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 , 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 Hölder‐regularity. We show that such Euler solutions are codimension‐ stable in the Sobolev space .