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Analysis of a singular Boussinesq model. (English) Zbl 1428.35367
Recently, a new singularity formation scenario for the 3D axi-symmetric Euler equation and the 2D inviscid Boussinesq system has been proposed by G. Luo and T. Y. Hou [Proc. Natl. Acad. Sci. USA 111, No. 36, 12968–12973 (2014; Zbl 1431.35115)] based on extensive numerical simulations. As the first step to understanding this scenario, models with a simplified sign-definite Biot-Savart law and forcing have recently been studied by K. Choi et al. [Commun. Pure Appl. Math. 70, No. 11, 2218–2243 (2017; Zbl 1377.35218); Commun. Math. Phys. 334, No. 3, 1667–1679 (2015; Zbl 1309.35072)], T. Do et al. [J. Nonlinear Sci. 28, No. 6, 2127–2152 (2018; Zbl 1406.35249)], V. Hoang et al. [J. Differ. Equations 264, No. 12, 7328–7356 (2018; Zbl 1387.35066)], T. Y. Hou and P. Liu [Res. Math. Sci. 2, Paper No. 5, 26 p. (2015; Zbl 1320.35269)], and the first author and C. Tan [Adv. Math. 325, 34–55 (2018; Zbl 1382.35054)]. In this paper, the authors aim to bring back one of the complications encountered in the original equation, i.e., the sign changing kernel in the Biot-Savart law. This makes the analysis harder, as there are two competing terms in the fluid velocity integral whose balance determines the regularity properties of the solution. The equation the authors study is based on the CKY model introduced by Choi et al. [2015, loc. cit.]. They prove that finite time blow-up persists in a certain range of parameters.
MSC:
35Q35 PDEs in connection with fluid mechanics
76B03 Existence, uniqueness, and regularity theory for incompressible inviscid fluids
35B65 Smoothness and regularity of solutions to PDEs
35B44 Blow-up in context of PDEs
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