Existence and uniqueness of solutions of partial differential equations Updated 2024-12-15 +Created 1970-01-01
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 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 would like to fully understand the statements and motivations of each the problems!
Easy to understand the motivation:
- Navier-Stokes existence and smoothness is basically the only problem that is really easy to understand the statement and motivation :-)
- p versus NP problem
Hard to understand the motivation!
- Riemann hypothesis: a bunch of results on prime numbers, and therefore possible applications to cryptographyOf course, everything of interest has already been proved conditionally on it, and the likely "true" result will in itself not have any immediate applications.As is often the case, the only usefulness would be possible new ideas from the proof technique, and people being more willing to prove stuff based on it without the risk of the hypothesis being false.
- Yang-Mills existence and mass gap: this one has to do with finding/proving the existence of a more decent formalization of quantum field theory that does not resort to tricks like perturbation theory and effective field theory with a random cutoff valueThis is important because the best theory of light and electrons (and therefore chemistry and material science) that we have today, quantum electrodynamics, is a quantum field theory.