Exact, Rotational, Infinite Energy, Blowup Solutions to the 3-Dimensional 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 ${\hat{I}}^3>1$:%} [c]{c}% ρ=\max\{\frac{γ-1}{Kγ}[ C^{2}[ x^{2}% +y^{2}+z^{2}-(xy+yz+xz)] -\dot{a}(t)(x+y+z)+b(t)], 0\} ^{\frac{1}{γ-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 $\mathrm{div}\vec{u}=0$ for the solutions, the solutions also work for the 3-dimnsional incompressible Euler and Navier-Stokes equations.

9 pages; Key Words: Euler Equations, Exact Solutions, Rotational, Compressible, Incompressible, Symmetry Reductions, Blowup, Global Solutions, Infinite Energy, Navier-Stokes Equations, Free Boundary