What is needed to solve the Navier-Stokes equations' well-posedness problem?

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Discussion Overview

The discussion revolves around the well-posedness problem of the Navier-Stokes equations, focusing on the mathematical requirements and knowledge necessary to approach this problem. It includes considerations of both pure mathematics and fluid mechanics.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant suggests that knowledge of partial differential equations (PDEs) is essential to address the well-posedness problem.
  • Another participant proposes that a strong understanding of fluid mechanics would also be beneficial.
  • A later reply clarifies that the inquiry is specifically about the mathematical aspects, seeking to prove or provide a counter-example regarding the existence of smooth and globally defined solutions in three dimensions.
  • Several participants mention the monetary incentive associated with solving the problem, indicating its significance in the mathematical community.
  • There is a question raised about the level of mathematical background required to tackle the well-posedness problem.

Areas of Agreement / Disagreement

Participants express varying views on the necessary knowledge for addressing the problem, with some focusing on fluid mechanics and others emphasizing pure mathematics. The discussion remains unresolved regarding the specific requirements needed to approach the well-posedness problem.

Contextual Notes

Participants have not reached a consensus on the exact mathematical prerequisites or the nature of the solutions being sought, highlighting the complexity and depth of the problem.

Winzer
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What does it take to look at the well poseness problem of the Navier stokes equations?
Besides knowledge in PDEs.
 
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I guess that a good knowledge of fluid mechanics will be of great aid.
 
I'm sorry i should have clarified. I meant in the pure mathematical sense.
Prove or give a counter-example that:
In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalar pressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations
 
I could win $1 million if I can solve this. :biggrin:

http://www.claymath.org/millennium/
 
Last edited by a moderator:
matematikawan said:
I could win $1 million if I can solve this. :biggrin:

http://www.claymath.org/millennium/
Cool. It's agreed that if you solve this we'll split the $$$.

Anyway, what kind of insane math background does one need to attempt his journey?
 
Last edited by a moderator:

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