Shock formation for 2D Isentropic Euler equations with self-similar variables

Abstract: We study the 2D isentropic Euler equations with the ideal gas law. We exhibit a set of smooth initial data that give rise to shock formation at a single point near the planar symmetry. These solutions are associated with non-zero vorticity at the shock and have uniform-in-time 1/3-H\"older bound. Moreover, these point shocks are of self-similar type and share the same profile, which is a solution to the 2D self-similar Burgers equation. Our proof, following the 3D shock formation result of Buckmaster, Shkoller and Vicol, is based on the stable 2D self-similar Burgers profile and the modulation method.