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- W1644580470 abstract "An approach to some (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 Holder 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 Holder 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 $n to 0$." @default.
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- W1644580470 date "2015-05-06" @default.
- W1644580470 modified "2023-10-17" @default.
- W1644580470 title "Towards optimal regularity for the fourth-order thin film equation in $re^N$: Graveleau-type focusing self-similarity" @default.
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