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arXiv · 2606.10793

Global weak solutions of a one-sided degenerate Cahn-Hilliard model for traction-driven digit morphogenesis

Marvin Fritz
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Abstract

We study a one-sided degenerate Cahn-Hilliard equation with anisotropic traction flux, arising as a reduced continuum description of mechanically biased cell interactions in digit-forming organoids. The equation combines a one-sided degenerate mobility with a density-weighted anisotropic higher-order transport term. This traction term is not generated by the variational derivative of the Cahn-Hilliard energy and therefore produces sign-indefinite contributions in the energy balance. For nonnegative initial data, we prove the global-in-time existence of nonnegative weak solutions. The proof combines an energy estimate for the diffusive flux with a mobility-matched entropy method adapted to the vacuum degeneracy. A key point is that the entropy variable cancels the mobility, turning the anisotropic traction contribution into a coercive first-order term in the entropy inequality, while the energy estimate supplies a weighted control of the diffusive flux.

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Marvin Fritz. 2026-06-09. Global weak solutions of a one-sided degenerate Cahn-Hilliard model for traction-driven digit morphogenesis. https://arxiv.org/abs/2606.10793

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