{c}
{tag=Millennium Prize Problems}
{tag=Existence and uniqueness of solutions of partial differential equations}
{wiki=Navier–Stokes_existence_and_smoothness}

See also: <existence and uniqueness of solutions of partial differential equations>.

On December 19, 2025 a bet was revealed between two <DeepMind> guys that one of them would have a <Lean> proof to the problem:
* https://x.com/mhutter42/status/2001857421569032444
* https://www.reddit.com/r/singularity/comments/1prw1kv/former_deepmind_director_of_engineering_david/

\Video[https://www.youtube.com/watch?v=BaDxv5Z4LkU]
{title=A report on the Navier-Stokes Problem by Vladimir Šverak}
{description=2025.}


Navier-Stokes existence and smoothness

{tag=Existence and uniqueness}

If you have a <PDE> that models <physics>[physical phenomena], it is fundamental that:
* there must exist a solution for every physically valid initial condition, otherwise it means that the equation does not describe certain cases of reality
* the solution must be unique, otherwise how are we to choose between the multiple solutions?

Unlike for <ordinary differential equations> which have the https://en.wikipedia.org/wiki/Picard–Lindelöf_theorem[Picard–Lindelöf theorem], the existence and uniqueness of solution is not well solved for PDEs.

For example, <Navier-Stokes existence and smoothness> was one of the <Millennium Prize Problems>.


Ciro Santilli @cirosantilli 37

 Tagged: Existence and uniqueness of solutions of partial differential equations