American Institute of Mathematical Sciences

Advanced
April  2018, 38(4): 2047-2064. doi: 10.3934/dcds.2018083

Well-posedness of a model for the growth of tree stems and vines

Alberto Bressan , and  Michele Palladino

Department of Mathematics, Penn State University, University Park, PA, 16802, USA

* Corresponding author: Prof. Alberto Bressan

Received  April 2017 Revised  August 2017 Published  January 2018

Fund Project: The first author is supported by NSF grant DMS-1714237, "Models of controlled biological growth".

The paper studies a PDE model for the growth of a tree stem or a vine, having the form of a differential inclusion with state constraints. The equations describe the elongation due to cell growth, and the response to gravity and to external obstacles.

The main theorem shows that the evolution problem is well posed, until a specific "breakdown configuration" is reached. A formula is proved, characterizing the reaction produced by unilateral constraints. At a.e. time $t$, this is determined by the minimization of an elastic energy functional under suitable constraints.

Keywords: Evolution equations, integro-differential equations, one-sided constraints, optimality conditions, mathematical biology.
Mathematics Subject Classification: Primary: 49K99, 35Q92.
Citation: Alberto Bressan, Michele Palladino. Well-posedness of a model for the growth of tree stems and vines. Discrete & Continuous Dynamical Systems, 2018, 38 (4) : 2047-2064. doi: 10.3934/dcds.2018083
References:
[1]

A. Bressan, M. Palladino and W. Shen, Growth models for tree stems and vines, J. Differential Equations, 263 (2017), 2280-2316.  doi: 10.1016/j.jde.2017.03.047.  Google Scholar

[2]

L. Cesari, Optimization -Theory and Applications, Springer-Verlag, 1983.  Google Scholar

[3]

G. Colombo and V. Goncharov, The sweeping processes without convexity, Set-Valued Anal., 7 (1999), 357-374.  doi: 10.1023/A:1008774529556.  Google Scholar

[4]

G. Colombo and M. Monteiro Marques, Sweeping by a continuous prox-regular set, J. Differential Equations, 187 (2003), 46-62.  doi: 10.1016/S0022-0396(02)00021-9.  Google Scholar

[5]

O. Leyser and S. Day, Mechanisms in Plant Development, Blackwell Publishing, 2003. Google Scholar

[6]

J. J. Moreau, Evolution problems associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374.  doi: 10.1016/0022-0396(77)90085-7.  Google Scholar

[7]

R. Rossi and U. Stefanelli, An order approach to a class of quasivariational sweeping processes, Adv. Diff. Equat., 10 (2005), 527-552.   Google Scholar

[8]

R. B. Vinter, Optimal Control, Birkhäuser, Boston, 2000.  Google Scholar

show all references

References:
[1]

A. Bressan, M. Palladino and W. Shen, Growth models for tree stems and vines, J. Differential Equations, 263 (2017), 2280-2316.  doi: 10.1016/j.jde.2017.03.047.  Google Scholar

[2]

L. Cesari, Optimization -Theory and Applications, Springer-Verlag, 1983.  Google Scholar

[3]

G. Colombo and V. Goncharov, The sweeping processes without convexity, Set-Valued Anal., 7 (1999), 357-374.  doi: 10.1023/A:1008774529556.  Google Scholar

[4]

G. Colombo and M. Monteiro Marques, Sweeping by a continuous prox-regular set, J. Differential Equations, 187 (2003), 46-62.  doi: 10.1016/S0022-0396(02)00021-9.  Google Scholar

[5]

O. Leyser and S. Day, Mechanisms in Plant Development, Blackwell Publishing, 2003. Google Scholar

[6]

J. J. Moreau, Evolution problems associated with a moving convex set in a Hilbert space, J. Differential Equations, 26 (1977), 347-374.  doi: 10.1016/0022-0396(77)90085-7.  Google Scholar

[7]

R. Rossi and U. Stefanelli, An order approach to a class of quasivariational sweeping processes, Adv. Diff. Equat., 10 (2005), 527-552.   Google Scholar

[8]

R. B. Vinter, Optimal Control, Birkhäuser, Boston, 2000.  Google Scholar

Figure 1.  Left: at any point $\gamma(t,\sigma)$ along the stem, an infinitesimal change in curvature is produced as a response to gravity (or stems of other plants). The angular velocity is given by the vector $\omega(\sigma)$. This affects the position of all higher points along the stem. Right: At a given time $t$, the curve $\gamma(t,\cdot)$ is parameterized by $s\in [0,t]$. It is convenient to prolong this curve by adding a segment of length $T-t$ at its tip (dotted line, possibly entering inside the obstacle). This yields an evolution equation on a fixed functional space $H^2([0,T];\,\mathbb{R}^3)$.
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  For the two initial configurations on the left, the constrained growth equation (8) admits a unique solution. On the other hand, the two configurations on the right satisfy both (26) and (27) in (B). In such cases, the Cauchy problem is ill posed.
Figure Options

Download full-size image

Download as PowerPoint slide

$\Gamma_3$ is tangent to the boundary of the set $S$. Here the shaded region is the complement of $S$.">Figure 2.  Left: three configurations of the stem, relative to the obstacle. Right: in an abstract space, the first two configurations are represented by points $\gamma_1,\gamma_2$ on the boundary of the admissible set $S$ where the corresponding cones $\Gamma_1,\Gamma_2$ are transversal. On the other hand, $\gamma_3$ is a "breakdown configuration", satisfying all assumptions (26)-(27). Its corresponding cone $\Gamma_3$ is tangent to the boundary of the set $S$. Here the shaded region is the complement of $S$.
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17
[2]

Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409
[3]

Piermarco Cannarsa, Vilmos Komornik, Paola Loreti. One-sided and internal controllability of semilinear wave equations with infinitely iterated logarithms. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 745-756. doi: 10.3934/dcds.2002.8.747
[4]

Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057
[5]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Singular integro-differential equations with applications. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021051
[6]

Dariusz Idczak, Stanisław Walczak. Necessary optimality conditions for an integro-differential Bolza problem via Dubovitskii-Milyutin method. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2281-2292. doi: 10.3934/dcdsb.2019095
[7]

Ji Shu, Linyan Li, Xin Huang, Jian Zhang. Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020044
[8]

Yi Cao, Jianhua Wu, Lihe Wang. Fundamental solutions of a class of homogeneous integro-differential elliptic equations. Discrete & Continuous Dynamical Systems, 2019, 39 (3) : 1237-1256. doi: 10.3934/dcds.2019053
[9]

Yubo Chen, Wan Zhuang. The extreme solutions of PBVP for integro-differential equations with caratheodory functions. Conference Publications, 1998, 1998 (Special) : 160-166. doi: 10.3934/proc.1998.1998.160
[10]

Martin Bohner, Osman Tunç. Qualitative analysis of integro-differential equations with variable retardation. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021059
[11]

Ramasamy Subashini, Chokkalingam Ravichandran, Kasthurisamy Jothimani, Haci Mehmet Baskonus. Existence results of Hilfer integro-differential equations with fractional order. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 911-923. doi: 10.3934/dcdss.2020053
[12]

Tonny Paul, A. Anguraj. Existence and uniqueness of nonlinear impulsive integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2006, 6 (5) : 1191-1198. doi: 10.3934/dcdsb.2006.6.1191
[13]

Narcisa Apreutesei, Arnaud Ducrot, Vitaly Volpert. Travelling waves for integro-differential equations in population dynamics. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 541-561. doi: 10.3934/dcdsb.2009.11.541
[14]

Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977
[15]

Sertan Alkan. A new solution method for nonlinear fractional integro-differential equations. Discrete & Continuous Dynamical Systems - S, 2015, 8 (6) : 1065-1077. doi: 10.3934/dcdss.2015.8.1065
[16]

Eitan Tadmor, Prashant Athavale. Multiscale image representation using novel integro-differential equations. Inverse Problems & Imaging, 2009, 3 (4) : 693-710. doi: 10.3934/ipi.2009.3.693
[17]

Patricio Felmer, Ying Wang. Qualitative properties of positive solutions for mixed integro-differential equations. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 369-393. doi: 10.3934/dcds.2019015
[18]

Sebti Kerbal, Yang Jiang. General integro-differential equations and optimal controls on Banach spaces. Journal of Industrial & Management Optimization, 2007, 3 (1) : 119-128. doi: 10.3934/jimo.2007.3.119
[19]

Luis Barreira, Davor Dragičević, Claudia Valls. From one-sided dichotomies to two-sided dichotomies. Discrete & Continuous Dynamical Systems, 2015, 35 (7) : 2817-2844. doi: 10.3934/dcds.2015.35.2817
[20]

Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete & Continuous Dynamical Systems, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (136)
  • HTML views (261)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences