# American Institute of Mathematical Sciences

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 & Continuous Dynamical Systems - A, 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. [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. 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. [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.
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 $H^2([0,T];\,\mathbb{R}^3)$.
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.
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 & Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17 [2] 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 [3] 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 [4] Piermarco Cannarsa, Vilmos Komornik, Paola Loreti. One-sided and internal controllability of semilinear wave equations with infinitely iterated logarithms. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 745-756. doi: 10.3934/dcds.2002.8.747 [5] 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 [6] 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 [7] 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 [8] 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 [9] Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977 [10] 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 [11] 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 [12] Luis Barreira, Davor Dragičević, Claudia Valls. From one-sided dichotomies to two-sided dichotomies. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 2817-2844. doi: 10.3934/dcds.2015.35.2817 [13] Frederic Abergel, Remi Tachet. A nonlinear partial integro-differential equation from mathematical finance. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 907-917. doi: 10.3934/dcds.2010.27.907 [14] Kyudong Choi. Persistence of Hölder continuity for non-local integro-differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (5) : 1741-1771. doi: 10.3934/dcds.2013.33.1741 [15] Xu Chen, Jianping Wan. Integro-differential equations for foreign currency option prices in exponential Lévy models. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 529-537. doi: 10.3934/dcdsb.2007.8.529 [16] Jaan Janno, Kairi Kasemets. A positivity principle for parabolic integro-differential equations and inverse problems with final overdetermination. Inverse Problems & Imaging, 2009, 3 (1) : 17-41. doi: 10.3934/ipi.2009.3.17 [17] Eduardo Cuesta. Asymptotic behaviour of the solutions of fractional integro-differential equations and some time discretizations. Conference Publications, 2007, 2007 (Special) : 277-285. doi: 10.3934/proc.2007.2007.277 [18] Hermann Brunner. The numerical solution of weakly singular Volterra functional integro-differential equations with variable delays. Communications on Pure & Applied Analysis, 2006, 5 (2) : 261-276. doi: 10.3934/cpaa.2006.5.261 [19] Cyril Imbert, Sylvia Serfaty. Repeated games for non-linear parabolic integro-differential equations and integral curvature flows. Discrete & Continuous Dynamical Systems - A, 2011, 29 (4) : 1517-1552. doi: 10.3934/dcds.2011.29.1517 [20] Luis Silvestre. Hölder continuity for integro-differential parabolic equations with polynomial growth respect to the gradient. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1069-1081. doi: 10.3934/dcds.2010.28.1069

2016 Impact Factor: 1.099

## Metrics

• HTML views (154)
• Cited by (0)