| Abstract: |
| Realistic models involving differential equations under constraints present challenges to mathematical analysis and numerical approximation. The analytical foundations within the framework of variational inequalities and monotone operator theory are well known, especially for some well studied free boundary problems, but the results do not extend broadly beyond scalar models.
In this talk we present three selected applications in porous media which we frame as evolution under constraints; these account for multiple fluid and solid phases which interact with each other in particular ways specific to an application while experimental data and images provide inspiration for models. In particular, we discuss biofilm growth at the porescale, hysteresis in adsorption, and hydrate crystal formation. If time permits, we will also touch on phenomena in permafrost. These models involve nonlinear couplings in addition to the constraints imposed in equilibrium or in non-equilibrium. Our results address the challenges of modeling, well-posedness as well as rigorous numerical approximation and simulations illustrating the challenges. Among these, we argue that the framework of constraints is more practical than of the smooth phase-field approaches. This is joint work with many students and other collaborators to be named in the talk. |
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