Two global stability theorems for small-data $3D$ compressible Euler solutions: Unique maximal globally hyperbolic developments with gradient-blowup & smooth global existence in the entire spacetime

RESUMO

We prove a pair of complementary small-data stability theorems describing the global future and past structure of classical solutions to the isentropic, spherically symmetric $3D$ compressible Euler equations. We allow any equation of state with positive sound speed, except that our shock-formation results do not apply to the Chaplygin gas. Our first theorem concerns open sets of smooth Cauchy data that are perturbations of trivial data with vanishing velocity and constant positive density. The perturbed data have finite kinetic energy, and the perturbed density is equal to a positive constant plus a signed ``asymptotically flat'' tail. Our main theorem yields a complete description of the existence and uniqueness of the maximal globally hyperbolic development (MGHD) of the data. The boundary of the MGHD contains hypersurfaces that extend to spatial infinity on which our solutions develop gradient-singularities. This is the first MGHD existence-uniqueness-stability result for shock-forming solutions for any multi-$D$ quasilinear wave system. While prior works have yielded local portions of ``candidate'' MGHDs, neither the existence nor the uniqueness of an MGHD can be inferred from local considerations. Our second theorem considers analogous data with the opposite sign of the $1/r$ density tail. We prove global existence throughout spacetime, to both the future and past. The solution's asymptotic behavior towards null infinity is not linear, but rather is distorted by a logarithmic bending of the sound cones away from the flat Minkowski ones, leading to logarithmically enhanced dispersion. This is the first small-data global existence result in the entire spacetime for any $3D$ quasilinear wave system that fails to satisfy the null condition and the weak null condition. The main mechanism of stabilization is a global-in-spacetime rarefaction effect, tied to the $1/r$ tail.