Source: cirosantilli/existence-and-uniqueness-of-solutions-of-partial-differential-equations

= Existence and uniqueness of solutions of partial differential equations
{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>.

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