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On the implosion of a compressible fluid. II: Singularity formation. (English) Zbl 1497.35385

Summary: In this paper, which continues our investigation of strong singularity formation in compressible fluids, we consider the compressible three-dimensional Navier-Stokes and Euler equations. In a suitable regime of barotropic laws, we construct a set of finite energy smooth initial data for which the corresponding solutions to both equations implode (with infinite density) at a later time at a point, and completely describe the associated formation of singularity. An essential step in the proof is the existence of \(\mathcal{C}^\infty\) smooth self-similar solutions to the compressible Euler equations for quantized values of the speed constructed in our companion paper [ibid. 196, No. 2, 567–778 (2022; Zbl 1497.35384)]. All blow up dynamics obtained for the Navier-Stokes problem are of type II (non self-similar).

MSC:

35Q35 PDEs in connection with fluid mechanics
35B44 Blow-up in context of PDEs
76N06 Compressible Navier-Stokes equations
35C06 Self-similar solutions to PDEs
35B65 Smoothness and regularity of solutions to PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence

Citations:

Zbl 1497.35384
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References:

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