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

In summary, the conversation discusses the well-posedness problem of the Navier-Stokes equations, which is related to PDEs and fluid mechanics. The question of whether a vector velocity and scalar pressure field, both smooth and globally defined, can solve the equations is posed, with a potential $1 million prize for solving it. The conversation also touches on the background needed for such an endeavor.
  • #1
Winzer
598
0
What does it take to look at the well poseness problem of the Navier stokes equations?
Besides knowledge in PDEs.
 
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  • #2
I guess that a good knowledge of fluid mechanics will be of great aid.
 
  • #3
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
 
  • #4
I could win $1 million if I can solve this. :biggrin:

http://www.claymath.org/millennium/ [Broken]
 
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  • #5
matematikawan said:
I could win $1 million if I can solve this. :biggrin:

http://www.claymath.org/millennium/ [Broken]
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?
 
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1. What are the Navier-Stokes Equations?

The Navier-Stokes Equations are a set of partial differential equations that describe the motion of a fluid. They take into account factors such as viscosity, density, and pressure to determine the velocity and pressure fields of a fluid at any given point in space and time.

2. What is the significance of the Navier-Stokes Equations?

The Navier-Stokes Equations are important in the study of fluid mechanics and are widely used in engineering and scientific applications. They provide a fundamental understanding of how fluids behave and can be used to solve a variety of problems, from predicting the flow of air around an airplane wing to simulating ocean currents.

3. Who developed the Navier-Stokes Equations?

The Navier-Stokes Equations were first developed independently by French engineer Claude-Louis Navier and Irish mathematician George Gabriel Stokes in the 19th century. However, the equations were later modified and simplified by German mathematician and physicist Hermann von Helmholtz.

4. What are the limitations of the Navier-Stokes Equations?

The Navier-Stokes Equations are based on certain assumptions, such as the fluid being incompressible and the flow being steady and laminar. In reality, fluids can exhibit turbulent and unsteady behavior, which the equations cannot accurately model. Additionally, the equations are only valid for Newtonian fluids and do not take into account non-Newtonian behavior.

5. Are the Navier-Stokes Equations solved analytically or numerically?

The Navier-Stokes Equations can be solved analytically for simple cases, but in most practical applications, they are solved numerically using computational fluid dynamics (CFD) methods. This involves breaking the equations down into discrete elements and solving them using algorithms and computer simulations.

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