Gradient catastrophes and an infinite hierarchy of Hölder cusp-singularities for 1D Euler

We establish an infinite hierarchy of finite-time gradient catastrophes for smooth solutions of the 1D Euler equations of gas dynamics with nonconstant entropy. Specifically, for all integers $n⩾1$, we prove that there exist classical solutions, emanating from smooth, compressive, and nonvacuous initial data, which form cusp-type gradient singularities in finite time, in which the gradient of the solution has precisely $C^{0,\frac{1}{2n+1}}$ Hölder-regularity. We show that such Euler solutions are codimension-$(2n-2)$ stable in the Sobolev space $W^{2n+2,\infty }$.