The Navier-Stokes equations are a set of partial differential equations that describe the motion of fluids. They are notoriously difficult to solve, and one of the biggest challenges is understanding the conditions under which solutions remain smooth and well-behaved (global regularity).
The connection between the Navier-Stokes equations and turbulence lies in the fact that turbulent flows are often associated with the breakdown of smooth solutions to these equations. In other words, it is believed that the onset of turbulence is related to the formation of singularities in the Navier-Stokes equations. However, proving this rigorously is a major challenge, and the difficulty of establishing global regularity for the Navier-Stokes equations that hinders our complete understanding of turbulence.
If we could prove that smooth solutions always exist, it would imply that turbulence cannot arise from the equations themselves, and we would need to look for other explanations, such as instabilities in the initial or boundary conditions. Conversely, if we could find an example of a solution that develops a singularity, it would provide strong evidence that the Navier-Stokes equations are indeed capable of describing the onset of turbulence.
What’s unknown about turbulence?
The Navier-Stokes equations are a set of partial differential equations that describe the motion of fluids. They are notoriously difficult to solve, and one of the biggest challenges is understanding the conditions under which solutions remain smooth and well-behaved (global regularity).
The connection between the Navier-Stokes equations and turbulence lies in the fact that turbulent flows are often associated with the breakdown of smooth solutions to these equations. In other words, it is believed that the onset of turbulence is related to the formation of singularities in the Navier-Stokes equations. However, proving this rigorously is a major challenge, and the difficulty of establishing global regularity for the Navier-Stokes equations that hinders our complete understanding of turbulence.
If we could prove that smooth solutions always exist, it would imply that turbulence cannot arise from the equations themselves, and we would need to look for other explanations, such as instabilities in the initial or boundary conditions. Conversely, if we could find an example of a solution that develops a singularity, it would provide strong evidence that the Navier-Stokes equations are indeed capable of describing the onset of turbulence.
https://terrytao.wordpress.com/2007/03/18/why-global-regularity-for-navier-stokes-is-hard/