Appears in collection : Partial Differential Equations, Analysis and Geometry

We consider global-in-time evolution of irrotational, isentropic, compressible Euler flow in 3-D. We study a broad class of smooth Cauchy data, prescribed on an annulus and surrounded by a non-vacuum constant exterior state, without symmetry assumptions. By imposing a sufficient expansion condition on the initial data and using the nonlinear structure of the Euler equations, we show that the first-order transversal derivative of the normalized density decays as ⟨t⟩⁻¹ (log⟨t⟩ + 1)⁻¹, provided that the perturbation arising from the tangential derivatives can be properly controlled for all t by using a bootstrap argument. This enables us to construct global exterior solutions, including a rather general subclass forming rarefaction at null infinity. Our result applies to data with a total energy of any size, as it does not require smallness of the transversal derivatives of smooth data.