A Computational Investigation of the Finite-Time Blow-Up of the 3D Incompressible Euler Equations Based on the Voigt Regularization
This work provides a computational method to study blow-up in the 3D Euler equations, a fundamental problem in fluid dynamics, but the criteria are only sufficient and the approach is tested on a simpler model.
The authors computationally investigate two blow-up criteria for the 3D incompressible Euler equations using the Euler-Voigt regularization, proving a new criterion and testing robustness via 1D Burgers equation. The approach indirectly probes singularity formation by simulating the better-behaved regularized equations.
We report the results of a computational investigation of two blow-up criteria for the 3D incompressible Euler equations. One criterion was proven in a previous work, and a related criterion is proved here. These criteria are based on an inviscid regularization of the Euler equations known as the 3D Euler-Voigt equations, which are known to be globally well-posed. Moreover, simulations of the 3D Euler-Voigt equations also require less resolution than simulations of the 3D Euler equations for fixed values of the regularization parameter $α>0$. Therefore, the new blow-up criteria allow one to gain information about possible singularity formation in the 3D Euler equations indirectly; namely, by simulating the better-behaved 3D Euler-Voigt equations. The new criteria are only known to be sufficient for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well-known to occur.