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

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

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.

Citation: Alberto Bressan, Michele Palladino. Well-posedness of a model for the growth of tree stems and vines. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 2047-2064. doi: 10.3934/dcds.2018083
References:
[1]

A. BressanM. 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.

[2]

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

[3]

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

[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.

[5]

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

[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.

[7]

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

[8]

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

show all references

References:
[1]

A. BressanM. 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.

[2]

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

[3]

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

[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.

[5]

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

[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.

[7]

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

[8]

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

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 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 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 $S$.
[1]

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

[2]

Matthieu Alfaro, Pierre Gabriel, Otared Kavian. Confining integro-differential equations originating from evolutionary biology: Ground states and long time dynamics. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022120

[3]

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

[4]

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

[5]

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

[6]

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

[7]

Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems on degenerate integro-differential equations. Discrete and Continuous Dynamical Systems - S, 2022  doi: 10.3934/dcdss.2022025

[8]

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

[9]

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

[10]

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

[11]

Ramasamy Subashini, Chokkalingam Ravichandran, Kasthurisamy Jothimani, Haci Mehmet Baskonus. Existence results of Hilfer integro-differential equations with fractional order. Discrete and 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 and 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 and 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 and 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 and 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 and 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 and 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 and Management Optimization, 2007, 3 (1) : 119-128. doi: 10.3934/jimo.2007.3.119

[19]

Ji Shu, Linyan Li, Xin Huang, Jian Zhang. Limiting behavior of fractional stochastic integro-Differential equations on unbounded domains. Mathematical Control and Related Fields, 2021, 11 (4) : 715-737. doi: 10.3934/mcrf.2020044

[20]

Martin Bohner, Osman Tunç. Qualitative analysis of integro-differential equations with variable retardation. Discrete and Continuous Dynamical Systems - B, 2022, 27 (2) : 639-657. doi: 10.3934/dcdsb.2021059

2021 Impact Factor: 1.588

Metrics

  • PDF downloads (203)
  • HTML views (262)
  • Cited by (0)

Other articles
by authors

[Back to Top]