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Analysis of miscible displacement through porous media with vanishing molecular diffusion and singular wells. (English) Zbl 1390.35265

Summary: This article proves the existence of solutions to a model of incompressible miscible displacement through a porous medium with zero molecular diffusion and modelling wells by spatial measures. We obtain the solution by passing to the limit on problems indexed by vanishing molecular diffusion coefficients. The proof employs cutoff functions to excise the supports of the measures and the discontinuities in the permeability tensor, thus enabling compensated compactness arguments used by Y. Amirat and A. Ziani for the analysis of the problem with \(L^2\) wells [Z. Anal. Anwend. 23, No. 2, 335–351 (2004; Zbl 1072.35039)]. We give a novel treatment of the diffusion-dispersion term, which requires delicate use of the Aubin-Simon lemma to ensure the strong convergence of the pressure gradient, owing to the troublesome lower-order terms introduced by the localisation procedure.

MSC:

35Q35 PDEs in connection with fluid mechanics
74S05 Finite element methods applied to problems in solid mechanics
35D30 Weak solutions to PDEs

Citations:

Zbl 1072.35039
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References:

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