Growing into the Right Shape Alberto Bressan Department of - PowerPoint PPT Presentation

Growing into the Right Shape Alberto Bressan Department of Mathematics, Penn State University Center for Interdisciplinary Mathematics bressan@math.psu.edu Alberto Bressan (Penn State) growth models 1 / 47

PDE models in continuum physics Many geometric shapes found in Nature can be described in terms of PDEs Alberto Bressan (Penn State) growth models 2 / 47

catenary minimum surface vortex rollup shock waves Alberto Bressan (Penn State) growth models 3 / 47

However, many other interesting shapes are not found in PDE books Alberto Bressan (Penn State) growth models 4 / 47

leaf shapes Alberto Bressan (Penn State) growth models 5 / 47

flower shapes Alberto Bressan (Penn State) growth models 6 / 47

bone shapes Alberto Bressan (Penn State) growth models 7 / 47

Controlling the growth of living tissues For higher living forms (plants, animals), growing into the right shape is essential for survival How can Nature control growth, sometimes in an amazingly precise way? Can we write PDEs describing this feedback control mechanism? What is the simplest system of PDEs generating the shapes found in nature? “With four parameters I can fit an elephant, and with five I can make him wiggle his trunk” (John von Neumann) Alberto Bressan (Penn State) growth models 8 / 47

Controlling the growth of living tissues For higher living forms (plants, animals), growing into the right shape is essential for survival How can Nature control growth, sometimes in an amazingly precise way? Can we write PDEs describing this feedback control mechanism? What is the simplest system of PDEs generating the shapes found in nature? “With four parameters I can fit an elephant, and with five I can make him wiggle his trunk” (John von Neumann) Alberto Bressan (Penn State) growth models 8 / 47

Controlling the growth of living tissues For higher living forms (plants, animals), growing into the right shape is essential for survival How can Nature control growth, sometimes in an amazingly precise way? Can we write PDEs describing this feedback control mechanism? What is the simplest system of PDEs generating the shapes found in nature? “With four parameters I can fit an elephant, and with five I can make him wiggle his trunk” (John von Neumann) Alberto Bressan (Penn State) growth models 8 / 47

Two main settings One-dimensional curves, growing in R 3 (plant stems) Two-dimensional sets, growing in R 2 (leaves) numerical simulations (done) + analytical proofs (in progress) Alberto Bressan (Penn State) growth models 9 / 47

Two main settings One-dimensional curves, growing in R 3 (plant stems) Two-dimensional sets, growing in R 2 (leaves) numerical simulations (done) + analytical proofs (in progress) Alberto Bressan (Penn State) growth models 9 / 47

Stabilizing stem growth what kind of stabilizing feedback is used here? Alberto Bressan (Penn State) growth models 10 / 47

Growth in the presence of obstacles Are the growth equations still well posed, when an obstacle is present? What additional feedback produces curling around other branches? Alberto Bressan (Penn State) growth models 11 / 47

A model of stem growth (F. Ancona, A.B., O. Glass) New cells are born at the tip of the stem Their length grows in time, at an exponentially decreasing rate τ t = P (τ,τ) t = s 2 P( ,s ) τ 2 t = s 1 τ P( ,s ) 1 P ( t , s ) = position at time t of the cell born at time s d ℓ = | ∂ s P ( t , s ) | = (1 − e − α ( t − s ) ) ds Unit tangent vector to the stem: k ( t , s ) = ∂ s P ( t , s ) | ∂ s P ( t , s ) | Alberto Bressan (Penn State) growth models 12 / 47

Stabilizing growth in the vertical direction stem not vertical = ⇒ local change in curvature e − β ( t − s ) = stiffness factor, k ⊥ = ( − k 2 , k 1 ) k = ( k 1 , k 2 ), � s ∂ α e − α ( t − σ ) k ( t , σ ) d σ ∂ t P ( t , s ) = 0 � s µ e − β ( t − σ ) k 1 ( t , σ ) · � ⊥ (1 − e − α ( t − σ ) ) d σ + � P ( t , s ) − P ( t , σ ) 0 k (t,s) ⊥ e  P ( t 0 , s ) = P ( s ) s ∈ [0 , t 0 ] P(t,s) k (t, ) σ 2    k (t, ) σ � e � 1 P ss ( t , s ) = 0   �  P(t, ) σ � s = t 0 Alberto Bressan (Penn State) growth models 13 / 47

We say that the growth equation is stable in the vertical direction if for any initial time t 0 > 0 and every ε > 0 there exists δ > 0 such that � � � � � e 1 · ∂ � � ∂ s P ( t 0 , s ) � ≤ δ for all s ∈ [0 , t 0 ] � � � e 1 · P ( t , s ) � ≤ ε implies for all t > t 0 , s ∈ [0 , t ] P(t,s) e 2 P(t ,s) 0 e 1 x 1 0 ε Alberto Bressan (Penn State) growth models 14 / 47

Numerical simulations (Wen Shen) β = 0 . 1 β = 1 . 0 β = 2 . 5 stability is always achieved increasing the stiffness reduces oscillations Alberto Bressan (Penn State) growth models 15 / 47

Analytical results (F. Ancona, A.B., O. Glass) β = stiffening constant, µ = 1 (strength of response) If β 4 − β 3 − 4 ≥ 0, then the growth is stable in the vertical direction (non-oscillatory regime: β ≥ β ∗ ≈ 1 . 7485) If β ≥ β 0 for a suitable β 0 < 1, growth is still stable in the vertical direction (oscillatory regime) Stability apparently holds for all β > 0 (??) Alberto Bressan (Penn State) growth models 16 / 47

Alberto Bressan (Penn State) growth models 17 / 47

Growth with obstacles Alberto Bressan (Penn State) growth models 18 / 47

Ω Ω P(s) ~ P(s) (σ) P ω ( σ ) = additional bending of the stem caused the obstacle, at the point P ( σ ) � s � P ( s ) − P ( s ) = ω ( σ ) × ( P ( s ) − P ( σ )) d σ s ∈ [0 , t ] 0 Among all infinitesimal deformations that push the stem outside the obstacle, � t minimize the elastic energy: E = 1 e β ( t − σ ) | ω ( σ ) | 2 d σ 2 0 Alberto Bressan (Penn State) growth models 19 / 47

A cone of admissible reactions P(t) P(s ) ’ P(s) v(s) = { s ′ ∈ [0 , t ] ; . P ( s ′ ) ∈ ∂ Ω } χ ( t ) (contact set) � . v : [0 , t ] �→ R 3 ; Γ = there exists a positive measure µ supported on χ ( t ) such that � �� s � � e − β ( t − σ ) � �� n ( t , s ′ ) × P ( t , s ′ ) − P ( t , σ ) d µ ( s ′ ) � � � v ( s ) = × P ( t , s ) − P ( t , σ ) d σ 0 Alberto Bressan (Penn State) growth models 20 / 47

Motion in the presence of an obstacle x = f ( x ) , ˙ x ( t ) / ∈ Ω Ω ⊂ R n open, with smooth boundary f Lipschitz, Γ (x) N (x) Ω f f x x Ω Ω � ˙ x = f ( x ) if x / ∈ Ω differential inclusion: x ˙ ∈ f ( x ) + Γ( x ) if x ∈ ∂ Ω upper semicontinuity + convexity = ⇒ existence Alberto Bressan (Penn State) growth models 21 / 47

Well posedness (also with a moving obstacle) J. J. Moreau, Evolution problems associated with a moving convex set in a Hilbert space, J. Differential Equat. 26 (1977), 347–374. G. Colombo and V. Goncharov, The sweeping processes without convexity. Set-Valued Anal. 7 (1999), 357–374. G. Colombo and M. Monteiro Marques, Sweeping by a continuous prox-regular set. J. Differential Equat. 187 (2003), 46–62. R. Rossi and U. Stefanelli, An order approach to a class of quasivariational sweeping processes. Adv. Differential Equat. 10 (2005), 527–552. Alberto Bressan (Penn State) growth models 22 / 47

Continuous dependence If Γ( x ) = N Ω ( x ) = normal cone, then d dt | x 1 ( t ) − x 2 ( t ) | ≤ C | x 1 ( t ) − x 2 ( t ) | = ⇒ uniqueness, continuous dependence Γ (x) v 2 f x v 2 x x 2 2 x x 1 v v 1 Ω 1 Ω 1 Ω Possible approach when Γ( x ) � = N Ω ( x ): introduce a Riemann metric on the Hilbert space ( = R n or H 2 ( R ) ) which renders each Γ( x ) a normal cone Alberto Bressan (Penn State) growth models 23 / 47

Continuous dependence If Γ( x ) = N Ω ( x ) = normal cone, then d dt | x 1 ( t ) − x 2 ( t ) | ≤ C | x 1 ( t ) − x 2 ( t ) | = ⇒ uniqueness, continuous dependence Γ (x) v 2 f x v 2 x x 2 2 x x 1 v v 1 Ω 1 Ω 1 Ω Possible approach when Γ( x ) � = N Ω ( x ): introduce a Riemann metric on the Hilbert space ( = R n or H 2 ( R ) ) which renders each Γ( x ) a normal cone Alberto Bressan (Penn State) growth models 23 / 47

Well-posedness of the stem growth model with obstacle (A.B. - M.Palladino, work in progress) Solutions exist and are unique except if a (highly non-generic) breakdown configuration occurs Ω Ω Ω Ω good good bad bad Alberto Bressan (Penn State) growth models 24 / 47

Vines clinging to tree branches (A.B., M.Palladino, W.Shen) add a term which bends the stem toward the obstacle, at points which are sufficiently close (i.e., at a distance < δ 0 from the obstacle) k (t,s) η (s) ⊥ α k (t,s) δ P(t,s) 0 Ω s δ 0 � � . ψ ( x ) = η d ( x , Ω) In the case of a vine that clings to a branch of another tree, the evolution equation contains an additional term (= ⇒ bending toward the obstacle) �� s � e − β ( t − σ ) � � ∂ ∇ ψ ( P ( t , σ )) , k ⊥ ( t , σ ) k ⊥ ( t , s ) ∂ t k ( t , s ) = · · · + d σ 0 Alberto Bressan (Penn State) growth models 25 / 47

Numerical simulations (Wen Shen) Alberto Bressan (Penn State) growth models 26 / 47