# Existence and uniqueness of solutions of partial differential equations | ðŸ—– nosplit | â†‘ parent "Differential equation" | 104

If you have a PDE that models 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 Picardâ€“LindelÃ¶f 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.