Existence of solutions for a higher order non-local equation appearing in crack dynamics

TL;DR

This paper proves the existence of non-negative solutions for a non-local, higher order degenerate parabolic equation modeling hydraulic fractures, addressing challenges posed by nonlocality and higher order derivatives.

Contribution

It introduces a novel existence proof for solutions to a non-local, higher order degenerate parabolic equation related to crack dynamics, extending analysis beyond classical thin film equations.

Findings

Existence of non-negative solutions established.

The equation's nonlocal nature complicates energy estimates.

Solutions lack boundedness and continuity due to critical dimension.

Abstract

In this paper, we prove the existence of non-negative solutions for a non-local higher order degenerate parabolic equation arising in the modeling of hydraulic fractures. The equation is similar to the well-known thin film equation, but the Laplace operator is replaced by a Dirichlet-to-Neumann operator, corresponding to the square root of the Laplace operator on a bounded domain with Neumann boundary conditions (which can also be defined using the periodic Hilbert transform). In our study, we have to deal with the usual difficulty associated to higher order equations (e.g. lack of maximum principle). However, there are important differences with, for instance, the thin film equation: First, our equation is nonlocal; Also the natural energy estimate is not as good as in the case of the thin film equation, and does not yields, for instance, boundedness and continuity of the solutions (our case is critical in dimension $1$ in that respect).