ELEMENTS OF SIMILARITY AND SINGULARITY

We briefly introduce through examples taken from the physics of fluids and con-tinuous media, the concepts of self-similarity and singularity. We start with the elementary formulation of dimensional analysis and its application to several prob-lems in lubrication and nonlinear diffusion. We also treat the more subtle case of second type similarity laws, related to the appearance of anomalous dimensions us-ing a dynamical renormalization group approach. Singularities in physical systems are presented using the example of shock formation for the simple wave equation. The method of characteristics is explained and the similarity solutions are related to the weak solutions of quasilinear equations using the dissipationless limit of the Burgers equation. 1