2019/12/23 by Merle, Frank, Raphael, Pierre, Rodnianski, Igor +1
#34C37 #35Q35 #Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.1912.10998
We consider the barotropic Euler equations in dimension d>1 with decaying density at spatial infinity. The phase portrait of the nonlinear ode governing the equation for spherically symmetric self-similar solutions has been introduced in the pioneering work of Guderley. It allows to construct global profiles of the self-similar problem, which however turn out to be generically non-smooth across the associated light (acoustic) cone. In a suitable range of barotropic laws and for a sequence of quantized speeds accumulating to a critical value, we prove the existence of non-generic C^∞ self-similar solutions with suitable decay at infinity. The C^∞ regularity is used in a fundamental way in the companion papers \citeMRRSnls, \citeMRRSfluid to control the associated linearized operator, and construct finite energy blow up solutions of respectively the defocusing nonlinear Schrödinger equation in dimension 5≤ d≤9, and the isentropic ideal compressible Euler and Navier-Stokes equations in dimensions d=2,3.