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Blowup with vorticity control for a 2D model of the Boussinesq equations. (English) Zbl 1387.35066
Summary: We propose a system of equations with nonlocal flux in two space dimensions which is closely modeled after the 2D Boussinesq equations in a hyperbolic flow scenario. Our equations involve a vorticity stretching term and a non-local Biot-Savart law and provide insight into the underlying intrinsic mechanisms of singularity formation. We prove stable, controlled finite time blowup involving upper and lower bounds on the vorticity up to the time of blowup for a wide class of initial data.

##### MSC:
 35B44 Blow-up in context of PDEs 35Q35 PDEs in connection with fluid mechanics
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##### References:
 [1] Chae, D.; Constantin, P.; Wu, J., An incompressible 2D didactic model with singularity and explicit solutions of the 2D Boussinesq equations, J. Math. Fluid Mech., 16, 473, (2014) · Zbl 1307.35213 [2] Choi, K.; Hou, T. Y.; Kiselev, A.; Luo, G.; Šverák, V.; Yao, Y., On the finite-time blowup of a one-dimensional model for the three-dimensional axisymmetric Euler equations, Comm. Pure Appl. Math., 70, 11, 2218-2243, (November 2017) [3] Choi, K.; Kiselev, A.; Yao, Y., Finite time blow up for a 1d model of 2d Boussinesq system, Comm. Math. Phys., 334, 3, 1667-1679, (2015) · Zbl 1309.35072 [4] Constantin, P., On the Euler equations of incompressible fluids, Bull. Amer. Math. Soc., 44, 4, 603-621, (October 2007) [5] Constantin, P.; Fefferman, C., Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. Math. J., 42, 775, (1993), MR1254117 (95j:35169) · Zbl 0837.35113 [6] Fefferman, C.; Majda, A., Geometric constraints on potentially singular solutions for the 3-D Euler equations, Comm. Partial Differential Equations, 21, 559-571, (1996), MR1387460 (97c:35154) · Zbl 0853.35091 [7] Gordon, W. B., On the diffeomorphisms of Euclidean space, Amer. Math. Monthly, 79, 7, 755-759, (1972) · Zbl 0263.57015 [8] Hoang, V.; Radosz, M., Blowup with vorticity control for 1D model equations, (2016), preprint [9] Hou, T. Y.; Luo, G., Toward the finite-time blowup of the 3d axisymmetric Euler equations: a numerical investigation, Multiscale Model. Simul., 12, 4, 1722-1776, (2014) · Zbl 1316.35235 [10] Hou, T. Y.; Luo, G., Potentially singular solutions of the 3D axisymmetric Euler equations, Proc. Natl. Acad. Sci. USA, 111, 36, 12968-12973, (2014) [11] Majda, A.; Bertozzi, A., Vorticity and incompressible flow, (2002), Cambridge University Press · Zbl 0983.76001 [12] Marchioro, C.; Pulvirenti, M., Mathematical theory of incompressible non-viscous fluids, Appl. Math. Sci., vol. 96, (1994), Springer [13] Kiselev, A.; Ryzhik, L.; Yao, Y.; Zlatoš, A., Finite time singularity formation for the modified SQG patch equation, Ann. of Math., (2018), in press [14] Kiselev, A.; Šverák, V., Small scale creation for solutions of the incompressible two dimensional Euler equation, Ann. of Math. (2), 180, 3, 1205-1220, (2014) · Zbl 1304.35521 [15] Kiselev, A.; Tan, C., Finite time blow up in the hyperbolic Boussinesq system, (2016) [16] Tao, T., Finite time blowup for Lagrangian modifications of the three-dimensional Euler equation, (2016), preprint · Zbl 1397.35181
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