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

1105.4216May 21, 20119 pages; Key Words: Euler Equations, Exact Solutions, Rotational, Compressible, Incompressible, Symmetry Reductions, Blowup, Global Solutions, Infinite Energy, Navier-Stokes Equations, Free Boundary

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

Manwai Yuen

Mathematical Physics Analysis of PDEs

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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 $γ>1$ :% [c]c% ρ={(γ-1/Kγ)[ C²[ x²% +y²+z²-(xy+yz+xz)] -a(t)(x+y+z)+b(t)], 0} ^(1/γ-1) u₁=a(t)+C(y-z) u₂=a(t)+C(-x+z) u₃=a(t)+C(x-y). where a(t)=c₀+c₁t and b(t)=3c₀c₁t+3/2c₁²t²+c₂% with $C$ , $c_{0}$ , $c_{1}$

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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.

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