Towards optimal regularity for the fourth-order thin film equation in RNRN: Graveleau-type focusing self-similarity

An approach to some “optimal” (more precisely, non-improvable) regularity of solutions of the thin film equationut=−∇⋅(|u|n∇Δu)inRN×R+,u(x,0)=u0(x)inRN, where n∈(0,2)n∈(0,2) is a fixed exponent, with smooth compactly supported initial data u0(x)u0(x), in dimensions N≥2N≥2 is discussed. Namely, a precise exponent for the Hölder continuity with respect to the spatial radial variable |x||x| is obtained by construction of a Graveleau-type focusing self-similar solution. As a consequence, optimal regularity of the gradient ∇u   in certain LpLp spaces, as well as a Hölder 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=0n=0 and n>0n>0 are also plotted. In general, we claim that, even for arbitrarily small n>0n>0 and positive analytic initial data u0(x)u0(x), the solutions u(x,t)u(x,t) cannot be better than Cx2−ε-smooth, where ε(n)=O(n)ε(n)=O(n) as n→0n→0.