Towards optimal regularity for the fourth-order thin film equation in $\re^N$: Graveleau-type focusing self-similarity

TL;DR

This paper constructs Graveleau-type self-similar solutions to determine the optimal spatial regularity of solutions to the fourth-order thin film equation in multiple dimensions, revealing limitations on smoothness even for small nonlinearity.

Contribution

It provides the first precise exponent for Hölder continuity of solutions, establishing optimal regularity bounds using self-similar solutions, beyond standard methods.

Findings

Optimal Hölder exponent for spatial regularity is obtained.

Solutions cannot be smoother than $C_x^{2- ext{small}}$ for small n.

Constructed self-similar solutions demonstrate regularity limits.

Abstract

An approach to some "optimal" (more precisely, non-improvable) regularity of solutions of the thin film equation u_{t} = -\nabla \cdot(|u|^{n} \nabla \D u) in \ren \times \re_+, u(x,0)=u_0(x) in \re^N, where n in (0,2) is a fixed exponent, with smooth compactly supported initial data u_0(x), in dimensions $N \geq 2$ is discussed. Namely, a precise exponent for the H\"older continuity with respect to the spatial radial variable $|x|$ is obtained by construction of a Graveleau-type focusing self-similar solution. As a consequence, optimal regularity of the gradient $\nabla u$ in certain $L^p$ spaces, as well as a H\"older continuity property of solutions with respect to x and t, are derived, which cannot be obtained by classic standard methods of integral identities-inequalities. Several profiles for the solutions in the cases n=0 and n>0 are also plotted. In general, we claim that, even for arbitrarily small n>0 and positive analytic initial data u_0(x), the solutions u(x,t) cannot be better than $C_x^{2-\e}$-smooth, where $\e(n)=O(n)$ as $n \to 0$.