The stability of simple plane-symmetric shock formation for 3D compressible Euler flow with vorticity and entropy

Abstract

Consider a $1$D simple small-amplitude solution $({\rho }_{(bkg)},v_{(bkg)}^1)$ 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 $({\rho }_{(bkg)},v_{(bkg)}^1)$ as a plane-symmetric solution to the full compressible Euler equations in $3$D, we prove that the shock-formation mechanism for the solution $({\rho }_{(bkg)},v_{(bkg)}^1)$ 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. Moreover, relative to a set of geometric coordinates constructed out of an acoustic eikonal function, the fluid solution and the inverse foliation density function remain smooth up to the shock; the blowup of the solution's Cartesian coordinate partial derivatives is caused by a degeneracy between the geometric and Cartesian coordinates, signified by the vanishing of the inverse foliation density (i.e., the intersection of the characteristics).

104 pages, 2 figures; minor revisions, version accepted for publication

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Cited by in corpus (1)

• Rough sound waves in $3D$ compressible Euler flow with vorticity