The Poincaré conjecture was solved by Grigory Perelman in 2006, this article is about James Glimm's solution of the Navier-Stokes Equation - see the paper at https://arxiv.org/pdf/2212.14734.pdf
The mathematics in this article looks not all that complex. From the little that I understand about it, it is not a direct proof that the equations are stable, but that it is based on a definition of enthropy and a stochastic argument. That argument gives good reasons to assume that the equations are smooth, but I wonder if it proofs that it is smooth in all cases. In this regard, it is interesting to note that in a recent paper [0] it was shown that Euler equations (a simpler system than the Navier-Stokes equations) might be instable.
