Boundary layer formation in the transition from the Porous Media Equation to a Hele–Shaw flow

Abstract

Let um(x,t) be the solution to the Porous Media Equation, ut=Δum, in a domain Ω⊂ ℝn, with initial data um(x,0)=f(x) and boundary data umm(x,t)=g(x). Let vm≡umm. We prove the convergence as m goes to infinity of the pair (um,vm) to a pair (u∞,v∞) which is a weak solution of the Hele–Shaw problem with boundary data v∞=g and initial data u∞(x,0) = f̃(x), where f̃(x)̃ is the projection of the initial data f(x) into a ‘mesa’. We also prove the convergence of the positivity sets of the functions um to the positivity set of u∞. For large but finite m a boundary layer connecting the initial data f(x) and its projection f̃(x) appears. We analyze the convergence of solutions and positivity sets in this boundary layer by introducing a suitable time scale. All our results hold true also for the Cauchy problem (Ω=ℝn, no boundary data).