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

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 $S$.