Analysis of a one dimensional energy dissipating free boundary model with nonlinear boundary conditions. Existence of weak solutions

TL;DR

This paper introduces a one-dimensional free boundary model with nonlinear boundary conditions to study corrosion in nuclear waste repositories, proving the existence of maximal weak solutions over time.

Contribution

It develops a novel time semidiscrete scheme based on Wasserstein-like distances to establish the existence of solutions for a complex free boundary diffusion model.

Findings

Existence of solutions to the free boundary problem is proven.

A convergent scheme for approximating solutions is introduced.

Maximal weak solutions are shown to exist over time.

Abstract

This work is part of a general study on the long-term safety of the geological repository of nuclear wastes. A diffusion equation with a moving free boundary in one dimension is introduced and studied. The model describes some mechanisms involved in corrosion processes at the surface of carbon steel canisters in contact with a claystone formation. The main objective of the paper is to prove the existence of weak solutions to the problem which are maximal in time. For this, a time semidiscrete minimizing movements scheme based on a Wasserstein-like distance is introduced. The existence of solutions to the scheme is proved. Then, using a priori estimates, it is shown that as the time step goes to zero these solutions converge up to extraction towards a maximal weak solution to the free boundary model.