Blow-up criterion for 2-D Boussinesq equations in bounded domain

Abstract

We extend the results for 2-D Boussinesq equations from R 2 to a bounded domain Ω. First, as for the existence of weak solutions, we transform Boussinesq equations to a nonlinear evolution equation U t + A(t, U ) = 0. In stead of using the methods of fundamental solutions in the case of entire R 2 , we study the qualities of F (u, v) = (u · ∇)v to get some useful estimates for A(t, U ), which helps us to conclude the local-in-time existence and uniqueness of solutions. Second, as for blow-up criterions, we use energy methods, Sobolev inequalities and Gronwall inequality to control θ H s (Ω) and u H s (Ω) by ∇θ L ∞ (Ω) and ∇u L ∞ (Ω) . Furthermore, ∇θ L ∞ (Ω) can control ∇u L ∞ (Ω) by using vorticity transportation equations. At last, ∇θ M φ (Ω) can control ∇θ L ∞ (Ω) . Thus, we can find a blowup criterion in the form of lim t→T * t 0 ∇θ(·, τ) M φ (Ω) dτ = ∞.

On the degenerate boussinesq equations on surfaces

“…• [5,17,18,19,20,22,27,37,51,53] for local well-posedness, blowup criteria, explicit solutions and finite-time singularities for the degenerate case (i.e., ν = κ = 0); • [1,2,3,4,14,15,16,21,23,24,25,32,33,34,35,36,38,39,44,45,46,48,49,62] for global well-posedness and regularity for the non-degenerate and partially degenerate cases; • [41,42,47,52,59,60,61] for well-posedness and regularity with critical and supercritical dissipation; and • [10,26,44,54,58,62] for long-time behaviours.…”

Global regularity for 2D Boussinesq equations with partial viscosity in the half plane

“…Lai et al 15 showed the global regularity of the viscous case, and Zhao 16 to the thermal diffusive one. For the inviscid case $$$ \mu =\kappa =0 $$$, one can refer to Chae 6 and Hu and Jian 17 about the local existence and blow‐up criterion. Very recently, Chen and Hou 18 proved the finite time singularity for the 2D Boussinesq and 3D axisymmetric Euler equations in the presence of boundary for a class of $$$ {C}^{1,\alpha } $$$ initial data.…”

Boundary-layer effects for the 2-D Boussinesq equations with vanishing diffusivity limit in the half plane

“…Zhao [45] proved the global existence of classical solutions to the 2-D inviscid-diffusive Boussinesq equations with slip boundary conditions. On the other hand, the global regularity/singularity question for the 2-D Boussinesq equations with both zero diffusivity and zero viscosity, which possess many similarities to the 3-D axi-symmetric Euler equations with swirl away from the symmetric axis r = 0, is still an outstanding open problem in mathematical fluid mechanics, see, for example, [6][7][8]11,23] for studies in this direction. It is worth pointing out that if θ ε = 0, then the system (1.1)-(1.3) reduces to the Navier-Stokes equations for incompressible fluids, which have been extensively studied by a great number of mathematician in a large variety of contexts, see, for example, [9,14,29,39], etc.…”