# Article

 Title: The microstructure of Lipschitz solutions for a one-dimensional logarithmic diffusion equation  (English) Author: Schadewaldt, Nicole Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 52 Issue: 2 Year: 2011 Pages: 227-255 Summary lang: English . Category: math . Summary: 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 $u_x = d^+ \mathbbm{1}_A + d^- \mathbbm{1}_B$ 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. Keyword: logarithmic diffusion Keyword: one-dimensional Keyword: differential inclusion Keyword: microstructured Lipschitz solutions MSC: 34A05 MSC: 35B05 MSC: 35B65 . Date available: 2011-05-17T08:37:07Z Last updated: 2012-08-13 Stable URL: http://hdl.handle.net/10338.dmlcz/141500 . Reference: [1] Ambrosio L., Fusco N., Pallara D.: Functions of Bounded Variation and Free Discontinuity Problems.Oxford Mathematical Monographs, Clarendon Press, Oxford, 2000. Zbl 0957.49001, MR 1857292 Reference: [2] Ball J.M.: A version of the fundamental theorem of Young measures.in Partial Differential Equations and Continuum Models of Phase Transitions, M. Rascle, D. Serre and M. Slemrod, editors, Lecture Notes in Physics, 334, Springer, Berlin, 1989, pp. 207–215. MR 1036070 Reference: [3] Ball J.M., James R.D.: Proposed experimental test of a theory of fine microstructure and the two-well problem.Philosophical Transactions: Physical Sciences and Engineering, 338 (1992), no. 1650, 389–450. Reference: [4] Conti S., Dolzmann B., Kirchheim B.: Existence of Lipschitz minimizers for the three-well problem in solid-solid phase transitions.Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), 953–962. Zbl 1131.74037, MR 2371114 Reference: [5] Dacorogna B., Marcellini P.: Implicit Partial Differential Equations.Progress in Nonlinear Differential Equations and their Applications, 37, Birkhäuser, Basel, 1999. Zbl 0939.49013, MR 1702252 Reference: [6] DiBenedetto E., Herrero M.A.: Non-negative solutions of the evolution $p$-Laplacian equation. Initial traces and Cauchy problem when \$1

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