[11] |
J. Bertoin,
The asymptotic behavior of fragmentation processes, J. Eur. Math. Soc., 5 (2003), 395-416.
doi: 10.1007/s10097-003-0055-3. |
[12] |
J. Bertoin and A. R. Watson,
Probabilistic aspects of critical growth-fragmentation equations, Adv. in Appl. Probab., 48 (2016), 37-61.
doi: 10.1017/apr.2016.41. |
[13] |
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., 96 (2011), 334-362.
doi: 10.1016/j.matpur.2011.01.003. |
[14] |
B. Cloez,
Limit theorems for some branching measure-valued processes, Adv. in Appl. Probab., 49 (2017), 549-580.
doi: 10.1017/apr.2017.12. |
[15] |
O. Diekmann, H. J. A. M. Heijmans and H. R. Thieme,
On the stability of the cell size distribution, J. Math. Biol., 19 (1984), 227-248.
doi: 10.1007/BF00277748. |
[16] |
M. Doumic and M. Escobedo,
Time asymptotics for a critical case in fragmentation and growth-fragmentation equations, Kinet. Relat. Models, 9 (2016), 251-297.
doi: 10.3934/krm.2016.9.251. |
[17] |
M. Doumic and P. Gabriel,
Eigenelements of a general aggregation-fragmentation model, Math. Models Methods Appl. Sci., 20 (2010), 757-783.
doi: 10.1142/S021820251000443X. |
[18] |
M. Doumic, M. Hoffmann, N. Krell and L. Robert,
Statistical estimation of a growth-fragmentation model observed on a genealogical tree, Bernoulli, 21 (2015), 1760-1799.
doi: 10.3150/14-BEJ623. |
[19] |
K.-J. Engel and R. Nagel, One-parameter Semigroups for Linear Evolution Equations, vol. 194 of Graduate Texts in Mathematics, Springer-Verlag, New York, 2000. |
[20] |
M. Escobedo, S. Mischler and M. Rodriguez Ricard,
On self-similarity and stationary problem for fragmentation and coagulation models, Ann. Inst. H. Poincaré Anal. Non Linéaire, 22 (2005), 99-125.
doi: 10.1016/j.anihpc.2004.06.001. |
[21] |
P. Gabriel and F. Salvarani,
Exponential relaxation to self-similarity for the superquadratic fragmentation equation, Appl. Math. Lett., 27 (2014), 74-78.
doi: 10.1016/j.aml.2013.08.001. |
[22] |
G. Greiner and R. Nagel, Growth of cell populations via one-parameter semigroups of positive
operators, in Mathematics Applied to Science, Academic Press, Boston, MA, 1988, 79–105. |
[23] |
P. Gwiazda and E. Wiedemann,
Generalized entropy method for the renewal equation with measure data, Commun. Math. Sci., 15 (2017), 577-586.
doi: 10.4310/CMS.2017.v15.n2.a13. |
[24] |
B. Haas,
Asymptotic behavior of solutions of the fragmentation equation with shattering: an approach via self-similar Markov processes, Ann. Appl. Probab., 20 (2010), 382-429.
doi: 10.1214/09-AAP622. |
[25] |
A. J. Hall and G. C. Wake,
Functional-differential equations determining steady size distributions for populations of cells growing exponentially, J. Austral. Math. Soc. Ser. B, 31 (1990), 434-453.
doi: 10.1017/S0334270000006779. |
[26] |
H. J. A. M. Heijmans,
An eigenvalue problem related to cell growth, J. Math. Anal. Appl., 111 (1985), 253-280.
doi: 10.1016/0022-247X(85)90215-X. |
[27] |
P. Laurençot, B. Niethammer and J. J. L. Velázquez,
Oscillatory dynamics in Smoluchowski's coagulation equation with diagonal kernel, Kinet. Relat. Models, 11 (2018), 933-952.
doi: 10.3934/krm.2018037. |
[28] |
P. Laurençot and B. Perthame,
Exponential decay for the growth-fragmentation/cell-division equation, Commun. Math. Sci., 7 (2009), 503-510.
doi: 10.4310/CMS.2009.v7.n2.a12. |
[29] |
P. Michel, S. Mischler and B. Perthame,
General entropy equations for structured population models and scattering, C. R. Math. Acad. Sci. Paris, 338 (2004), 697-702.
doi: 10.1016/j.crma.2004.03.006. |
[30] |
P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration
on growth models, J. Math. Pures Appl. (9), 84 (2005), 1235–1260.
doi: 10.1016/j.matpur.2005.04.001. |
[31] |
S. Mischler and J. Scher,
Spectral analysis of semigroups and growth-fragmentation equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 33 (2016), 849-898.
doi: 10.1016/j.anihpc.2015.01.007. |
[32] |
K. Pakdaman, B. Perthame and D. Salort, Adaptation and fatigue model for neuron networks and large time asymptotics in a nonlinear fragmentation equation, J. Math. Neurosci., 4 (2014), Art. 14, 26 pp.
doi: 10.1186/2190-8567-4-14. |
[33] |
B. Perthame, Transport Equations in Biology, Frontiers in Mathematics, Birkhäuser Verlag, Basel, 2007. |
[34] |
B. Perthame and L. Ryzhik,
Exponential decay for the fragmentation or cell-division equation, J. Differential Equations, 210 (2005), 155-177.
doi: 10.1016/j.jde.2004.10.018. |
[35] |
J. Sinko and W. Streifer,
A model for populations reproducing by fission, Ecology, 52 (1971), 330-335.
doi: 10.2307/1934592. |
[36] |
C. Villani, Hypocoercivity, Mem. Amer. Math. Soc., 202 (2009), ⅳ+141pp.
doi: 10.1090/S0065-9266-09-00567-5. |
[37] |
A. A. Zaidi, B. Van Brunt and G. C. Wake, Solutions to an advanced functional partial differential equation of the pantograph type, Proc. A., 471 (2015), 20140947, 15pp.
doi: 10.1098/rspa.2014.0947. |
[38] |
A. A. Zaidi, B. van Brunt and G. C. Wake,
A model for asymmetrical cell division, Math. Biosc. Eng., 12 (2015), 491-501.
doi: 10.3934/mbe.2015.12.491. |


Left: peak in

Left: for the peak as initial condition. Right: for the smooth initial condition.


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