On smooth self similar solutions to the compressible Euler equations

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 $5\le d\le 9$, and the isentropic ideal compressible Euler and Navier-Stokes equations in dimensions d=2,3.

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References in corpus (1)

• Large global solutions for energy supercritical nonlinear wave equations on \R^{3+1}

Cited by in corpus (2)

• Gradient blow-up for dispersive and dissipative perturbations of the Burgers equation
• Shock Formation of 3D Euler-Poisson System for Electron Fluid with Steady Ion Background