Power geometry in nonlinear partial differential equations

Abstract

Power Geometry (PG) is a new calculus developing the differential calculus and aimed at nonlinear problems. The main concept of PG is the study of nonlinear problems in logarithms of original coordinates. Then many relations nonlinear in the original coordinates become linear. The algorithms of PG are based on these linear relations. They allow to simplify equations, to resolve their singularities (including singular perturbations), to isolate their first approximations, and to find asymptotic forms and asymptotic expansions of their solutions. In particular, they give simple methods to identify the equations and systems as quasihomogeneous, and then to introduce for them self-similar coordinates. As an application, we consider the stationary spatial axially symmetric flow of the viscous compressible heat conducting gas around a semi-infinite needle. Other application: finding blow-up solutions.