A moving boundary problem for periodic Stokesian Hele–Shaw flows

Abstract

This paper is concerned with the motion of an incompressible, viscous fluid in a Hele-Shaw cell. The free surface is moving under the influence of gravity and the fluid is modelled using a modified Darcy law for Stokesian fluids.We combine results from the theory of quasilinear elliptic equations, analytic semigroups and Fourier multipliers to prove existence of a unique classical solution to the corresponding moving boundary problem.

On periodic Stokesian Hele-Shaw flows with surface tension

“…Comparing it to ref. [4], the inclusion of surface tension effects is the new important feature of this work. The Newtonian case μ ≡ constant, treated in refs.…”

“…ref. [4], that (A 1 ) and (A 2 ) hold, provided there exist positive constants m μ and M μ such that Thus, if (V 1 ) and (V 2 ) hold, then Q is a uniformly elliptic quasilinear operator in R 2 . Let us now consider the moving boundary problem…”

“…[4] is the inclusion of surface tension effects. From the analytical point of view this implies that the full problem (1) is governed by a nonlinear evolution equation of third order, whereas the problem studied in ref.…”

Well-Posedness, Instabilities, and Bifurcation Results for the Flow in a Rotating Hele–Shaw Cell

“…∂ Ψ(γ, 0) ∈ Lis(h 4+α 0,e (S), h 1+α 0,e (S)). We shall derive this result from the following theorem which characterizes Fourier multiplication operators between the classical Hölder spaces, and which can be found in [2] for r = s, and in [11] for arbitrary r, s ∈ (0, ∞) \ N. Theorem 6.2. Let r, s ∈ (0, ∞) \ N and (M k ) k∈Z ⊂ C a sequence satisfying (i) sup k∈Z\{0} |k| r−s |M k | < ∞,…”

“…The smooth functions are dense in H r (S) and the Sobolev embedding H m+s (S) d → h m+α (S), (6.8) holds for all m ∈ N, α ∈ [0, 1] and s > 3/2 (see [11,Proposition 3.1]). Let A : C 1+α (S) → C 4+α (S) be the operator given by…”