Is Schwartz Space a Viable Basis for Understanding PDEs?

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In summary, the conversation discusses the use of Schwartz spaces in relation to PDEs. The participants debate whether Schwartz spaces are a suitable basis for modeling PDEs, with one person mentioning that they are intermediate between general and nuclear spaces. The conversation also touches on the use of Schwartz spaces in modeling the Navier-Stokes equations, with one person mentioning that it may be possible to use Schwartz spaces for initial Schwartz Class data. However, there is uncertainty about the applicability of Schwartz spaces for all PDEs.
  • #1
greentea28a
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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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  • #2
I learned all about Schwartz space in a book called Nuclear and Conuclear Spaces, Herni Hogbe-Nlend, Chapter 1.
 
  • #3
I don't understand what you mean by the "basis" or "origins" of PDEs. What, exactly are you looking for?
 
  • #4
PDEs model physical systems.
All systems are subjected to nonlinear turbulence.
I am wondering if Schwartz space is suitable for modeling general PDEs.
 
  • #5
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.
 
  • #6
Do you know if Schwartz space fits the Navier-Stokes equations?
 
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  • #7
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.
 
  • #8
I have no idea what you're trying to get at besides advertising for a book and author.
 

1. What is a PDE?

A PDE stands for Partial Differential Equation. It is a mathematical equation that involves partial derivatives of a function with respect to multiple independent variables. It is used to model various physical phenomena in fields such as physics, engineering, and economics.

2. What is the difference between an ordinary differential equation (ODE) and a PDE?

An ODE involves derivatives of a function with respect to a single independent variable, while a PDE involves derivatives with respect to multiple independent variables. ODEs are used to model processes that change over a single independent variable, such as time, while PDEs are used to model processes that involve multiple independent variables, such as space and time.

3. What is the basis of PDEs?

The basis of PDEs lies in the fundamental laws of physics, such as conservation of mass, energy, and momentum. PDEs are derived from these laws and are used to describe physical phenomena in a mathematical form. They provide a powerful tool for understanding and predicting the behavior of complex systems.

4. What are the main types of PDEs?

There are several types of PDEs, including elliptic, hyperbolic, and parabolic equations. Elliptic PDEs are used to model steady-state problems, hyperbolic PDEs are used to model wave-like phenomena, and parabolic PDEs are used to model diffusion and heat transfer processes.

5. How are PDEs solved?

PDEs can be solved analytically or numerically. Analytical solutions involve finding an exact mathematical expression for the function that satisfies the equation. This is only possible for a limited number of PDEs. Numerical solutions involve using computational methods to approximate the solution to the PDE. This is the most common approach for solving complex PDEs.

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