Equations of hydrodynamic type: exact solutions, reduction of order, transformations, and nonlinear stability/unstability

TL;DR

This paper investigates hydrodynamic type equations derived from Navier-Stokes and boundary layer equations, introducing a transformation to reduce equation order and proving the nonlinear instability of many solutions using a novel exact method.

Contribution

It presents a Crocco-type transformation to reduce equation order and introduces a new exact method to prove the nonlinear instability of solutions.

Findings

Many solutions of Navier-Stokes equations are inherently unstable.

A new exact method for analyzing nonlinear stability is developed.

The instability results have implications for understanding fluid dynamics phenomena.

Abstract

Systems of hydrodynamic type equations derived from the Navier-Stokes equations and the boundary layer equations are considered. A transformation of the Crocco type reducing the equation order for the longitudinal velocity component is described. The issues of nonlinear stability of the obtained solutions are studied. It is found that a specific feature of many solutions of the Navier-Stokes equations is instability. The nonlinear instability of solutions is proved by a new exact method, which may be useful for the analysis of other nonlinear physical models and phenomena.