Preprint

Inserted: 7 jul 2026
Last Updated: 7 jul 2026

Year: 2026

Abstract:

We extend the validity of well-known integral inequalities for the thin-film equation to the range $n \in (\frac12,3)$ in any space dimension $N$. Consequently, known existence results on strong solutions to the thin-film equation for spatial dimensions $N < 4$ as well as recent results by Cornalba, Fischer and Maringova-Kokavcova on their H\¨older continuity for $N = 2$ are extended to the range $n \in ( \frac12, 3)$. We also show that the above-mentioned range is optimal and extend some dissipation relations involving weighted Dirichlet energies to arbitrary spatial dimensions.

Keywords: free boundary problems, thin-film equation, Integral inequalities, Higher-order degenerate parabolic equations, Lubrication theory

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