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.
Wellposedness and singularity formation beyond the Yudovich class
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abstract
We introduce a local-in-time existence and uniqueness class for solutions to the 2d Euler equation with unbounded vorticity. Furthermore, we show that solutions belonging to this class can develop stronger singularities in finite time, meaning that they experience finite time blow up and exit the wellposedness class. Such solutions may be continued as weak solutions (potentially non-uniquely) after the singularity. While the general dynamics of 2d Euler solutions beyond the Yudovich class will certainly not be so tame, studying such solutions gives a way to study singular phenomena in a more controlled setting.
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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.