Whether the 3D incompressible Navier-Stokes equations will have a global smooth solution for all smooth, finite energy initial data is a Millennium Prize problem. One of the main difficulties of this problem is that the Navier-Stokes equations are actually a system of semilinear heat equations rather than a single equation. In this paper, we discover a remarkable hidden symmetry of the 3D incompressible Navier-Stokes equations. Under this symmetric reduction, the system reduces to a single scalar semilinear heat equation. The symmetry also holds for the 3D incompressible Euler equations.
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Lai et al. (2024) studied this question.
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