Locally bounded linear differential operators

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SUMMARY

The discussion centers on locally bounded linear differential operators within the context of the Schwartz space of smooth functions on sigma-compact manifolds. The participant connects this abstract concept to the Navier-Stokes equations, suggesting that understanding these operators is crucial for grasping the mathematical framework behind the equations. The mention of bornology, particularly from Hogbe-Nlend's works, emphasizes the necessity of understanding boundedness in functional analysis. The conversation highlights the need for clarity in defining specific problems when discussing advanced mathematical concepts.

PREREQUISITES
  • Understanding of locally bounded linear differential operators
  • Familiarity with the Schwartz space of smooth functions
  • Knowledge of sigma-compact manifolds
  • Basic concepts of bornology from functional analysis
NEXT STEPS
  • Study the properties of locally bounded linear differential operators
  • Explore the relationship between the Navier-Stokes equations and functional analysis
  • Investigate Hogbe-Nlend's works on bornology and functional analysis
  • Learn about Nuclear Convex Bornological Spaces and their applications
USEFUL FOR

Mathematicians, theoretical physicists, and graduate students focusing on functional analysis, particularly those interested in the mathematical foundations of fluid dynamics and the Navier-Stokes equations.

greentea28a
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The following is a problem statement.

locally bounded (or locally (weakly) compact) differential operators of the Schwartz space of smooth functions on a sigma-compact manifold

I realize this is very abstract. I expect the solution to be just as abstract.

Thanks in advance.
 
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That's not a question, and we're not in the habit of doing people's work for them anyway. What is the problem you're working on, and what have you done so far?
 
I believe the above problem is the core of the Navier-Stokes equation. It describes the inner working of the NS equations from a mathematical operator's point-of-view.

I have studied bornology from Hogbe-Nlend's books Bornology and Functional Analysis; Nuclear and Conuclear Spaces. According to Wikipedia, bornology is the minimum amount of structure to address boundedness of sets and functions.

The next closest thing I have come across is Nuclear Convex Bornological Spaces.
 
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greentea28a said:
I believe the above problem is the core of the Navier-Stokes equation.

What problem? You just said a kind of operator. It would be like me starting a thread and saying "bounded linear function". What are you trying to prove?
 

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