For 2D Navier-Stokes on a torus, non-uniqueness holds with velocity gradients in C([0,T], H^p) for every exponent 0 < p < 1, making p = 1 the sharp threshold between non-uniqueness and uniqueness in the vorticity path space.
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Pathological solutions of Navier-Stokes equations on $\mathbb{T}^2$ with gradients in Hardy spaces
For 2D Navier-Stokes on a torus, non-uniqueness holds with velocity gradients in C([0,T], H^p) for every exponent 0 < p < 1, making p = 1 the sharp threshold between non-uniqueness and uniqueness in the vorticity path space.