On the blow-up of some complex solutions of the 3-d Navier-Stokes Equations: Theoretical Predictions and Computer simulations
For mathematicians studying Navier-Stokes regularity, this provides numerical insights into blow-up mechanisms, but the work is incremental as it builds on existing theoretical results.
The authors analyze complex-valued solutions of the 3D Navier-Stokes equations that blow up in finite time, identifying two types with different divergence rates. Computer simulations reveal energy and enstrophy concentration near singular points, with the rest of the fluid remaining quiet, features not predicted by theory.
We consider some complex-valued solutions of the Navier-Stokes equations in $R^{3}$ for which Li and Sinai proved a finite time blow-up. We show that there are two types of solutions, with different divergence rates, and report results of computer simulations, which give a detailed picture of the blow-up for both types. They reveal in particular important features not, as yet, predicted by the theory, such as a concentration of the energy and the enstrophy around a few singular points, while elsewhere the "fluid" remains quiet.