For every integer n>=1, smooth 1D Euler data can form a pre-shock cusp with Holder exponent 1/(2n+1), and the set of such data is a codimension-(2n-2) Banach manifold in W^{2n+2,infinity}.
Plane algebraic curves
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Gradient catastrophes and an infinite hierarchy of H\"older cusp-singularities for 1D Euler
For every integer n>=1, smooth 1D Euler data can form a pre-shock cusp with Holder exponent 1/(2n+1), and the set of such data is a codimension-(2n-2) Banach manifold in W^{2n+2,infinity}.