On global dynamics of $3$-D irrotational compressible fluids

Abstract

We consider global-in-time evolution of irrotational, isentropic, compressible Euler flow in $3$-D, for a broad class of $H^4$ classical Cauchy data without assuming symmetry, prescribed on an annulus surrounded by a constant state in the exterior. By giving a sufficient expansion condition on the initial data and using the nonlinear structure of the compressible Euler equations, we show that the decay rate of the first order transversal derivative of the normalized density is better than that of the same derivative of a free wave, provided that the perturbation arising from the tangential derivatives can be properly controlled for all $t$ by using a bootstrap argument. Building on this critical analysis, we construct global exterior solutions in $H^4$ for the broad class of data, with a rather general subclass forming rarefaction at null infinity. Our result does not require smallness on the transversal derivatives of classical data, thus applies to data with a total energy of any size.