The vanishing surface tension limit for the Hele-Shaw problem

Hyung Ju Hwang,Marco Antonio Fontelos,Youngmin Oh

doi:10.3934/dcdsb.2016108

Abstract

In this paper, we show the existence of solutions of the Hele-Shaw problem in two dimensions in the presence of surface tension and for a general class of initial data. The limit problem from nonzero to zero surface tension will also be investigated. In the case of injection and when volume conservation holds, for a sufficiently small surface tension, we prove the existence and uniqueness of perturbed solutions with nonzero surface tension near solutions with zero surface tension. We also show that solutions with nonzero surface tension exist up to a finite time before a possible singularity occurs in which solutions with zero surface tension are well defined. In addition, in the finite time interval, we prove that the solutions with nonzero surface tension approach the solutions with zero surface tension as the surface tension coefficient goes to zero. In the case of suction, for sufficiently small surface tension, we prove the existence of perturbed solutions near solutions with zero surface tension in any initially smooth domains. In this case, the local existence time depends on the surface tension coefficient.

Highlights

The classical Hele-Shaw problem originates from a device called the Hele-Shaw cell [32]
Where Q is a constant which represents the strength of the sink/source, γ the surface tension coefficient, κ the curvature, v the velocity of the fluid, p the pressure on ∂Ω (t), and δ = δ (x) the Dirac delta function located at x = 0
If Q < 0, the domain Ω (t) of the fluid expands in time, and if Q > 0, the domain Ω (t) of the fluid shrinks in time

Summary

Introduction

The classical Hele-Shaw problem originates from a device called the Hele-Shaw cell [32]. On ζ = eiθ ∈ ∂B10 (0) = {|z| = 1, z ∈ C}, where ∂tf (t, ζ) and f (t, ζ) denote the derivatives with respect to t and ζ respectively It is a nonlinear equation, if an initial domain is the disk Br0 (0) with radius r > 0, there exists a trivial solution of (1.4):. Notice that by [28], in the case Q ≤ 0 for γ = 0, we can assure the existence of a unique local in time solution f 0 (t, ζ) for a connected bounded domain with a smooth initial boundary. The linear operator L and nonlinear terms have similar in structure to the right-hand side of (1.12) We will use this to show that a unique solution of the linear problem satisfies (1.14), and extend this result to the nonlinear problem.

Then we can apply to get

Note that

We can easily deduce