Evolution of the two-dimensional Boussinesq system. (English) Zbl 1358.35197

Tang, Tai-Man; Xie, Ye

Dyn. Partial Differ. Equ. 13, No. 4, 333-350 (2016).

Summary: The smooth evolutions along the trajectories of the main physical quantities of the two dimensional Boussinesq system with viscousity and thermal diffusivity not both non-zero are studied. Specifically, for a spatially \(H^m\) solution with \(m>4\) (only \(m>3\) is needed for some result), quantities including the speed, vorticity, temperature gradient and their stretching rates are shown to evolve smoothly along the trajectories. Conclusions on their evolutions are obtained. Results on some of the stretching rates give information on the evolutions of the relative sizes of some basic quantities. When the viscousity and thermal diffusivity are zero, it is not known if smooth solutions exist globally and we study the dichotomy between finite time singularity and the long time behaviors of the main quantities. If either the viscousity or thermal diffusivity is non-zero, it is known that smooth solutions are global and this investigation provides some information about them by describing the dynamics of the main quantities.

MSC:

35Q86	PDEs in connection with geophysics
76B99	Incompressible inviscid fluids
35B40	Asymptotic behavior of solutions to PDEs
35Q35	PDEs in connection with fluid mechanics
80A20	Heat and mass transfer, heat flow (MSC2010)
76N99	Compressible fluids and gas dynamics, general
35B44	Blow-up in context of PDEs
86A05	Hydrology, hydrography, oceanography
35B65	Smoothness and regularity of solutions to PDEs

Keywords:

Boussinesq equations; asymptotic behavior; blow up; dynamics; global smooth solution