Analysis and PDE
Millennium Prize Problem
Status: Open
Status last verified
Navier–Stokes Existence and Smoothness
Do smooth, physically reasonable solutions of the 3D incompressible Navier–Stokes equations exist for all time, or can they break down in finite time?
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Suggested experiments
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1D viscous Burgers equation
Solve u_t + u u_x = ν u_xx with a finite-difference scheme and watch a steep front form as ν gets small — a toy model of how nonlinearity sharpens gradients.
Establishes: The behaviour of your numerical scheme on this 1D model.
Does not establish: Anything about the 3D Navier–Stokes equations.