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Non-uniqueness of Weak Solutions to Hyperviscous Navier-Stokes Equations -- On Sharpness of J.-L. Lions Exponent

Using the convex integration technique for the three-dimensional Navier-Stokes equations introduced by T. Buckmaster and V. Vicol, it is shown the existence of non-unique weak solutions for the 3D Navier-Stokes equations with fractional hyperviscosity $(-Δ)^θ$, whenever the exponent $θ$ is less than J.-L. Lions&#39; exponent $5/4$, i.e., when $θ< 5/4$.