(I,J) similar solutions to Euler and Navier-Stokes equations

TL;DR

This paper introduces the (I,J) similar method for solving incompressible Euler and Navier-Stokes equations, providing explicit solutions including singular, smooth, and piecewise solutions, and analyzing their properties and limits.

Contribution

It develops the (I,J) similar method for Euler and Navier-Stokes equations, yielding new explicit solutions and insights into their behaviors and limits.

Findings

Derived explicit (I,J) similar solutions including twin wave and singular solutions.

Identified plane wave and constant vector as special solutions.

Showed viscosity limit solutions may not converge to Euler solutions.

Abstract

In this paper we introduce (I,J) similar method for incompressible two and three dimensional Euler equations and Navier-Stokes equations, obtain a series of explicit (I,J) similar solutions to the incompressible two dimensional Euler equations, they include all of twin wave solutions, some new singularity solutions, and some global smooth solutions with finite energy. We also discover that twin wave solution and affine solution to two dimensional incompressible Euler equations are respectively plane wave and constant vector. Finally, we supply some explicit piecewise smooth solutions to incompressible three dimensional Euler and an example to incompressible three dimensional Navier-Stokes equations which indicates that viscosity limit of a solution to Navier-Stokes equations does not need to be a solution to Euler equations.