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