### Abstract

Multiscale analysis of a degenerate pseudoparabolic variational inequality, modelling the two-phase flow with dynamical capillary pressure in a perforated domain, is the main topic of this work. Regularisation and penalty operator methods are applied to show the existence of a solution of the nonlinear degenerate pseudoparabolic variational inequality defined in a domain with microscopic perforations, as well as to derive a priori estimates for solutions of the microscopic problem. The main challenge is the derivation of a priori estimates for solutions of the variational inequality, uniformly with respect to the regularisation parameter and to the small parameter defining the scale of the microstructure. The method of two-scale convergence is used to derive the corresponding macroscopic obstacle problem.

Original language | English |
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Pages (from-to) | 44-75 |

Number of pages | 32 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 469 |

Issue number | 1 |

Early online date | 28 Aug 2018 |

DOIs | |

Publication status | Published - 1 Jan 2019 |

### Keywords

- Degenerate nonlinear PDEs
- Homogenization
- Obstacle problems
- Penalty operator method
- Pseudoparabolic inequalities
- Two-scale convergence

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics