Asymptotic Convergence of the Stefan Problem to Hele-shaw
نویسندگان
چکیده
We discuss the asymptotic behaviour of weak solutions to the Hele-Shaw and one-phase Stefan problems in exterior domains. We prove that, if the space dimension is greater than one, the asymptotic behaviour is given in both cases by the solution of the Dirichlet exterior problem for the Laplacian in the interior of the positivity set and by a singular, radial and self-similar solution of the Hele-Shaw flow near the free boundary. We also show that the free boundary approaches a sphere as t→∞, and give the precise asymptotic growth rate for the radius.
منابع مشابه
N ov 2 01 5 Nonlinear diffusion equations as asymptotic limits of Cahn – Hilliard systems Pierluigi Colli Dipartimento di Matematica , Università di Pavia and IMATI C . N . R . Pavia
An asymptotic limit of a class of Cahn–Hilliard systems is investigated to obtain a general nonlinear diffusion equation. The target diffusion equation may reproduce a number of well-known model equations: Stefan problem, porous media equation, Hele-Shaw profile, nonlinear diffusion of singular logarithmic type, nonlinear diffusion of Penrose–Fife type, fast diffusion equation and so on. Namely...
متن کاملA study of a Stefan problem governed with space–time fractional derivatives
This paper presents a fractional mathematical model of a one-dimensional phase-change problem (Stefan problem) with a variable latent-heat (a power function of position). This model includes space–time fractional derivatives in the Caputo sense and time-dependent surface-heat flux. An approximate solution of this model is obtained by using the optimal homotopy asymptotic method to find the solu...
متن کاملLong-time behavior for the Hele-Shaw-Cahn-Hilliard system
We study the long time behavior of the Hele-Shaw-Cahn-Hilliard system (HSCH) which models two phase incompressible Darcian flow in porous media with matched density but arbitrary viscosity contrast. We demonstrate that the ω-limit set of each trajectory is a single stationary solution of the system via Lojasiewicz-Simon type technique. Moreover, a rate of convergence has been established. Event...
متن کاملPhase-field model for Hele-Shaw flows with arbitrary viscosity contrast. I. Theoretical approach.
We present a phase-field model for the dynamics of the interface between two inmiscible fluids with arbitrary viscosity contrast in a rectangular Hele-Shaw cell. With asymptotic matching techniques we check the model to yield the right Hele-Shaw equations in the sharp-interface limit, and compute the corrections to these equations to first order in the interface thickness. We also compute the e...
متن کاملThe Hele-Shaw asymptotics for mechanical models of tumor growth
Models of tumor growth, now commonly used, present several levels of complexity, both in terms of the biomedical ingredients and the mathematical description. The simplest ones contain competition for space using purely fluid mechanical concepts. Another possible ingredient is the supply of nutrients through vasculature. The models can describe the tissue either at the level of cell densities, ...
متن کامل