Evolution of the two-dimensional Boussinesq system.

*(English)*Zbl 1358.35197Summary: 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 |