Mathematics Colloquia and Seminars

Return to Colloquia & Seminar listing

We introduce the concept of pre-shock  solutions for hyperbolic systems of conservation laws, and the problem of constructive shock formation. We then construct an infinite hierarchy of initially-smooth solutions of the 1D Euler equations  which form finite-time gradient singularities. Specifically, for all integers $n\geq 1$, we prove that there exist classical solutions, emanating from  smooth, compressive,  and non-vacuous initial data, which form a cusp-type gradient singularity in finite time,  in which the gradient of the solution has precisely  $C^{0,\frac{1}{2n+1}}$ Holder-regularity. We show that such Euler solutions are codimension-$(2n-2)$  stable in the Sobolev space $W^{2n+2,\infty}$. This is joint work with S. Shkoller and V. Vicol.