Asymptotic Forms of Solutions to System of Nonlinear Partial Differential Equations

Abstract

Here, we considerably develop the methods of power geometry for a system of partial differential equations and apply them to two different fluid dynamics problems: computing the boundary layer on a needle in the first approximation and computing the asymptotic forms of solutions to the problem of evolution of the turbulent flow. For each equation of the system, its Newton polyhedron and its hyperfaces with their normals and truncated equations are calculated. To simplify the truncated systems, power-logarithmic transformations are used and the truncated systems are further extracted. Here, we propose algorithms for computing unimodular matrices of power transformations for differential equations. Results: (1) the boundary layer on the needle is absent in liquid, while in gas it is described in the first approximation; (2) the solutions to the problem of evolution of turbulent flow have eight asymptotic forms, presented explicitly.