The Navier-Stokes equations are among the most important — and most challenging — equations in physics. For centuries, scientists and engineers have used them to describe the motion of fluids, from turbulent air and ocean currents to blood flowing through our bodies. But mathematicians couldn’t prove whether their solutions always remain smooth, a mystery that became one of seven $1 million Millennium Prize Problems.

So, what makes the Navier-Stokes equations so difficult? What happens when their solutions blow up? And how could a new OpenAI proof finally resolve the problem?

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