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

✍ Scribed by O. Gil; F. Quirós

• Publisher: Elsevier Science
• Year: 2003
• Tongue: English
• Weight: 169 KB
• Volume: 20
• Category: Article
• ISSN: 0294-1449
• DOI: 10.1016/s0294-1449(02)00016-1

✦ Synopsis

Let u m (x, t) be the solution to the Porous Media Equation, u t = u m , in a domain ⊂ R n , with initial data u m (x, 0) = f (x) and boundary data u m m (x, t) = g(x). Let v m ≡ u m m . We prove the convergence as m goes to infinity of the pair (u m , v m ) 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) = f (x), where f (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 u m to the positivity set of u ∞ . For large but finite m a boundary layer connecting the initial data f (x) and its projection f (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 ( = R n , no boundary data).