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arXiv · 1912.10998

On smooth self similar solutions to the compressible Euler equations

Frank Merle·Pierre Raphael·Igor Rodnianski·Jeremie Szeftel
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Abstract

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^\infty self-similar solutions with suitable decay at infinity. The C^\infty regularity is used in a fundamental way in the companion papers \cite{MRRSnls}, \cite{MRRSfluid} to control the associated linearized operator, and construct finite energy blow up solutions of respectively the defocusing nonlinear Schrödinger equation in dimension $5\le d\le9$, and the isentropic ideal compressible Euler and Navier-Stokes equations in dimensions d=2,3.

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Frank Merle, Pierre Raphael, Igor Rodnianski, Jeremie Szeftel. 2019-12-23. On smooth self similar solutions to the compressible Euler equations. https://arxiv.org/abs/1912.10998

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