TY - JOUR

T1 - Towards optimal regularity for the fourth-order thin film equation in RN

T2 - Graveleau-type focusing self-similarity

AU - Alvarez-Caudevilla, P.

AU - Evans, J. D.

AU - Galaktionov, V. A.

PY - 2015/11/15

Y1 - 2015/11/15

N2 - An approach to some “optimal” (more precisely, non-improvable) regularity of solutions of the thin film equation
View the MathML sourceut=−∇⋅(|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 View the MathML sourceCx2−ε-smooth, where ε(n)=O(n)ε(n)=O(n) as n→0n→0.

AB - An approach to some “optimal” (more precisely, non-improvable) regularity of solutions of the thin film equation
View the MathML sourceut=−∇⋅(|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 View the MathML sourceCx2−ε-smooth, where ε(n)=O(n)ε(n)=O(n) as n→0n→0.

UR - http://dx.doi.org/10.1016/j.jmaa.2015.06.027

U2 - 10.1016/j.jmaa.2015.06.027

DO - 10.1016/j.jmaa.2015.06.027

M3 - Article

VL - 431

SP - 1099

EP - 1123

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 2

ER -