"Better" is subjective and dependent on the problem. In most cases, Navier-Stokes is probably what one should be using. As an example of when not to use Navier-Stokes, if the density is low enough you'd probably want something like the Boltzmann equation. But that typically adds a lot of complexity and computational cost, so it's best to not use the Boltzmann equation unless strictly necessary. See this for an example of a computational implementation of the Boltzmann equation: https://en.wikipedia.org/wiki/Direct_simulation_Monte_Carlo
> Also, do you think it's possible to improve the equations in case a proof for the singularities can be found?
If there are possible singularities, they don't seem to be causing issues in practice, so I'm not sure anything needs to be done.
But yes, the equations can be fixed if necessary by using a different viscosity law. As I said in my other post, it was proved a long time ago that 3D Navier-Stokes with a particular more realistic viscosity law has smooth and unique solutions. Plus, in many (if not most) instances the Navier-Stokes equations are not used directly because the computational cost is prohibitively high, and instead approximations are used. I'm not aware of proofs that the approximate equations have smooth and unique solutions but I think they are more likely to as in effect they add a lot of additional viscosity, which usually helps.
> Since you are researching fluid dynamics, what is in your opinion the most interesting development in the field right now, theoretical or practical?
Sorry, but there isn't a lot that I like going on right now. You can skip the next paragraph if you don't want the rant.
Rant: Fluid dynamics as a whole is distracted by things that mostly don't matter like running simulations on bigger and bigger supercomputers (as if sheer size compensates for the fact that they aren't computing something that matters), or doing experiments with fancy equipment involving things like lasers and X-rays (motivated more by trying a new technique than measuring something that matters; do you see the theme?), rather than trying to measure something that matters or understand the physics. I've been told by many others that developments using machine learning right now are very exciting, but I don't agree because those models seem extremely fragile to me. Seems to be the bandwagon effect there. Another thing I dislike about "data-driven" approaches is that they aren't really data-driven... they tend to ignore the huge amount of previously published literature and data in favor of running new experiments and simulations, often duplicating previous experiments but worse than was done before. Seems to be similar to "not invented here" syndrome: "not measured here".
So what do I actually like that's going on right now? I like research that tries to tie together a lot of the hard data and theory that has been published over the past 100+ years and find gaps. That's rare. Most review articles compile conclusions based on a small amount of data rather than compiling the data itself and then drawing conclusions from that. The common behavior leads to a whole host of issues like Simpson's paradox. Computers now make compiling a ton of data from the open literature easier than ever, but you see few people doing it.
Here's an example of someone taking advantage of the huge amount of data (open access): https://doi.org/10.1115/1.4046795
And I've done the same sort of thing myself: https://engrxiv.org/preprint/view/740