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Bubble growth in porous media. (English) Zbl 0667.35074
Let \(R^ N\), \(N\geq 2\), be occupied by a porous medium saturated with an incompressible viscous fluid. A bubble D is formed by injection of a fluid of negligible viscosity and it grows occupying at each time t a set D(t), since fluid is withdrawn at constant rate Q at infinity. The problem is then \[ \Delta u=0\quad in\quad R^ N\setminus D(t);\quad u=0\quad in\quad D(t) \] \[ u=0,\quad u_ t-| \nabla u|^ 2=0\quad on\quad \partial D(t). \] The condition at infinity is \[ \nabla [u(x,t)-Q\Gamma_ N(r)]=O(r^{-N})\quad as\quad r\to \infty \] where \(\Gamma_ N=-(\log r)/2\pi\) for \(N=2\) and \(\Gamma_ N=(r^{2-N})/(N- 2)\omega_ N\) for \(N>2,\omega_ N\) being the area of the unit sphere in \(R^ N.\)
The main question is to determine the class of possible D(0) such that the solution exists for all \(t>0\). In the paper the set \(\Sigma =R^ N\setminus D(\infty)\) is prescribed and it is discussed whether or not a solution D(t) exists with a final position \(D(\infty)=R^ N\setminus \Sigma\). In the case where \(\Sigma\) is empty it is proved that any possible solutions, either weak or classical, have D(t) of ellipsoidal shape.
Reviewer: M.Primicerio

MSC:
35R35 Free boundary problems for PDEs
76S05 Flows in porous media; filtration; seepage
35B40 Asymptotic behavior of solutions to PDEs
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
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