Mykola Yaremenko, On the thin-film equation: Renormalized solutions in one space dimension and non-uniqueness of weak solutions in higher dimensions, Eur. J. Math. Appl. 6 (2026), Article ID 7.
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Volume 6 (2026), Article ID 7
https://doi.org/10.28919/ejma.2026.6.7
Published: 31/07/2026
Abstract:
We develop a complete theory of renormalized and entropy solutions for the degenerate fourth-order thin-film equation $\partial_t u = -\nabla\cdot(u^n\nabla\Delta u)$ in one space dimension. Existence, uniqueness, and the fundamental role of the entropy dissipation are established through a rigorous regularisation procedure. We then analyse the step to several space dimensions $d\ge 2$. The natural weighted energy estimate $\iint u^n\vert{}\nabla\Delta u\vert{}^2 < \infty$ does not provide an unweighted $H^2$ bound, and the one-dimensional compactness arguments collapse. Known existence theorems for $d\ge 2$ only yield very weak solutions for restricted exponents $0 < n < 2$. As a novel contribution we prove that even for those exponents the weak formulation alone is insufficient to guarantee uniqueness: we exhibit a pair of distinct weak solutions emanating from the same Lipschitz initial datum. One is a stationary state with a positive contact angle, the other is a time-evolving entropy solution (obtained as the limit of the standard zero-contact-angle regularisation) that spreads instantly. The counterexample is rigorous, valid for any $d\ge 2$ and $0 < n < 2$, and shows that an additional selection principle (like an entropy condition) is necessary to restore uniqueness. An outlook on possible selection criteria and open problems concludes the paper.
How to Cite:
Mykola Yaremenko, On the thin-film equation: Renormalized solutions in one space dimension and non-uniqueness of weak solutions in higher dimensions, Eur. J. Math. Appl. 6 (2026), Article ID 7. https://doi.org/10.28919/ejma.2026.6.7