Boundary layer models of the Hou-Luo scenario. (English) Zbl 1470.35270
Summary: Finite time blow up vs global regularity question for 3D Euler equation of fluid mechanics is a major open problem. Several years ago, G. Luo and T. Y. Hou [Multiscale Model. Simul. 12, No. 4, 1722–1776 (2014; Zbl 1316.35235)] proposed a new finite time blow up scenario based on extensive numerical simulations. The scenario is axi-symmetric and features fast growth of vorticity near a ring of hyperbolic points of the flow located at the boundary of a cylinder containing the fluid. An important role is played by a small boundary layer where intense growth is observed. Several simplified models of the scenario have been considered, all leading to finite time blow up [K. Choi et al., Commun. Math. Phys. 334, No. 3, 1667–1679 (2015; Zbl 1309.35072); ibid. 70, No. 11, 2218–2243 (2017; Zbl 1377.35218); V. Hoang et al., J. Differ. Equations 264, No. 12, 7328–7356 (2018; Zbl 1387.35066); A. Kiselev and C. Tan, Adv. Math. 325, 34–55 (2018; Zbl 1382.35054); T. Y. Hou and P. Liu, Res. Math. Sci. 2, Paper No. 5, 26 p. (2015; Zbl 1320.35269); A. Kiselev and H. Yang, ibid. 6, No. 1, Paper No. 13, 16 p. (2019; Zbl 1428.35367)]. In this paper, we propose two models that are designed specifically to gain insight in the evolution of fluid near the hyperbolic stagnation point of the flow located at the boundary. One model focuses on analysis of the depletion of nonlinearity effect present in the problem. Solutions to this model are shown to be globally regular. The second model can be seen as an attempt to capture the velocity field near the boundary to the next order of accuracy compared with the one-dimensional models such as [Choi et al., loc. cit.]. Solutions to this model blow up in finite time.
MSC:
| 35Q31 | Euler equations |
| 35B44 | Blow-up in context of PDEs |
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35B07 | Axially symmetric solutions to PDEs |
| 76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |
Citations:
Zbl 1316.35235; Zbl 1309.35072; Zbl 1377.35218; Zbl 1387.35066; Zbl 1382.35054; Zbl 1320.35269; Zbl 1428.35367References:
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