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Finite time blow up for a 1D model of 2D Boussinesq system. (English) Zbl 1309.35072
Summary: The 2D conservative Boussinesq system describes inviscid, incompressible, buoyant fluid flow in a gravity field. The possibility of finite time blow up for solutions of this system is a classical problem of mathematical hydrodynamics. We consider a 1D model of the 2D Boussinesq system motivated by a particular finite time blow up scenario. We prove that finite time blow up is possible for the solutions to the model system.

##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76R10 Free convection 35B44 Blow-up in context of PDEs
##### Keywords:
Boussinesq system; buoyant fluid flow; blow up
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##### References:
 [1] Beale, J.T.; Kato, T.; Majda, A., Remarks on the breakdown of smooth solutions for the 3-D Euler equations, Commun. Math. Phys., 94, 61-66, (1984) · Zbl 0573.76029 [2] Cao, C.; Wu, J., Global regularity for the 2D anisotropic Boussinesq equations with vertical dissipation, Arch Ration. Mech. Anal., 208, 985-1004, (2013) · Zbl 1284.35140 [3] Chae, D., Global regularity for the 2D Boussinesq equations with partial viscosity terms, Adv. Math., 203, 497-513, (2006) · Zbl 1100.35084 [4] Chae, D.; Nam, H., Local existence and blow-up criterion for the Boussinesq equations, Proc. Roy. Soc. Edinburgh Sect. A, 127, 935-946, (1997) · Zbl 0882.35096 [5] Constantin, P.; Fefferman, C., Direction of vorticity and the problem of global regularity for the Navier-Stokes equations, Indiana Univ. Math. J., 42, 775-789, (1993) · Zbl 0837.35113 [6] Choi, K., Hou, T., Kiselev, A., Luo, G., Sverak, V., Yao, Y.: On the finite-time blowup of a 1D model for the 3D axisymmetric Euler equations, preprint arXiv:1407.4776 · Zbl 1377.35218 [7] Constantin P., Fefferman C., Majda A.: Geometric constraints on potentially singular solutions for the 3-D Euler equation. Commun. PDE 21:559-571 (1996) · Zbl 0853.35091 [8] Deng, J.; Hou, T.Y.; Yu, X., Geometric properties and non-blowup of 3D incompressible Euler flow, Commun. PDEs, 30, 225-243, (2005) · Zbl 1142.35549 [9] E, W.; Shu, C., Samll-scale structures in Boussinesq convection, Phys. Fluids, 6, 49-58, (1994) · Zbl 0822.76087 [10] Gibbon, J.D., The three-dimensional Euler equations: where do we stand?, Physica D, 237, 1894-1904, (2008) · Zbl 1143.76389 [11] Grauer, R.; Sideris, T.C., Numerical computation of 3D incompressible ideal fluids with swirl. phys, Rev. Lett., 67, 3511-3514, (1991) [12] Hou, T.; Li, C., Global well-posedness of the viscous Boussinesq equations. discrete contin, Dyn. Syst., 12, 1-12, (2005) · Zbl 1274.76185 [13] Hou, T.Y.; Li, R., Dynamic depletion of vortex stretching and non-blowup of the 3-D incompressible Euler equations, J. Nonlinear Sci., 16, 639-664, (2006) · Zbl 1370.76015 [14] Hou, T.Y.; Li, R., Blowup or no blowup? the interplay between theory and numerics, Physica D, 237, 1937-1944, (2008) · Zbl 1143.76390 [15] Hou, T., Luo, G.: Potentially Singular Solutions of the 3D Incompressible Euler Equations, preprint arXiv:1310.0497 · Zbl 1284.35140 [16] Hou, T., Luo, G.: On the finite-time blow up of a 1D model for the 3D incompressible Euler equations, preprint arXiv:1311.2613 [17] Kerr, R.M., Evidence for a singularity of the three-dimensional incompressible Euler equations, Phys. Fluids A, 5, 1725-1746, (1993) · Zbl 0800.76083 [18] Kiselev, A., Sverak, A.: Small scale creation for solutions of the incompressible two dimensional Euler equation, preprint arXiv:1310.4799, to appear in Ann. Math. · Zbl 0825.76121 [19] Kufner A.: Weighted Sobolev Spaces. Wiley, New York (1985) · Zbl 0567.46009 [20] Majda A., Bertozzi A.: Vorticity and Incompressible Flow. Cambridge University Press, Cambridge (2002) · Zbl 0983.76001 [21] Pumir, A.; Siggia, E.D., Development of singular solutions to the axisymmetric Euler equations, Phys. Fluids A, 4, 1472-1491, (1992) · Zbl 0825.76121 [22] Yudovich, V.I., Eleven great problems of mathematical hydrodynamics, Mosc. Math. J., 3, 711-737, (2003) · Zbl 1061.76003
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