I will present two complementary small-data stability theorems for the 3D irrotational and isentropic compressible Euler equations in spherical symmetry. We consider the Cauchy problem with initial data that are perturbations of a non-vacuum constant state.

In the first theorem, we treat all equations of state except that of the Chaplygin gas. For an open set of initial data with tails at infinity, we construct a maximal globally hyperbolic development (MGHD), both to the future and to the past, provide a complete description of its global structure, and prove its uniqueness. The uniqueness is a substantive part of the result, since Eperon, Reall, and Sbierski proved that MGHDs need not be unique for general quasilinear hyperbolic equations. The boundary of this MGHD consists of three types of components: creases where singularities first occur in positive and negative time, singular boundaries where gradient blowup occurs, and Cauchy horizons emanating from the creases. This is the first MGHD existence-uniqueness-stability result for shock-forming solutions for any multi-dimensional quasilinear hyperbolic wave-like system.

In the second theorem, we consider another open set of initial data whose tails have the opposite sign from those in the first theorem. We prove global existence both to the future and to the past. This is the first small-data global existence result in the entire spacetime for any 3D quasilinear hyperbolic wave-like system that fails to satisfy the null condition and the weak null condition.

This is joint work with Leonardo Abbrescia and Jared Speck.