Lemniscates do not survive Laplacian growth
D. Khavinson, M. Mineev-Weinstein, M. Putinar, R. Teodorescu
aa r X i v : . [ m a t h . C V ] D ec LEMNISCATES DO NOT SURVIVE LAPLACIANGROWTH
D. KHAVINSON, M. MINEEV-WEINSTEIN, M. PUTINAR,AND R. TEODORESCU Introduction
Many moving boundary processes in the plane, e.g., solidification,electrodeposition, viscous fingering, bacterial growth, etc., can be math-ematically modeled by the so-called Laplacian growth [9, 13]. In a nut-shell, it can be described by the equation(1.1) V ( z ) = ∂ n g Ω( t ) ( z, ζ ) , where V is the normal component of the velocity of the boundary ∂ Ω( t )of the moving domain Ω( t ) ⊂ R ≃ C , z ∈ ∂ Ω( t ), t is time, ∂∂n denotesthe normal derivative on ∂ Ω( t ) and g Ω( t ) ( z, ζ ) is the Green functionfor the Laplace operator in the domain Ω( t ) with a unit source at thepoint ζ ∈ Ω( t ). Equation (1.1) can be elegantly rewritten as the area-preserving diffeomorphism(1.2) ℑ (¯ z t z θ ) = 1 , where ℑ denotes the imaginary part of a complex number, ∂ Ω( t ) := { z := z ( t, θ ) } is the moving boundary parametrized by w = e iθ on theunit circle and the conformal mapping from, say, the exterior of theunit disk D + := {| w | > } onto Ω( t ) with the normalization z ( ∞ ) = ζ , z ′ ( ∞ ) > Laplacian growth or the
Polubarinova -Galin equation in modern literature, was first derived by Polubarinova-Kochina [11] and Galin [7] in 1945, as a description of secondary oilrecovery processes.This equation is known to be integrable [10], and as such possessesan infinte number of conserved quantities. More precisely, it admits
Part of this work was done during the first and third authors’ visit to LANL andwas supported by the LDRD project 20070483 “Minimal Description of Complex2D Shapes” at LANL. Also, D. Khavinson and M. Putinar gratefully acknowledgepartial support by the National Science Foundation. conserved moments c n = R Ω( t ) z n dx dy , where n runs over either all non-negative or all non-positive integers depending on whether domainsΩ( t ) are finite or infinite. At the same time (1.2) admits an impressivenumber of closed-form solutions.For the background, history, generalizations, references, connectionsto the theory of quadrature domains and other branches of mathemat-ical physics we refer the reader to [4, 8–10, 12, 13] and the referencestherein.In section § n : Γ t := {| P ( z, t ) | = 1 } , P ( z, t ) = a ( t ) n Q j =1 ( z − λ j ( t )), where a ( t ) is real-valued, is destroyed instantlyunder the Laplacian growth process described in (1.1), with Ω( t ) = {| P ( z, t ) | > } , ζ = ∞ , unless n = 1, λ ( t ) = const and { Γ t = ∂ Ω( t ) } is simply a family of concentric circles. Here the roots λ j ( t ) of P ( z, t )are all assumed to be inside Ω ′ t := {| P ( z, t ) | < } , so Ω and Ω ′ aresimply connected.This result shows that unlike quadrature domains (cf. [4,12]) that arepreserved under the Laplacian growth process, lemniscates for which allthe roots of the defining polynomial are in Ω ′ t are instantly destroyed,except for the trivial case of concentric circles. This, incidentally, agreeswith a well-known fact — cf. [5] — that lemniscates which are alsoquadrature domains must be circles. The proof of the theorem for thecase of Laplacian growth is given in § § § V ( z ) = χ ( z ) ∂ n g Ω( t ) ( z, ζ ) , with χ ( z ) is a bounded, real, positive function on Γ t . Invariance undertime-reversal is defined here in the following way: if the boundary Γ t + dt is the image of Γ t under a map f ( t,dt ) : z t ∈ Γ t z t + dt ∈ Γ t + dt , then f ( t + dt, − dt ) ◦ f ( t,dt ) = I .We conclude with a few remarks in § Destruction of Lemniscates
Theorem 2.1.
Suppose that a family of moving boundaries Γ t , (where t > is time), produced by a Laplacian growth process, is a family ofpolynomial lemniscates {| P ( z, t ) | = 1 } , where P ( z, t ) = a ( t ) n Q j =1 [ z − λ j ( t )] , EMNISCATES DO NOT SURVIVE LAPLACIAN GROWTH 3 and all λ j ( t ) are assumed to be inside Γ t . Then, n = 1 and λ = const ,i.e., Γ t is a family of concentric circles.Proof. Let Ω t = { z : | P ( z, t ) | > } , D + = {| w | > } . The function ϕ ( t ) : Ω t → D + , w = ϕ ( z, t ) = n p P ( z, t ), where we choose the branchfor the n − th root so that ϕ ′ ( t, ∞ ) >
0, maps Ω t conformally onto D + , ϕ ( t, ∞ ) = ∞ . It is useful to note that on Γ t , P ( z, t ) = w n , | w | = 1 anddoes not depend on t . This is because for any two moments of time t, τ ,we have w ( t )( . ) = w ( τ ) ◦ κ ( t, τ )( . ), where κ is a M¨obius automorphismof the disk. In our case, κ ( t, τ )( ∞ ) = ∞ , so κ ( t, τ )( z ) = e iα z, α ∈ R ,but since it also fixes the argument at ∞ , κ is the identity.Therefore, we have (where, as is customary, we denote the partial t -derivative by a “dot”):(2.1) ˙ P + P ′ z ˙ z = 0 . Since ϕ ( t ) maps Γ t onto the unit circle, we have z ( t ) = Ψ( t, w ), whereΨ( t, w ) = ϕ − ( t, z ). We also have on Γ t , by differentiating P ( z ( w ) , t ) = w n with respect to w ,(2.2) P ′ z · z w = nw n − or(2.3) wz w = nw n P ′ z = nPP ′ z . From (2.1), conjugating, we infer(2.4) ˙ z = − ˙ PP ′ z . Parametrize the unit circle by w = e iθ , 0 ≤ θ ≤ π . Then, from (2.3),it follows that we have on Γ t (since ( z ( t ) = z ( w, t ) = z ( w ( θ, t )))),(2.5) 1 i z θ := ∂zi∂θ = z w w = nPP ′ z . Combining (2.4) and (2.5) yields ( ℜ stands for the real part):(2.6) ℜ (cid:18) ˙ z i z θ (cid:19) = ℜ − ˙ PP ′ z · nPP ′ z ! . Also,(2.7) ℜ (cid:18) ˙ z ∂zi∂θ (cid:19) = ℑ (cid:0) ˙ zz θ (cid:1) = 1 , KHAVINSON, MINEEV-WEINSTEIN, PUTINAR, AND TEODORESCU where in the last equality we used the hypothesis that the lemniscatesΓ t := { z ( t, θ ) } satisfy the main equation (1.2) of Laplacian growthprocesses— cf. [8, § n ℜ ˙ PP ′ z iz θ ! = ℜ ˙ PP ′ z PP ′ z ! = − n . Or, we can rewrite (2.8) as(2.9) ℜ (cid:16) ˙ P P (cid:17) = − n | P ′ z | . Thus, we are finally arriving at(2.10) ddt (cid:0) | P | (cid:1) = − n | P ′ z | . Therefore, (2.10) holds on the lemniscates Γ t = {| P ( z, t ) | = 1 } thatare assumed to be interfaces of a Laplacian growth process. Now thetheorem follows from the following. Lemma 2.1.
Let t be the time variable, P ( z, t ) = a ( t ) n Q ( z − λ i ( t )) , be a “flow” of n -degree polynomials. Assume that the lemniscates Γ t := {| P ( z, t ) | = 1 } all have connected interiors {| P ( z, t ) | < } and ageneralized equation (2.10) holds on Γ t ; i.e., (2.11) ddt (cid:0) | P ( z, t ) | (cid:1) − c ( t ) | P ′ z ( z, t ) | = 0 , where the function c ( t ) is real-valued, depends on t only and, hence, isa constant on Γ t . Then, n = 1 , λ = λ ( t ) = const and Γ t is a familyof concentric circles centered at λ .Proof of the Lemma. Our hypothesis implies that all polynomials | P ( z, t ) | − ddt (cid:0) | P ( z, t ) | (cid:1) − c ( t ) | P ′ z ( z, t ) | = B ( t ) (cid:0) | P ( z, t ) | − (cid:1) . Equation (2.12) holds for all z ∈ C and for an interval of time t , and foreach t , both sides are real-analytic functions in z and z . Hence, we can“polarize” (2.12), i.e., replace z by an independent complex variable ξ . (This is due to a simple observation: real-analytic functions of twovariables are nothing else but restrictions of holomorphic functions in z, ξ -variables to the plane { ξ = z } . Hence, if two real-analytic functionscoincide on that plane, they coincide in C as well.) Denoting by P the polynomial whose coefficients are obtained from P by complex EMNISCATES DO NOT SURVIVE LAPLACIAN GROWTH 5 conjugation, we have (2.12) in a “polarized” form holding for ( z, ξ ) ∈ C :(2.13) ddt (cid:0) P ( z, t ) P ( ξ, t ) (cid:1) − c ( t ) (cid:16) P ′ z ( z, t ) · (cid:0) P (cid:1) ′ ξ ( ξ, t ) (cid:17) = B ( t ) (cid:0) P ( z, t ) P ( ξ, t ) − (cid:1) . Now let us denote by k j the multiplicity of the root λ j ( t ) of the polyno-mial P ( z, t ), so that there are m ≤ n distinct roots and P mj =1 k j = n .Since P ( z, t ) = a ( t ) m Y ( z − λ j ( t )) k j ,P ( ξ, t ) = ¯ a ( t ) m Y (cid:16) ξ − λ j ( t ) (cid:17) k j , dividing by P ( z, t ) P ( ξ, t ) we obtain:(2.14) 2 ℜ (cid:18) ˙ aa (cid:19) − m X k j ˙ λ j ( t ) z − λ j ( t ) + k j ˙ λ j ( t ) ξ − λ j ( t ) ! − c ( t ) " m X k j z − λ j ( t ) · " m X k j ξ − λ j ( t ) = B ( t ) (cid:18) − P ( z, t ) P ( ξ, t ) (cid:19) . Integrating (2.14) along a small circle centered at λ j ( t ), so that it doesnot enclose other zeros of P , yields for all ξ :(2.15) − k j ˙ λ j ( t ) − c ( t ) m X k i k j ξ − ¯ λ i ( t ) ! = − B ( t ) P ( ξ, t ) q j , where q j = k j − (cid:0) ∂∂z (cid:1) k j − h ( z − λ j ) kj P ( z,t ) i z = λ j . Letting ξ → ∞ in (2.15)implies that ˙ λ j ( t ) = 0 for all j = 1 , . . . , n . In other words, the “nodes” λ j ( t ) of all the lemniscates Γ t are fixed, i.e. do not move with time.So,(2.16) P ( z, t ) = a ( t ) n Y ( z − λ j ) = a ( t ) Q ( z ) . KHAVINSON, MINEEV-WEINSTEIN, PUTINAR, AND TEODORESCU
Substituting (2.16) into (2.13), we obtain(2.17) ddt (cid:0) | a | (cid:1) Q ( z ) Q ( ξ ) − c ( t ) | a | Q ′ z (cid:0) Q (cid:1) ′ ξ = B ( t ) (cid:0) | a | Q ( z ) Q ( ξ ) − (cid:1) . Comparing the leading terms (i.e., the coefficients at z n ξ n ) in (2.17)yields(2.18) ddt (cid:0) | a | (cid:1) = B ( t ) | a | . Therefore,(2.19) c ( t ) | a | Q ′ z (cid:0) Q (cid:1) ′ ξ = B ( t ) , and thus deg Q ′ z = 0, i.e., n = deg P = 1. The proofs of the Lemmaand the Theorem are now complete. (cid:3) Extending the theorem to growth processes invariantunder time reversal
First, let us note that any boundary Γ t is an equipotential line of thelogarithmic potential(3.1) Φ( z ) = log | P n ( z, λ i ( t )) | . The boundary velocity of the general growth process defined in (1.3)can now be expressed as ~V ( z ) = χ ( z ) ~ ∇ Φ , z ∈ Γ t , χ ( z ) ∈ R + .As indicated in the Introduction, invariance under time-reversal isdefined here in the following way: if the boundary Γ t + dt is the image ofΓ t under a map f ( t,dt ) : z t ∈ Γ t z t + dt ∈ Γ t + dt , then f ( t + dt, − dt ) ◦ f ( t,dt ) = I . That means that the normal at z t + dt ∈ Γ t + dt must be parallel to thenormal at z t ∈ Γ t , which shows that Γ t + dt is perpendicular at everypoint to gradient lines of Φ, and is therefore a level line of Φ. Thedisplacement of the point z t becomes z t + dt − z t = χ ( z ) ~ ∇ Φ( z t ) dt. Denoting by ~E = ~ ∇ Φ = 2 ¯ ∂ Φ = 2 · P ′ n ( z, λ i ( t )) P n ( z, λ i ( t ))the gradient of the logarithmic potential and by ~r = z t , conservationof the normal (or gradient) direction becomes ~E ( ~r + χ ~E ( ~r ) dt ) = µ ( z ) ~E ( ~r ) , EMNISCATES DO NOT SURVIVE LAPLACIAN GROWTH 7 where µ ( z ) = 1 + m ( z ) dt, m ( z ) = O (1) , m ( z ) ∈ R , so after expandingin the infinitesimal time interval dt ,( ~E · ~ ∇ ) ~E ( ~r ) = m ( z ) χ ( z ) ~E ( ~r ) . Remark . The proportionality relation indicated above carries alsothe following physical significance: the dynamical system that we studyis of frictional type, where the acceleration field (proportional to theforce, or gradient of Green’s function) is also proportional to the veloc-ity . In other words, the transport derivative (or Lie derivative) of thevelocity field must be parallel to the velocity itself: L ~V ~V = [ i ~V ◦ d − d ◦ i ~V ] ~V = ( ~V · ~ ∇ ) ~V = χ [ χ ( ~E · ~ ∇ ) ~E + ( ~E · ~ ∇ χ ) ~E ]is parallel to ~V and therefore, to ~E .In complex notation, using the fact that ( ~E · ~ ∇ ) = ¯ E ¯ ∂ + E∂ , weobtain P ′ n ( z, λ i ( t )) P n ( z, λ i ( t )) = δ ( z ) P ′ n ( z, λ i ( t )) P n ( z, λ i ( t )) · (cid:18) P ′ n ( z, λ i ( t )) P n ( z, λ i ( t )) (cid:19) ′ , δ ( z ) ∈ R , which (after multiplying both sides by E ( z )) reduces to (cid:20) P ′ n ( z, λ i ( t )) P n ( z, λ i ( t )) (cid:21) − (cid:18) P ′ n ( z, λ i ( t )) P n ( z, λ i ( t )) (cid:19) ′ ∈ R , or(3.2) ℑ (cid:26)(cid:20) P n ( z, λ i ( t )) P ′ n ( z, λ i ( t )) (cid:21) ′ (cid:27) = 0 , ( ∀ ) z ∈ Γ t . We note that, since E ( z ) = 2 ¯ P ′ n / ¯ P n is the gradient of the Green’sfunction for Ω t and Ω t is simply connected, it cannot vanish anywherein Ω t ∪ Γ t , so all the zeros of P ′ n ( z ), denoted by ξ k , k = 1 , , . . . , n − ′ t . Then (cid:20) P n ( z, λ i ( t )) P ′ n ( z, λ i ( t )) (cid:21) ′ = 1 n + n − X k =1 A k ( z − ξ k ) , ξ k ∈ Ω ′ t , with A k constants. The imaginary part of this expression coincideswith the imaginary part of an analytic function in Ω t , that is boundedthere, so the condition (3.2) can only be satisfied if the function is aconstant. Since at z → ∞ it vanishes, it follows that (cid:20) P n ( z, λ i ( t )) P ′ n ( z, λ i ( t )) (cid:21) ′ = 1 n , which means that boundaries Γ t can only be concentric circles. KHAVINSON, MINEEV-WEINSTEIN, PUTINAR, AND TEODORESCU Concluding Remarks (1) It is plausible that the result can be extended to rational lemnis-cates Γ t := {| R ( z, t ) | = 1 } , where R ( z, t ) are rational functionsof degree n where all the zeros are inside Γ, while all poles arein the unbounded component of C \ Γ t .(2) It is well-known that arbitrary “shapes”, i.e. Jordan curves canbe arbitrarily close approximated by both lemniscates (Hilbert’stheorem – cf. [14]) and quadrature domains [1]. At the sametime our results imply that there are fundamental differencesbetween these two classes of curves. We think it is interestingto pursue these observations in greater depth.(3) From the argument in § { Γ t } t> evolves by the flow along the velocityfield V ( z ) according to (1.3). Assuming the invariance undertime-reversal, the argument of § χ =const, i.e. the process is that of Laplacian growth. Invokingnow well-known results on standard Hele-Shaw flows, we canat once conclude, e.g., that the process (1.3) continues for alltimes t >
0, i.e., the curves { Γ t } move out to infinity such that ∪ t>t Γ t = C \ Ω t , if and only if the initial curve Γ is an ellipseand all the curves { Γ t } are also ellipses homotetic with Γ -cf. [3], also cf. [6]. References [1] S. R. Bell,
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Department of Mathematics & Statistics, University of South Florida,Tampa, FL 33620-5700, USA
E-mail address : [email protected] Los Alamos National Laboratory, MS-365, Los Alamos, NM 87545,USA
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E-mail address : [email protected] Department of Mathematics & Statistics, University of South Florida,Tampa, FL 33620-5700, USA
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