in this
article of Quanta Magazine, it says in isometric embedding of $C^k$ of flow, if k>1/3, it's a energy-conserving smooth flow, but if k<1/3, it becomes energy-dissipating turbulence. It also says in the isometric embedding of $C^r$ of any manifold with a fixed boundary, if r>3/2, the manifold can be crumpled without buckling, but if r<3/2, the manifold can only be crumpled with buckling.