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Section: New Results

Variational approach for multiphase flows

In [66], C. Cancès, T. O. Gallouët, and L. Monsaingeon show that the equations governing two-phase flows in porous media have a formal gradient flow structure. The goal of the longer contribution [20] is then twofold. First, it extends the variational interpretation of [66] to the case where an arbitrary number of phases are in competition to flow within a porous medium. Second, we provide rigorous foundations to our claim. More precisely, the convergence of a minimizing movement scheme à la Jordan, Kinderlehrer, and Otto  [86] is shown in [20], providing by the way a new existence result for multiphase flows in porous media. The result relies on advances tools related to optimal transportation  [94], [93].

Based on the previous work, Clément Cancès, Daniel Matthes, and Flore Nabet derived in [46] a model of degenerate Cahn-Hilliard type for the phase segregation in incompressible multiphase flows. The model is obtained as the Wasserstein gradient flow of a Ginzburg-Landau energy with the constraint that the sum of the volume fractions must stay equal to 1. The resulting model differs from the classical degenerate Cahn-Hilliard model (see  [97], [77]) and is closely related to a model proposed by Weinan E and collaborators  [76], [89]. Besides the derivation of the model, the convergence of a minimizing movement scheme is proven in [46].