What is the basis of PDEs?

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

HallsofIvy

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I don't understand what you mean by the "basis" or "origins" of PDEs. What, exactly are you looking for?
 
PDEs model physical systems.
All systems are subjected to nonlinear turbulence.
I am wondering if Schwartz space is suitable for modeling general PDEs.
 

MarneMath

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

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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

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

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