Prove the existence of smooth solutions to a certain model

In summary, "smooth solutions" in mathematics and science refer to solutions that are infinitely differentiable and have no discontinuities or sharp changes. To "prove the existence" of smooth solutions, a scientist must show that there is at least one solution that satisfies all necessary conditions. This is important for providing a solid mathematical foundation and allowing for accurate predictions and further research. However, there may be limitations or assumptions when proving the existence of smooth solutions, which should be clearly stated. Common techniques used for proving the existence of smooth solutions include the method of continuity, the method of compactness, and the method of contraction mapping. These techniques involve concepts such as continuity, compactness, and fixed point theorems.
  • #1
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You don't need to solve this, just tell me what problem it is... although if you do solve it you'll be a million dollars richer.

Prove the existence of smooth solutions to a certain model of incompressible fluid dynamics. This is the ... problem?
 
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  • #2
It's the Navier-stokes problem and Clay Mathematics institute offers 1 million dollars prize to solve it or 6 other different problems (possibly 5 because the Poincare conjecture seems to have been solved recently)
 
  • #3
point for meteor!
 

What is the meaning of "smooth solutions" in this context?

In mathematics and science, "smooth solutions" refer to solutions that are infinitely differentiable and have no discontinuities or sharp changes. In other words, these solutions are continuous and have no abrupt changes in their behavior.

What does it mean to "prove the existence" of smooth solutions?

In order to prove the existence of smooth solutions, a scientist must show that there is at least one solution to the given model that satisfies all the necessary conditions. This can be done through various mathematical techniques and proofs.

Why is it important to prove the existence of smooth solutions?

Proving the existence of smooth solutions is important because it provides a solid mathematical foundation for understanding and analyzing a particular model. It also allows for accurate predictions and further research to be conducted based on these solutions.

Are there any limitations or assumptions when proving the existence of smooth solutions?

Yes, there can be limitations or assumptions when proving the existence of smooth solutions. These may include assumptions about the behavior of certain variables or the boundaries of the model. It is important for a scientist to clearly state any limitations or assumptions in their proof.

What are some common techniques used to prove the existence of smooth solutions?

Some common techniques used to prove the existence of smooth solutions include the method of continuity, the method of compactness, and the method of contraction mapping. These techniques involve mathematical concepts such as continuity, compactness, and fixed point theorems.

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