Existence and uniqueness of solutions of ordinary 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>.


Existence and uniqueness of solutions of partial differential equations

{wiki}

Existence and uniqueness results are fundamental in <mathematics> because we often define objects by their properties, and then start calling them "the object", which is fantastically convenient.

But calling something "the object" only makes sense if there exists exactly one, and only one, object that satisfies the properties.

One particular context where these come up very explicitly is in solutions to <differential equations>, e.g. <existence and uniqueness of solutions of partial differential equations>.


Ciro Santilli @cirosantilli 37

 Tagged: Existence and uniqueness