Global regularity problem for the 2D Boussinesq equations.

摘要

The Boussinesq equations are a system of nonlinear partial differential equations which model the thermal convection and geostrophic flows. One major issue concerning the Boussinesq equations is whether or not their classical solutions are always global in time. This thesis mainly focus on the global regularity issue of the 2D Boussinesq equations with vertical dissipations.;For the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion, we prove that the Lq-norm of the vertical velocity v for any 1 < q < infinity is globally bounded and that the Linfinity -norm of v controls any possible breakdown of classical solutions. We also show that an extra thermal diffusion given by the fractional Laplace (-Delta)delta for delta > 0 would guarantee the global regularity of classical solutions.;Next we are able to show that the vertical velocity v of any classical solution in the Lebesgue space Lq with 2 ≤ q < infinity is bounded by C 1 q for C1 independent of q. This bound significantly improves the previous exponential bound. In addition, we prove that, if v satisfies 0Tsup q≥2v ˙,t 2Lqq dt < infinity, then the associated solution of the 2D Boussinesq equations preserve its smoothness on [0, T]. In particular, vL q ≤ C2 q implies global regularity. We also investigate the uniqueness of global weak solutions.;We also settle the global regularity issue of the 2D Boussinesq equation with vertical diffusivity and viscosity only in the second equation of the velocity field, namely with kappathetayy and nuDeltav.