Is Schwartz Space a Viable Basis for Understanding PDEs?

  • Context: Graduate 
  • Thread starter Thread starter greentea28a
  • Start date Start date
  • Tags Tags
    Basis Pdes
Click For Summary

Discussion Overview

The discussion revolves around the viability of Schwartz space as a basis for understanding partial differential equations (PDEs). Participants explore the relationship between Schwartz spaces, their properties, and their applicability to modeling physical systems described by PDEs, particularly in the context of nonlinear turbulence and the Navier-Stokes equations.

Discussion Character

  • Exploratory
  • Debate/contested
  • Technical explanation

Main Points Raised

  • One participant questions whether there is a gap in knowledge regarding the origins of PDEs and suggests Schwartz space as a potential basis.
  • Another participant references a specific book on nuclear spaces to support their understanding of Schwartz space.
  • A participant expresses confusion about the terms "basis" and "origins" in relation to PDEs, seeking clarification on the inquiry.
  • It is noted that PDEs model physical systems, which are influenced by nonlinear turbulence, raising the question of Schwartz space's suitability for general PDE modeling.
  • One participant argues that while Schwartz space may not universally apply due to specific properties required of functions, it can be used for some PDEs.
  • A question is posed regarding the applicability of Schwartz space to the Navier-Stokes equations.
  • A participant, while not an expert, suggests that the Navier-Stokes equations do not necessarily require Schwartz space unless the initial data is from a Schwartz class, indicating a conditional relationship.
  • Another participant expresses skepticism about the discussion, implying that it may be an attempt to promote a book rather than a substantive inquiry.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the applicability of Schwartz space to PDEs, with some arguing for its limited use and others questioning the relevance of the discussion itself. Multiple competing views remain regarding the relationship between Schwartz space and specific PDEs.

Contextual Notes

There are limitations in understanding the specific properties of functions required for Schwartz space and how these relate to the broader context of PDEs. The discussion also reflects varying levels of expertise among participants, which may influence the depth of the arguments presented.

greentea28a
Messages
11
Reaction score
0
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.
 
Last edited:
Physics news on Phys.org
I learned all about Schwartz space in a book called Nuclear and Conuclear Spaces, Herni Hogbe-Nlend, Chapter 1.
 
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.
 
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?
 
Last edited:
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.
 
I have no idea what you're trying to get at besides advertising for a book and author.
 

Similar threads

Graduate How Can I Bridge the Gap Between Undergraduate and Graduate Level PDEs?
  • · Replies 1 ·
Replies
1
Views
2K
Graduate Does the quantum space of states have countable or uncountable basis?
  • · Replies 67 ·
3
Replies
67
Views
8K
Undergrad Why should a Fourier transform not be a change of basis?
  • · Replies 43 ·
2
Replies
43
Views
8K
Graduate Rigged Hilbert Spaces In Quantum Mechanics
  • · Replies 19 ·
Replies
19
Views
5K
Undergrad "Exploring Time Perception Across Different Regions of Space"
  • · Replies 20 ·
Replies
20
Views
2K
Understanding inner product space and matrix representations of Operat
  • · Replies 16 ·
Replies
16
Views
3K
Graduate Reproducing Kernel Hilbert Spaces
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Understanding Ewald's sphere in the context of X-Ray diffraction
  • · Replies 3 ·
Replies
3
Views
6K
High School Questions regarding expansion, the vacuum in space, and black holes
  • · Replies 6 ·
Replies
6
Views
2K
Graduate Double Dual of Vector Space: Is V** Always Same as V?
  • · Replies 14 ·
Replies
14
Views
3K