Exact, rotational, infinite energy, blowup solutions to the 3-dimensional Euler equations

In this Letter, we construct a new class of blowup or global solutions with elementary functions to the 3-dimensional compressible or incompressible Euler and Navier–Stokes equations. And the corresponding blowup or global solutions for the incompressible Euler and Naiver–Stokes equations are also given. Our constructed solutions are similar to the famous Arnold–Beltrami–Childress (ABC) flow. The obtained solutions with infinite energy can exhibit the interesting behaviors locally. Furthermore, due to divu→=0 for the solutions, the solutions also work for the 3-dimensional incompressible Euler and Navier–Stokes equations.

► We construct a new class of solutions to the 3D compressible or incompressible Euler and Navier–Stokes equations. ► The constructed solutions are similar to the famous Arnold–Beltrami–Childress flow. ► The solutions with infinite energy can exhibit the interesting behaviors locally.