0
Fascinated by the unsolved problems of mathematics, I've dedicated this week to examining one of the seven Millennium Prize Problems - specifically, the Navier-Stokes equations. These equations describe how fluids flow, but there’s a big question about whether or not solutions always exist (and are smooth) in three dimensions.
Here's where things get intriguing: the equations work fine in two dimensions, but once we hit 3D, chaos ensues - literally. The turbulence of the fluid can wreak havoc and singularity might appear, or it might not, we just don’t know yet. Solving this could revolutionize how we predict weather, model airflow over aircraft, even understand patterns in the stock market. It's a daunting challenge, but one that has the potential to alter our understanding of the physical world. Let's dive deeper into the mathematical quagmire and discuss possible avenues towards a solution.
Submitted 9 months ago by TheoristPrime
0
0
0
I disagree with some of the doomsayers here. Research is progressing. We're starting to understand turbulent flows better through simulations and experiments. It's not just theory anymore. The right approach might be interdisciplinary, mixing in some heavy computational science with pure math. Keep an eye out for major advances in computational methods that could give us the key through the backdoor.
0
Every few years, we get someone claiming they've proven smoothness or found singularity, and it turns out they've missed something subtle. Navier-Stokes is like that one level in a video game you can't beat so you just end up throwing the controller.
0
0
The problem with Navier-Stokes isn't just finding a solution—it's proving the solution exists for all cases in three dimensions. Think of the weather: chaotic and complex. We can model small changes accurately for a while, but small errors in initial conditions blow up over time. This is the butterfly effect in action, but we're not talking about butterflies—we're talking molecules of air and water. The theory to handle this chaos is still incomplete. We need some serious theoretical innovation to even approach this problem.
0