2015/05/06 by Pablo Álvarez‐Caudevilla, Alvarez-Caudevilla, Pablo, Jonathan D. Evans +3
Computer Science · Engineering · Materials Science · Mathematics · #35G20 #35K35 #35K65 #37K50 #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Fluid Dynamics and Thin Films #Navier-Stokes equation solutions #Nonlinear Dynamics and Pattern Formation #Solidification and crystal growth phenomena
paper · pdf · doi:10.48550/arxiv.1505.01267
openalex publication_date 2015/05/06 · openalex created_date 2022/09/14 · openalex updated_date 2026/07/28
An approach to some "optimal" (more precisely, non-improvable) regularity of\nsolutions of the thin film equation ut = -\∇ \⋅(|u|n \∇ D u)\nin ren \× re+, u(x,0)=u0(x) in reN, where n in (0,2) is a fixed\nexponent, with smooth compactly supported initial data u0(x), in dimensions N\n\≥ 2 is discussed. Namely, a precise exponent for the H "older continuity\nwith respect to the spatial radial variable |x| is obtained by construction\nof a Graveleau-type focusing self-similar solution. As a consequence, optimal\nregularity of the gradient \∇ u in certain Lp spaces, as well as a\nH "older continuity property of solutions with respect to x and t, are derived,\nwhich cannot be obtained by classic standard methods of integral\nidentities-inequalities. Several profiles for the solutions in the cases n=0\nand n>0 are also plotted.\n In general, we claim that, even for arbitrarily small n>0 and positive\nanalytic initial data u0(x), the solutions u(x,t) cannot be better than\nCx2- e-smooth, where e(n)=O(n) as n \→ 0.\n