| Abstract: |
| For the oil activities mathematical modelisation, the exchange between the rock-shop and the outside are organized through the wells of injection and production, or are the result of the natural activity of oil slick expansion. As the size of the deposit is about hectometer and wells ones are about decimeters, it is not realistic to give conditions on boundary neglectible regions. The terms of sources and wells modeling require the introduction of singularities, which raises analytical and numerical problems. The case of data measures had been studied by several authors. We propose a more general approach allowing considering very irregular data, as for example distributions data. For that purpose, we work in the framework of the so-called generalized Sobolev algebra, constructed from the classical Sobolev spaces by S. Bernard and S.P. Nuiro. This talk is devoted to set and solve a non-linear degenerate PDE problem involving very irregular data, with Robin-type boundary conditions, in the context of generalized Sobolev-type functions. This algebra is more appropriate than the others, because only $H^1\cap L^\infty$ estimates are required. The well-posedness and some qualitative properties of the solutions are proved. |
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