Is Schwartz Space a Viable Basis for Understanding PDEs?

[greentea28a]

Is there a hole in knowledge as to the origins of PDEs?

If there is a void, is Schwartz space a suitable basis?

Schwartz spaces are intermediate between general spaces and nuclear spaces.
Infra-Schwartz spaces are intermediate between Schwartz spaces and reflexive spaces.

 

[greentea28a]

I learned all about Schwartz space in a book called Nuclear and Conuclear Spaces, Herni Hogbe-Nlend, Chapter 1.

 

[HallsofIvy]

I don't understand what you mean by the "basis" or "origins" of PDEs. What, exactly are you looking for?

 

[greentea28a]

PDEs model physical systems.
All systems are subjected to nonlinear turbulence.
I am wondering if Schwartz space is suitable for modeling general PDEs.

 

[MarneMath]

I would wager the general answer is no since a Schwartz space requires a special property of a function's derivative that not all functions may have. If you're asking can you use a Schwartz space for some PDE's, the answer is yes.

 

[greentea28a]

Do you know if Schwartz space fits the Navier-Stokes equations?

 

[MarneMath]

I'm not an expert regarding PDE's or methods dealing with them, so I don't want to give you a wrong answer but I'll give a minimum answer that you probably know if you are asking these questions. I don't believe Navier-Stokes must be in a Schwartz Space unless the initial data is a Schwartz Class. So with that said if you want to look at the N-S equation via a Schwartz Class you can do so. You can probably even extract that information to gather information on global properties for initial Schwartz Class data.

 

[MarneMath]

I have no idea what you're trying to get at besides advertising for a book and author.