The Navier-Stokes equations describe how fluids move and are fundamental in fields like physics and engineering. The challenge of proving the existence and smoothness of solutions to these equations means showing that, under certain conditions, solutions not only exist but also remain well-behaved over time without developing singularities or discontinuities. This problem is one of the seven Millennium Prize Problems established by the Clay Mathematics Institute. A solution to the Navier-Stokes existence and smoothness problem would advance our understanding of fluid dynamics and could have significant implications in various scientific and practical applications.