Understand
Consider a $1$D simple small-amplitude solution $(\rho_{(bkg)}, v^1_{(bkg)})$ to the isentropic compressible Euler equations which has smooth initial data, coincides with a constant state outside a compact set, and forms a shock in finite time.
- Viewing $(\rho_{(bkg)}, v^1_{(bkg)})$ as a plane-symmetric solution to the full compressible Euler equations in $3$D, we prove that the shock-formation mechanism for the solution $(\rho_{(bkg)}, v^1_{(bkg)})$ is stable against all sufficiently small and compactly supported perturbations.
- In particular, these perturbations are allowed to break the symmetry and have non-trivial vorticity and variable entropy.
- Our approach reveals the full structure of the set of blowup-points at the first singular time: within the constant-time hypersurface of first blowup, the solution's first-order Cartesian coordinate partial derivatives blow up precisely on the zero level set of a function that measures the inverse foliation density of a family of characteristic hypersurfaces.