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:
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