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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formed by injection of a fluid of negligible viscosity and it grows occupying at each time t a set D(t), since fluid is withdrawn at constant rate Q at infinity.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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