The microstructure of Lipschitz solutions for a one-dimensional logarithmic diffusion equation

Nicole Schadewaldt

Commentationes Mathematicae Universitatis Carolinae (2011)

  • Volume: 52, Issue: 2, page 227-255
  • ISSN: 0010-2628

Abstract

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We consider the initial-boundary-value problem for the one-dimensional fast diffusion equation u t = [ sign ( u x ) log | u x | ] x on Q T = [ 0 , T ] × [ 0 , l ] . For monotone initial data the existence of classical solutions is known. The case of non-monotone initial data is delicate since the equation is singular at u x = 0 . We ‘explicitly’ construct infinitely many weak Lipschitz solutions to non-monotone initial data following an approach to the Perona-Malik equation. For this construction we rephrase the problem as a differential inclusion which enables us to use methods from the description of material microstructures. The Lipschitz solutions are constructed iteratively by adding ever finer oscillations to an approximate solution. These fine structures account for the fact that solutions are not continuously differentiable in any open subset of Q T and that the derivative u x is not of bounded variation in any such open set. We derive a characterization of the derivative, namely u x = d + 1 A + d - 1 B with continuous functions d + > 0 and d - < 0 and dense sets A and B , both of positive measure but with infinite perimeter. This characterization holds for any Lipschitz solution constructed with the same method, in particular for the ‘microstructured’ Lipschitz solutions to the one-dimensional Perona-Malik equation.

How to cite

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Schadewaldt, Nicole. "The microstructure of Lipschitz solutions for a one-dimensional logarithmic diffusion equation." Commentationes Mathematicae Universitatis Carolinae 52.2 (2011): 227-255. <http://eudml.org/doc/246936>.

@article{Schadewaldt2011,
abstract = {We consider the initial-boundary-value problem for the one-dimensional fast diffusion equation $u_t = [\operatorname\{sign\}(u_x) \log |u_x|]_x$ on $Q_T=[0,T]\times [0,l]$. For monotone initial data the existence of classical solutions is known. The case of non-monotone initial data is delicate since the equation is singular at $u_x=0$. We ‘explicitly’ construct infinitely many weak Lipschitz solutions to non-monotone initial data following an approach to the Perona-Malik equation. For this construction we rephrase the problem as a differential inclusion which enables us to use methods from the description of material microstructures. The Lipschitz solutions are constructed iteratively by adding ever finer oscillations to an approximate solution. These fine structures account for the fact that solutions are not continuously differentiable in any open subset of $Q_T$ and that the derivative $u_x$ is not of bounded variation in any such open set. We derive a characterization of the derivative, namely $u_x = d^+ \mathbb \{1\}_A + d^- \mathbb \{1\}_B$ with continuous functions $d^+>0$ and $d^-<0$ and dense sets $A$ and $B$, both of positive measure but with infinite perimeter. This characterization holds for any Lipschitz solution constructed with the same method, in particular for the ‘microstructured’ Lipschitz solutions to the one-dimensional Perona-Malik equation.},
author = {Schadewaldt, Nicole},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {logarithmic diffusion; one-dimensional; differential inclusion; microstructured Lipschitz solutions; logarithmic diffusion; differential inclusion; microstructured Lipschitz solution},
language = {eng},
number = {2},
pages = {227-255},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {The microstructure of Lipschitz solutions for a one-dimensional logarithmic diffusion equation},
url = {http://eudml.org/doc/246936},
volume = {52},
year = {2011},
}

TY - JOUR
AU - Schadewaldt, Nicole
TI - The microstructure of Lipschitz solutions for a one-dimensional logarithmic diffusion equation
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2011
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 52
IS - 2
SP - 227
EP - 255
AB - We consider the initial-boundary-value problem for the one-dimensional fast diffusion equation $u_t = [\operatorname{sign}(u_x) \log |u_x|]_x$ on $Q_T=[0,T]\times [0,l]$. For monotone initial data the existence of classical solutions is known. The case of non-monotone initial data is delicate since the equation is singular at $u_x=0$. We ‘explicitly’ construct infinitely many weak Lipschitz solutions to non-monotone initial data following an approach to the Perona-Malik equation. For this construction we rephrase the problem as a differential inclusion which enables us to use methods from the description of material microstructures. The Lipschitz solutions are constructed iteratively by adding ever finer oscillations to an approximate solution. These fine structures account for the fact that solutions are not continuously differentiable in any open subset of $Q_T$ and that the derivative $u_x$ is not of bounded variation in any such open set. We derive a characterization of the derivative, namely $u_x = d^+ \mathbb {1}_A + d^- \mathbb {1}_B$ with continuous functions $d^+>0$ and $d^-<0$ and dense sets $A$ and $B$, both of positive measure but with infinite perimeter. This characterization holds for any Lipschitz solution constructed with the same method, in particular for the ‘microstructured’ Lipschitz solutions to the one-dimensional Perona-Malik equation.
LA - eng
KW - logarithmic diffusion; one-dimensional; differential inclusion; microstructured Lipschitz solutions; logarithmic diffusion; differential inclusion; microstructured Lipschitz solution
UR - http://eudml.org/doc/246936
ER -

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