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

Graveleau-type focusing self-similarity

P. Alvarez-Caudevilla, J. D. Evans, V. A. Galaktionov

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Abstract

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.
Original languageEnglish
Pages (from-to)1099-1123
Number of pages25
JournalJournal of Mathematical Analysis and Applications
Volume431
Issue number2
Early online date18 Jun 2015
DOIs
Publication statusPublished - 15 Nov 2015

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Thin Film Equation
Fourth-order Equations
Hölder Continuity
Self-similarity
Regularity
Exponent
Thin films
Regularity of Solutions
Self-similar Solutions
Thin Films
Gradient
Profile
Standards

Cite this

@article{0e97717502c84111a131e7f9ab107318,
title = "Towards optimal regularity for the fourth-order thin film equation in RN: Graveleau-type focusing self-similarity",
abstract = "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 H{\"o}lder 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 H{\"o}lder 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.",
author = "P. Alvarez-Caudevilla and Evans, {J. D.} and Galaktionov, {V. A.}",
year = "2015",
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language = "English",
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T1 - Towards optimal regularity for the fourth-order thin film equation in RN

T2 - Graveleau-type focusing self-similarity

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AU - Evans, J. D.

AU - Galaktionov, V. A.

PY - 2015/11/15

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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.

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