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

Toai Luong; Steve Wise

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arXiv · 2404.13482

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

Toai Luong·Steve Wise

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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.

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Toai Luong, Steve Wise. 2024-04-20. A Nonnegative Weak Solution to the Phase Field Crystal Model with Degenerate Mobility. https://arxiv.org/abs/2404.13482

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