On turning waves for the inhomogeneous Muskat problem: a computer-assisted proof (1311.0430v2)

Abstract: We exhibit a family of graphs that develop turning singularities (i.e. their Lipschitz seminorm blows up and they cease to be a graph, passing from the stable to the unstable regime) for the inhomogeneous, two-phase Muskat problem where the permeability is given by a nonnegative step function. We study the influence of different choices of the permeability and different boundary conditions (both at infinity and considering finite/infinite depth) in the development or prevention of singularities for short time. In the general case (inhomogeneous, confined) we prove a bifurcation diagram concerning the appearance or not of singularities when the depth of the medium and the permeabilities change. The proofs are carried out using a combination of classical analysis techniques and computer-assisted verification.

Turning Waves in Inhomogeneous Muskat Problems

This paper addresses the development of turning singularities in the inhomogeneous Muskat problem, where differing permeabilities between two incompressible fluids create complex dynamics that are rigorously analyzed. The primary focus is on the evolution of fluid interfaces and how variations in the permeability and boundary conditions influence the emergence of singularities.

The paper presents a computer-assisted proof of the existence of singularities in the inhomogeneous two-phase Muskat problem with permeability modeled by a step function. The authors explore a range of boundary conditions and permeability types to determine their roles in forming these turning singularities.

Key Contributions

1. Singularity Formation Analysis: The paper highlights how varying permeability profiles can trigger turning singularities, where previously smooth interfaces become unstable. The Lipschitz seminorm blows up, causing the interface to cease being a graph and shift from a stable to an unstable regime.
2. Bifurcation Diagrams and Stability: A detailed bifurcation analysis illustrates how changes in medium depth and permeabilities affect singularity formation, providing a comprehensive understanding of the confined and unconfined regimes. The authors demonstrate cases where boundary constraints induce singularities or where changes in permeabilities can either prevent or promote them.
3. Computer-Assisted Verification: Using interval arithmetic and advanced computational techniques, the authors validate the existence of singularities rigorously. This involves transforming complex multidimensional integrals to more tractable forms and leveraging adaptive algorithms to ensure numerical stability and precision.

Implications and Future Research

The findings in this paper carry significant implications for understanding fluid dynamics in porous media, particularly relevant to modeling aquifers, oil wells, and geothermal reservoirs. The computer-assisted approach provides a template for handling complex mathematical proofs where traditional analytical techniques fall short. This contributes to developing more reliable predictive models for real-world applications in engineering and environmental sciences.

Theoretically, the exploration opens new avenues for investigating non-linear PDEs governing fluid dynamics. Future research could expand on this by exploring alternative modeling assumptions, incorporating more complex boundary conditions, and examining the effect of varying fluid viscosities.

Moreover, the analysis paves the way for more computational advancements in proving mathematical phenomena. The use of rigorous computation alongside classical analysis methods offers a robust framework that could be applied to a wide array of mathematical and engineering challenges beyond the scope of fluid dynamics.

In conclusion, this paper combines theoretical insights with computational prowess to tackle the intricate inhomogeneous Muskat problem, contributing to a deeper understanding of wave behavior in porous media and setting the stage for future inquiry into complex fluid interfaces.