The microstructure of Lipschitz solutionsfor a one-dimensional logarithmic diffusion equation

Commentationes mathematicae Universitatis Carolinae

Nicole Schadewaldt¹ 

关键词: logarithmic diffusion;    one-dimensional;    differential inclusion;    microstructured Lipschitz solutions;   

DOI  :  

学科分类：物理化学和理论化学

来源: Univerzita Karlova v Praze * Matematicko-Fyzikalni Fakulta / Charles University in Prague, Faculty of Mathematics and Physics

【 摘 要 】

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^-

【 授权许可】

CC BY