The Navier-Stokes existence and smoothness problem is one of the most important open problems in modern mathematics. Ya. Sinai and D. Li have proposed a renormalization approach to constructing a counter-example to existence. In this approach, existence of a blow-up solution (a solution whose energy becomes infinite in finite time) is equivalent to existence of fixed point of an appropriate operator in some functional space. We will explain a computer assited technique which can be conjecturally used to prove existence of such a fixed point for 3D NS equations, and describe our numerical evidence for a fixed point in the setting of a 1D version of NS.
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