One strategy mathematicians have pursued to do that is to first relax just how descriptive they require solutions to the equations to be. When mathematicians study equations like Navier-Stokes, they sometimes start by broadening their definition of what counts as a solution.
If you think of a smooth solution as a mathematical image of a fluid down to infinitely fine resolution, weak solutions are like the 32-bit, or 16-bit, or 8-bit version of that picture.
Solutions to the Navier-Stokes equations correspond to real physical events, and physical events happen in just one way. If the equations give you multiple possible solutions, they’ve failed.
Nonunique Leray solutions would mean that, according to the rules of Navier-Stokes, the exact same fluid from the exact same starting conditions could end up in two distinct physical states, which makes no physical sense and implies that the equations aren’t really describing what they’re supposed to describe.
“Their result is certainly a warning, but you could argue it’s a warning for the weakest notion of weak solutions. There are many layers on which you could still hope for much better behavior” in the Navier-Stokes equations, said De Lellis.