Abstract
An approach to some “optimal” (more precisely, nonimprovable) regularity of solutions of the thin film equation
View the MathML sourceut=−∇⋅(un∇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 xx is obtained by construction of a Graveleautype focusing selfsimilar 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 View the MathML sourceCx2−εsmooth, where ε(n)=O(n)ε(n)=O(n) as n→0n→0.
Original language  English 

Pages (fromto)  10991123 
Number of pages  25 
Journal  Journal of Mathematical Analysis and Applications 
Volume  431 
Issue number  2 
Early online date  18 Jun 2015 
DOIs  
Publication status  Published  15 Nov 2015 
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Jonathan Evans
 Department of Mathematical Sciences  Reader
 EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
 Institute for Mathematical Innovation (IMI)
 Centre for Nonlinear Mechanics
 EPSRC Centre for Doctoral Training in Advanced Automotive Propulsion Systems (AAPS CDT)
Person: Research & Teaching, Affiliate staff