The Navier-Stokes Equation
Describes the evolution of a fluid. Quanta magazine has a nice article on mathematicians' attempts to probe the limits of the equation, and more importantly, for me, a lovely video of a Kelvin-Helmholtz instability evolving under the equation. Mathematicians have long suspected that there might be something dodgy about the equation, and there is big money, a Millenium Prize, riding on the conjecture that the equation doesn't always have consistent solutions.
Quanta notes that these potential problems don't bother physicists, but doesn't bother to say why. The more fundamental reason is that physicists know that the N-S is not a faithful description of nature. If you look at a fluid in close detail, it becomes a seething mass of individual particles, not infinitely divisible fluid elements. There is also another fact: because the NS has chaotic solutions, their predictivity is always limited in practice, whether it is in theory or not.