Nonbranching weak and starshaped strong solutions for Hele-Shaw dynamics

Abstract

We consider multidimensional weak and strong Hele-Shaw dynamics (t) of an advancing/receding viscous fluid injected/removed through a single finite point into/from a bounded domain (0). A class of weak solutions is shown to preserve local uniqueness in both directions. Then we also consider strong solutions (t), and show that if (0) is starshaped with respect to a small ball centered on the point of injection, then the evolution (t) exists for all time.

Hele-Shaw Model for Melting/Freezing with Two Dendrits

“…Following to [11] and [12] we rewrite the corresponding Hele-Shaw problem as a couple of two problems, namely, the time-independent Riemann-Hilbert problem for a doubly connected domain and an evolution problem reformulated as an abstract Cauchy-Kovalevsky problem. In contrast to the case of simply connected domain in our situation there is no explicit formula for the Riemann-Hilbert problem (see, e.g., [13]- [15], [16], cf., also [7]).…”

“…The study of complex-analytic Hele-Shaw model going back to [5] and to [6] (cf. also the recent monograph [7]). This approach was applied to melting/freezing process in [8]- [10].…”

“…By freezing variable t we omit it in all components of (4)- (6). Relation (7), in principal (see [22]), guarantees the uniqueness of the solution to (4) (hence of the solution to (1) with respect to ∂f ∂t (z, t)). Moreover, an additional condition (3) yields the single-valuedness of this solution.…”

From the Hele-Shaw Experiment to Integrable Systems: A Historical Overview

“…This occurs when a simply connected domain Ω(T ) for some 0 < T < ∞ can be obtained as a result of a strong simply connected dynamics Ω(t), or else, as a result of a weak dynamics G(t), where G(t) for t < T is multiply connected with some holes to be filled in as t → T − . In [32] it was shown that for a simply connected initial domain the classical solution does not branch backward in time (if it exists).…”

Classical and Stochastic Laplacian Growth

“…These statements are based on the known regularity results for the obstacle problem [76], [77], [80], [79], [193], [431] combined with geometric results (relevant parts of Theorem 4.7.2 remain true [243]). Under certain assumptions, like strong starlikeness of the initial domain, it is possible even to prove the existence globally in time in the well-posed time direction (injection) [248]. Compare also [100], [154].…”