Exact, Rotational, Infinite Energy, Blowup Solutions to the 3-Dimensional Euler Equations

TL;DR

This paper constructs explicit rotational blowup solutions for 3D Euler and Navier-Stokes equations, revealing new infinite energy solutions with local interesting behaviors and extending to incompressible and compressible flows.

Contribution

The paper introduces a new class of exact, rotational blowup solutions for 3D Euler and Navier-Stokes equations, including infinite energy solutions similar to ABC flow.

Findings

Constructed explicit blowup solutions with elementary functions.

Solutions exhibit infinite energy and local interesting behaviors.

Extended solutions to incompressible and compressible Euler equations.

Abstract

In this paper, we construct a new class of blowup solutions with elementary functions to the 3-dimensional compressible or incompressible Euler and Navier-Stokes equations. In detail, we obtain a class of global rotational exact solutions for the compressible fluids with $\gamma>1$:%} [c]{c}% \rho=\max\{\frac{\gamma-1}{K\gamma}[ C^{2}[ x^{2}% +y^{2}+z^{2}-(xy+yz+xz)] -\dot{a}(t)(x+y+z)+b(t)], 0\} ^{\frac{1}{\gamma-1}} u_{1}=a(t)+C(y-z) u_{2}=a(t)+C(-x+z) u_{3}=a(t)+C(x-y). where a(t)=c_{0}+c_{1}t and b(t)=3c_{0}c_{1}t+{3/2}c_{1}^{2}t^{2}+c_{2}% with $C$, $c_{0}$, $c_{1}$ and $c_{2}$ are arbitrary constants; And the corresponding blowup or global solutions for the incompressible Euler equations are also given. Our constructed solutions are similar to the famous Arnold-Beltrami-Childress (ABC) flow. The solutions with infinite energy can exhibit the interesting behaviors locally. Besides, the corresponding global solutions are also given for the compressible Euler equations. Furthermore, due to $\operatorname{div}\vec{u}=0$ for the solutions, the solutions also work for the 3-dimnsional incompressible Euler and Navier-Stokes equations.