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Merle, Frank; Rapha\'’el, Pierre; Rodnianski, Igor; Szeftel, Jeremie

On the implosion of a compressible fluid. I: Smooth self-similar inviscid profiles. (English) Zbl 1497.35384

Ann. Math. (2) 196, No. 2, 567-778 (2022).
Summary: In this paper and its sequel, we construct a set of finite energy smooth initial data for which the corresponding solutions to the compressible three-dimensional Navier-Stokes and Euler equations implode (with infinite density) at a later time at a point, and we completely describe the associated formation of singularity. This paper is concerned with existence of smooth self-similar profiles for the barotropic Euler equations in dimension \(d\geq 2\) 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 us to construct global profiles of the self-similar problem, which however turn out to be generically non-smooth across the associated 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 \(\mathcal{C}^\infty\) self-similar solutions with suitable decay at infinity. The \(\mathcal{C}^\infty\) regularity is used in a fundamental way in our companion paper [ibid. 196, No. 2, 779–889 (2022; Zbl 1497.35385)] in the analysis of the associated linearized operator and leads, in turn, to the construction of finite energy blow up solutions of the compressible Euler and Navier-Stokes equations in dimensions \(d=2,3\).

Cited in 1 Review
Cited in 21 Documents

MSC:

35Q35 PDEs in connection with fluid mechanics
35Q31 Euler equations
34C37 Homoclinic and heteroclinic solutions to ordinary differential equations
35C06 Self-similar solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
76Q05 Hydro- and aero-acoustics
76N10 Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics

Keywords:

self-similar profile; Euler equations

Citations:

Zbl 1497.35385
Cite Review PDF
Full Text: DOI HAL

References:

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