Lemniscates do not survive Laplacian growth

where V is the normal component of the velocity of the boundary ∂Ω(t)

of the moving domain Ω(t) ⊂ R 2 ≃ C, z ∈ ∂Ω(t), t is time, ∂ ∂n denotes the normal derivative on ∂Ω(t) and g Ω(t) (z, ζ) is the Green function for the Laplace operator in the domain Ω(t) with a unit source at the point ζ ∈ Ω(t). Equation (1.1) can be elegantly rewritten as the areapreserving diffeomorphism

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 the unit circle and the conformal mapping from, say, the exterior of the unit disk D + := {|w| > 1} onto Ω(t) with the normalization z(∞) = ζ, z ′ (∞) > 0.

The equation (1.2), named 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 oil recovery processes.

This equation is known to be integrable [10], and as such possesses an infinte number of conserved quantities. More precisely, it admits z n dx dy, where n runs over either all nonnegative or all non-positive integers depending on whether domains Ω(t) are finite or infinite. At the same time (1.2) admits an impressive number of closed-form solutions.

For the background, history, generalizations, references, connections to the theory of quadrature domains and other branches of mathematical physics we refer the reader to [4, 8-10, 12, 13] and the references therein.

In section §2 of this paper, we show that any continuous chain of polynomial lemniscates of order n:

, where a(t) is real-valued, is destroyed instantly under the Laplacian growth process described in (1.1), with Ω(t) = {|P (z, t)| > 1}, ζ = ∞, unless n = 1, λ 1 (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)| < 1}, so Ω and Ω ′ are simply connected.

This result shows that unlike quadrature domains (cf. [4,12]) that are preserved under the Laplacian growth process, lemniscates for which all the roots of the defining polynomial are in Ω ′ t are instantly destroyed, except for the trivial case of concentric circles. This, incidentally, agrees with a well-known fact -cf. [5] -that lemniscates which are also quadrature domains must be circles. The proof of the theorem for the case of Laplacian growth is given in §2.

In §3 we extend the result of §2 to all the growth processes that are invariant under time-reversal and for which the boundary velocity is given by

with χ(z) is a bounded, real, positive function on Γ t . Invariance under time-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) :

We conclude with a few remarks in §4.

Theorem 2.1. Suppose that a family of moving boundaries Γ t , (where t > 0 is time), produced by a Laplacian growth process, is a family of

and all λ j (t) are assumed to be inside Γ t . Then, n = 1 and λ 1 = const, i.e., Γ t is a family of concentric circles.

Proof.

, where we choose the branch for 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 and does not depend on t. This is because for any two moments of time t, τ , we have

, where κ is a Möbius automorphism of 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) Ṗ + P ′ z ż = 0. Since ϕ(t) maps Γ t onto the unit circle, we have z(t) = Ψ(t, w), where Ψ(t, w) = ϕ -1 (t, z). We also have on Γ t , by differentiating P (z(w), t) = w n with respect to w,

From (2.1), conjugating, we infer

Parametrize the unit circle by w = e iθ , 0 ≤ θ ≤ 2π. Then, from (2.3), it follows that we have on Γ t (since (z(t) = z(w, t) = z(w(θ, t)))),

Combining (2.4) and (2.5) yields (ℜ stands for the real part):

where in the last equality we used the hypothesis that the lemniscates Γ t := {z(t, θ)} satisfy the main equation (1.2) of Laplacian growth processes-cf. [8, §4]. Hence, (2.7), (2.4) and (2.5) imply that

Or, we can rewrite (2.8) as

(2.9)

Thus, we are finally arriving at

Therefore, (2.10) holds on the lemniscates Γ t = {|P (z, t)| = 1} that are assumed to be interfaces of a Laplacian growth process. Now the theorem follows from the following.

Equation (2.12) holds for all z ∈ C and for an interval of time t, and for each 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 f

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