Global wellposedness of an inviscid 2D Boussinesq system with nonlinear thermal diffusivity

Abstract

Abstract. We consider a two-dimensional inviscid Boussinesq system with temperature-dependent thermal diffusivity. We prove global wellposedness of strong solutions for arbitrarily large initial data in Sobolev spaces.

Existence of solutions to heat equations with hysteresis coupled with Navier–Stokes equations in 2D domains

“…Larios-Lunasin-Titi [16] proved well-posedness for an anisotropic Boussinesq equation. Li-Xu [17] established existence and uniqueness for an inviscid Boussinesq equation with temperature-dependent thermal diffusivity. Miao-Zheng [18] gave an existence result for a Boussinesq equation with horizontal dissipation.…”

“…17) where we have used the Gagliardo-Nirenberg inequality in(5.17). Therefore, it follows from (5.9)-(5.11) and (5.17) that {dθ n /dt} is bounded in L 2 (0, T ; H),(5.18) {θ n } is bounded in L ∞ (0, T ; V ),(5.19){Δθ n } is bounded in L 2 (0, T ; H), (5.20) {∇θ n } is bounded in L 4 0, T ; L 4 (Ω) ,(5.21) where we have used the Gagliardo-Nirenberg inequality in (5.21).…”

Global well‐posedness of the 2D Euler‐Boussinesq system with stratification effects

“…Fortunately, the 2D Euler equations in provide the uniformly estimates (Theorem ) in about the vorticity ω := ∂ 1 u 2 − ∂ 2 u 1 , which also leads to the key estimate in — θ is Hölder continuous (Lemma ), by using the De Giorgi method in . Another point is different from the method in ; thanks to the local well‐posedness theory and the logarithmic Sobolev inequality, we need only to prove that ∇ θ is bounded in in order to extend the local solution to the global one (Theorem ), which will be guaranteed by the fact that θ is Hölder continuous as shown previously (Proposition ).…”

Persistence of regularity for the viscous Boussinesq equations with zero diffusivity

“…The well-posedness (local and global) as well as persistence of regularity have recently attracted attention of many mathematicians [2,3,5,9,10,12,13,15,17,23,24,26,28,30], starting with Chae [7] and Hou and Li [20] who proved the global existence of regular solutions for smooth initial data. More precisely, Chae showed the global existence for (u 0 , ρ 0 ) ∈ H 3 × H 3 while Hou and Li proved the same for initial datum in H 3 × H 2 .…”