On some geometrical aspects of the potential structure of the equations of evolution: The case of Navier-Stokes

Federal Research Center, Institute of Applied Mathematics, M. V. Keldysh Institute of the Russian Academy of Sciences - Miusskaya sq. 4, 125047, Moscow, Russian Federation and National Institute of Plasma Physics (INFIP), Consejo Nacional de Investigaciones Cientificas y Tecnicas (CONICET), Universidad de Buenos Aires - Buenos Aires, Argentina

^(a) diego777jcl@gmail.com (corresponding author)

Received: 14 May 2022
Accepted: 20 June 2022

Abstract

In this paper we discuss the potential structure of the evolution equations, in particular Navier-Stokes. To this end, the method of prolongation of Wahlquist H. D. and Estabrook F. B., J. Math. Phys., 16 (1975) 1 is introduced and the most general potential for the flow velocity is found, expressing everything in terms of the representative differential forms of the system of equations. Steady-flow and self-similar solutions and conditions are presented and briefly discussed, as well as the most general solution when a general transformation similar to the one given by Cole is introduced into the original system. In this theoretical context, the solution can be associated with a damped acoustic wave. Consequently, a useful application area for the present work is certainly in nonlinear acoustics, as we discuss briefly at the end of this letter.