Short-time shock continuation conjecture for asymptotically self-similar Euler shocks

For $\rho$ the density and $u=(u_r,u_\uptheta)$ the velocity solving the two-dimensional isentropic compressible Euler equations, suppose the solution constructed in Theorem general forms a cusp of Hölder regularity $C^{1/3}$ at its first blowup time $t=T_*$. Shock continuation conjecture. These solutions can be continued for short time as propagating, piecewise smooth, discontinuous (possibly non-unique) shock profiles that solve the Euler equations on either side of a time-dependent curve of discontinuity, with the evolution of the shock governed by the Rankine–Hugoniot conditions. The case $\gamma=3$ is established for all time in the paper; the conjecture concerns the corresponding short-time continuation for general $\gamma>1$.