This paper aims at the global regularity of classical solutions to the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion. We prove that the L r -norm of the vertical velocity v for any 1 < r < ∞ is globally bounded and that the L ∞ -norm of v controls any possible breakdown of classical solutions. In addition, we show that an extra thermal diffusion given by the fractional Laplace (− ) δ for δ > 0 would guarantee the global regularity of classical solutions.
This paper furthers the study of Adhikari et al. (2010) [2] on the global regularity issue concerning the 2D Boussinesq equations with vertical dissipation and vertical thermal diffusion. It is shown here that the vertical velocity v of any classical solution in the Lebesgue space L q with 2 q < ∞ is bounded by C 1 q for C 1 independent of q. This bound significantly improves the previous exponential bound. In addition, we prove that, if v satisfies T 0 sup q 2 v(·,t) 2 L q q dt < ∞, then the associated solution of the 2D Boussinesq equations preserve its smoothness on [0, T ]. In particular, v L q C 2 √ q implies global regularity.
Abstract. The two-dimensional (2D) incompressible Euler equations have been thoroughly investigated and the resolution of the global (in time) existence and uniqueness issue is currently in a satisfactory status. In contrast, the global regularity problem concerning the 2D inviscid Boussinesq equations remains widely open. In an attempt to understand this problem, we examine the damped 2D Boussinesq equations and study how damping affects the regularity of solutions. Since the damping effect is insufficient in overcoming the difficulty due to the "vortex stretching", we seek unique global small solutions and the efforts have been mainly devoted to minimizing the smallness assumption. By positioning the solutions in a suitable functional setting (more precisely the homogeneous Besov spaceB 1 ∞,1 ), we are able to obtain a unique global solution under a minimal smallness assumption.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.