A Nonnegative Weak Solution to the Phase Field Crystal Model with Degenerate Mobility

TL;DR

This paper proves the existence of a nonnegative weak solution to the phase field crystal model with degenerate mobility, extending previous results and ensuring energy dissipation, which is crucial for modeling crystalline materials.

Contribution

It establishes the existence of a nonnegative weak solution for the phase field crystal model with degenerate mobility, a novel extension of prior work with non-degenerate mobilities.

Findings

Existence of weak solutions with non-degenerate mobility.

Existence of nonnegative weak solutions with degenerate mobility.

Verification of energy dissipation inequality for the solutions.

Abstract

Phase field crystal is a model used to describe the behavior of crystalline materials at the mesoscale. In this study, we investigate the well-posedness of a phase field crystal equation subject to a degenerate mobility $M(u)$ that equals zero for $u\leq 0$. First, we prove the existence of a weak solution to a phase field crystal equation with non-degenerate cutoff mobility. Then, assuming that the initial data $u_0(x)$ is positive, we establish the existence of a nonnegative weak solution to the degenerate case. Such solution is the limit of solutions corresponding to non-degenerate mobilities. We also verify that such a weak solution satisfies an energy dissipation inequality.