Abstract
We study non-univalent solutions of the Polubarinova-Galin equation, describing the time evolution of the conformal map from the unit disk onto a Hele-Shaw blob of fluid subject to injection at one point. In particular, we tackle the difficulties arising when the map is not even locally univalent, in which case one has to pass to weak solutions developing on a branched covering surface of the complex plane.
One major concern is the construction of this Riemann surface, which is not given in ad
Results & Lemmas (22)
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Proposition 9.1
Proposition 9.1 below). The rightmost member of (2.11) will be used when we need to be explicit about the orders of the poles. The…
Proposition 9.1 below). The rightmost member of (2.11) will be used when we need to be explicit about the orders of the poles. The convention then is that ζ1, . . . , ζℓare distinct and nj ≥1. Thus n = Pℓ j=1 nj, and in the full sequence ζ1, . . . , ζn, the tail ζℓ+1, . . . , ζn will be repetitions of (some of) the ζ1, . . . , ζℓaccording to their orders. In equations (2.10) and (2.11), ℓand the nj are the same. One can easily express the L¨owner-Kufarev equation (2.7) directly in terms of g, in
Lemma 3.1.
Lemma 3.1. Let f(·, t) t∈I ⊂Onorm(D), where I ⊂[0, ∞) is any interval. Then the following are equivalent. (i) f(·, t) is a subordination…
Lemma 3.1. Let {f(·, t)}t∈I ⊂Onorm(D), where I ⊂[0, ∞) is any interval. Then the following are equivalent. (i) {f(·, t)} is a subordination chain on I. (ii) There exists a Riemann surface M, a nonconstant analytic function p : M →C (‘covering map’) and univalent analytic functions ˜f(·, t) : D →M (t ∈I) (‘liftings’ of the f(·, t)) such that 12
Lemma 3.2.
Lemma 3.2. For f, g ∈Onorm(D), f ≺g implies νf ≤νg (almost every- where).
Lemma 3.2. For f, g ∈Onorm(D), f ≺g implies νf ≤νg (almost every- where).
Theorem 3.1.
Theorem 3.1. Let I ∋t 7→f(·, t) ∈Onorm(D) be smooth on some time interval I and assume that f ′ ̸= 0 on ∂D on this interval. Then for q(t)…
Theorem 3.1. Let I ∋t 7→f(·, t) ∈Onorm(D) be smooth on some time interval I and assume that f ′ ̸= 0 on ∂D on this interval. Then for q(t) ≥0 the following are equivalent. (i) f(ζ, t) solves the L¨owner-Kufarev equation (2.7). (ii) f(ζ, t) solves the Polubarinova-Galin equation (2.1) and ˙f(ω, t) = 0 for every root ω ∈D of f ′(ω, t) = 0. (iii) f(ζ, t) solves the Polubarinova-Galin equation (2.1) and {f(·, t)} is a subordination chain.
Theorem 3.2.
Theorem 3.2. Let I ∋t 7→f(·, t) ∈Onorm(D) be smooth on some time interval I and assume that f ′ ̸= 0 on ∂D on this time interval. Then f(ζ,…
Theorem 3.2. Let I ∋t 7→f(·, t) ∈Onorm(D) be smooth on some time interval I and assume that f ′ ̸= 0 on ∂D on this time interval. Then f(ζ, t) solves the Polubarinova-Galin equation (2.1) if and only if ˙f(ζ, t) = ζf ′(ζ, t) (P(ζ, t) + R(ζ, t)) , (3.7) where P is the Poisson integral (2.8) and where R(ζ, t) is any function of the form R(ζ, t) = −i Im X ωj∈D rj X k=1 2Bjk(t) (−ωj(t))k +
Theorem 3.2
Theorem 3.2 holds for general solutions of the Polubarinova-Galin equa- tion, but will become of particular interest when we discuss…
Theorem 3.2 holds for general solutions of the Polubarinova-Galin equa- tion, but will become of particular interest when we discuss rational solutions in Section 9. 4 Weak solutions 4.1 Preliminaries and definition Some of our main results will be formulated in terms of variational inequality weak solutions, just called weak solutions for short, which are expressed in terms of time independent test functions which are subharmonic in the domains Ω(t). We shall also need general smooth test functi
Lemma 4.1.
Lemma 4.1. For any smooth evolution t 7→f ∈Onorm(D) and any smooth function Ψ(ζ, t) which satisfies (4.1) we have d dt Z D Ψ(ζ, t)|f ′(ζ,…
Lemma 4.1. For any smooth evolution t 7→f ∈Onorm(D) and any smooth function Ψ(ζ, t) which satisfies (4.1) we have d dt Z D Ψ(ζ, t)|f ′(ζ, t)|2 dm(ζ) = Z ∂D Ψ(ζ, t)Re h ˙f(ζ, t)ζf ′(ζ, t) i dθ, (4.2)
Corollary 4.2.
Corollary 4.2. For any smooth evolution t 7→f(·, t) ∈Onorm(D)) and any smooth function Φ in C we have d dt Z C Φ(z)νf(·,t)(z)dm(z) = Z 2π 0…
Corollary 4.2. For any smooth evolution t 7→f(·, t) ∈Onorm(D)) and any smooth function Φ in C we have d dt Z C Φ(z)νf(·,t)(z)dm(z) = Z 2π 0 Φ(f(ζ, t))Re h ˙f(ζ, t)ζf ′(ζ, t) i dθ. (4.3)
Corollary 4.3.
Corollary 4.3. For any solution t 7→f(·, t) ∈Onorm(D) of the Polubarinova- Galin equation (2.1) with q(t) ≥0, νf(·,t) is an increasing…
Corollary 4.3. For any solution t 7→f(·, t) ∈Onorm(D) of the Polubarinova- Galin equation (2.1) with q(t) ≥0, νf(·,t) is an increasing function of t. Specializing (2.1), on the other hand, to subharmonic and harmonic test functions (which we then denote h) we obtain, in view of the mean-value properties satisfied by such functions: 18
Corollary 4.4.
Corollary 4.4. Let t 7→f(·, t) ∈Onorm(D) solve the Polubarinova-Galin equation (2.1) with q(t) ≥0. Then d dt Z C hνf(·,t)dm ≥2πq(t)h(0) for…
Corollary 4.4. Let t 7→f(·, t) ∈Onorm(D) solve the Polubarinova-Galin equation (2.1) with q(t) ≥0. Then d dt Z C hνf(·,t)dm ≥2πq(t)h(0) for any h which is subharmonic in a neighborhood of supp νf. If h is har- monic, equality holds. As a particular case we get the relevant version of moment conservation. Keeping the rightmost member of (2.4) as definition of the harmonic moments in the non-univalent case, so that Mk(t) = 1 2πi
Corollary 4.5.
Corollary 4.5. Whenever s ≤t and h is subharmonic in a neighborhood of supp νf(·, t) we have, when f solves the Polubarinova-Galin equation…
Corollary 4.5. Whenever s ≤t and h is subharmonic in a neighborhood of supp νf(·, t) we have, when f solves the Polubarinova-Galin equation (2.1), Z C hνf(·,t)dm − Z C hνf(·,s)dm ≥2π(Q(t) −Q(s))h(0), (4.4) where Q is the accumulated source (see (2.6)). This corollary connects to a well-established notion of (variational in- equality) weak solution for the Hele-Shaw problem with a source of strength q(t) ≥0 at the origin. We formulate the definition first for domains (or open sets) in C. It will la
Lemma 3.1.
Lemma 3.1. Most of the previous formulas have simple formulations on M, for example (2.2) generalizes to d dt Z ˜Ω(t) h d ˜m = 2πq(t)h(˜0),…
Lemma 3.1. Most of the previous formulas have simple formulations on M, for example (2.2) generalizes to d dt Z ˜Ω(t) h d ˜m = 2πq(t)h(˜0), (5.7) for h harmonic in a neighborhood of ˜Ω(t), and where ˜Ω(t) = ˜f(D, t), f = p ◦˜f : D →M →C. For subharmonic h we have inequality ≥. Thus on integrating (5.7) with respect to t we arrive at the natural notion of weak solution on the Riemann surface M. 25
Proposition 5.1.
Proposition 5.1. Let t 7→f(·, t) ∈Onorm(D) be a smooth evolution on some time interval and assume that f ′ ̸= 0 on ∂D on this time…
Proposition 5.1. Let t 7→f(·, t) ∈Onorm(D) be a smooth evolution on some time interval and assume that f ′ ̸= 0 on ∂D on this time interval. Then f(·, t) solves the Polubarinova-Galin equation (2.1) if and only if d dt Z D h(ζ, t)|f ′(ζ, t)|2 dm(ζ) = 2πq(t)h(0, t). (5.22) for every function h(·, t) ∈O(D) which satisfies (5.19) (equivalently, (4.1) or (5.21)), and it solves the L¨owner-Kufarev equation (2.7) if and only if moreover (3.5) holds (equivalently, f(·, t) is a subordination chain).
Proposition 5.2.
Proposition 5.2. A family f(·, t) ∈Onorm(D): 0 ≤t ≤T represents a weak solution as in Definition 5.1 (with I = [0, T]) if an only if it is a…
Proposition 5.2. A family {f(·, t) ∈Onorm(D) : 0 ≤t ≤T} represents a weak solution as in Definition 5.1 (with I = [0, T]) if an only if it is a subordination chain as in Definition 3.2 and (5.25) holds for 0 ≤s < t ≤T. 6 Compatibility between balayage and cover- ing maps The family of branched covering surfaces over C form a partially ordered set in a natural way. Within in each of the surfaces one can perform partial balayage, sweeping to the area form lifted from C. Thus we have two kinds of pro
Proposition 6.1.
Proposition 6.1. With p: ˜ M →M a nonconstant proper analytic map be- tween two Riemann surfaces, where M = C, let ˜µ be a measure with…
Proposition 6.1. With p : ˜ M →M a nonconstant proper analytic map be- tween two Riemann surfaces, where M = C, let ˜µ be a measure with compact 34
Theorem 7.1.
Theorem 7.1. Let f(·, 0) ∈Onorm(D) be given, together with q(t) ≥0 (0 ≤ t < ∞) such that Q(t) →∞as t →∞. Then, under the assumption that…
Theorem 7.1. Let f(·, 0) ∈Onorm(D) be given, together with q(t) ≥0 (0 ≤ t < ∞) such that Q(t) →∞as t →∞. Then, under the assumption that Conjecture 1.1 (or Conjecture 7.3 below) is true, there exists a Riemann surface M, a nonconstant holomorphic function p : M →C and a point ˜0 ∈M with p(˜0) = 0 such that the following assertions hold. (i) f(·, 0) factorizes over M, i.e., there exists a univalent function ˜f(·, 0) : D →M with ˜f(0, 0) = ˜0 such that f(·, 0) = p( ˜f(·, 0)). (ii) On setting ˜Ω(0)
Lemma 7.1.
Lemma 7.1. Let f ∈Onorm(D) and let 0 < r < 1. Then the following are equivalent. (i) f extends to be meromorphic in D(0, 1 r) with poles…
Lemma 7.1. Let f ∈Onorm(D) and let 0 < r < 1. Then the following are equivalent. (i) f extends to be meromorphic in D(0, 1 r) with poles only at the reflected (in ∂D) zeros of g, more precisely so that fg∗∈O(D(0, 1 r) \ D). (ii) For every number ρ with r < ρ < 1 there exists a constant Cρ such that | Z D h|g|2dm| ≤Cρ sup D(0,ρ) |h| (h ∈O(D)). (7.1)
Lemma 7.2.
Lemma 7.2. Let ˜Ω(·, t) = ˜f(D, t) be a simply connected weak solution on a Riemann surface M with projection p: M →C and let f(ζ, t) = p(…
Lemma 7.2. Let ˜Ω(·, t) = ˜f(D, t) be a simply connected weak solution on a Riemann surface M with projection p : M →C and let f(ζ, t) = p( ˜f(ζ, t)). Assume q(t) ≥0 and that for a certain 0 < r < 1 the equivalent conditions in Lemma 7.1 hold for f = f(·, 0). Then they hold with the same r for all f(·, t), t > 0.
Theorem 9.1.
Theorem 9.1. Under the assumption that g has only simple zeros, that g and g∗have no common zeros (in particular g has no zero on ∂D), and…
Theorem 9.1. Under the assumption that g has only simple zeros, that g and g∗have no common zeros (in particular g has no zero on ∂D), and that in addition (9.10) holds, the Polubarinova-Galin equation (2.1), or (9.3), gives the following rational dynamics for g: d dt log ωk = −C −2Ak ωk − m X j=1, j̸=k 2(Ak + Aj) ωk −ωj + n
Theorem 9.2.
Theorem 9.2. Given g(ζ, 0) of the form (2.11) such that no two zeros of g(ζ, 0) are related by ωk = ω∗ j, then for exactly one choice of…
Theorem 9.2. Given g(ζ, 0) of the form (2.11) such that no two zeros of g(ζ, 0) are related by ωk = ω∗ j, then for exactly one choice of R(ζ, t), namely that given by (9.10), there exists a solution g(ζ, t) of (9.3) which remains on the original rational form (2.11). Necessary and sufficient condition for this rational solution to also solve the L¨owner-Kufarev equation (2.7) is that R(ζ, t) = 0. This occurs precisely under the condition that whenever g(ζ, t) has a zero ωk in D, the reflected point
Lemma 7.1
Lemma 7.1 can be made more explicit and ends up with a quadrature formula for h ∈O(D). Specifically we get 1 π Z D h|g|2dm = 1 2πi Z ∂D hf…
Lemma 7.1 can be made more explicit and ends up with a quadrature formula for h ∈O(D). Specifically we get 1 π Z D h|g|2dm = 1 2πi Z ∂D hf ∗df = X Res D (hf ∗gdζ) +
Proposition 9.1.
Proposition 9.1. Let f ∈Onorm(D). Then g = f ′ is a rational function if and only if there exist αj, γj, akj, cj, r, ℓ, nj so that the…
Proposition 9.1. Let f ∈Onorm(D). Then g = f ′ is a rational function if and only if there exist αj, γj, akj, cj, r, ℓ, nj so that the quadrature identity 1 π Z D h|g|2dm = r X j=1 cj Z γj hgdζ + ℓ
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