Asymptotic properties of the Boussinesq Equations with Dirichlet Boundary Conditions

Abstract

We address the asymptotic properties for the Boussinesq equations with vanishing thermal diffusivity in a bounded domain with no-slip boundary conditions. We show the dissipation of the L 2 norm of the velocity and its gradient, convergence of the L 2 norm of Au, and an o(1)-type exponential growth for A 3/2 u L 2 . We also obtain that in the interior of the domain the gradient of the vorticity is bounded by a polynomial function of time.

Growth of curvature and perimeter of temperature patches in the 2D Boussinesq equations

“…Indeed, considering (1.1) in a bounded domain, Ju [16] showed ρ H 1 ≲ e ct 2 , which was further improved to an exponential bound e ct in T 2 by Kukavica-Wang [21]. Subsequently, Kukavica-Massatt-Ziane [20] achieved a slightly better upper bound ρ H 2 ≲ C ϵ e ϵt for any small ϵ > 0. In addition to these upper bounds on growth rates, an interesting lower bound was obtained by Brandolese-Schonbek [4] proving that in R 2 , the kinetic energy u L 2 must grow faster than c(1 + t) 1/4 as t → ∞, provided that the initial density ρ 0 does not have a zero average.…”

“…When considering the Boussinesq system in an unbounded domain, obtaining such a result for general initial data becomes nontrivial. The challenge arises from the fact that while the proof provided in [2,20] relies on a uniform bound of the kinetic energy, the kinetic energy in an unbounded domain may not be uniformly bounded in time in general. Although the total energy is inferred to be bounded from (2.4), it remains possible for the kinetic energy to increase indefinitely throughout the evolution without a lower bound of the potential energy.…”

“…Besides these quantitative analyses of norm growth, one might expect some asymptotic behaviour due to the damping effect induced by viscosity. In this direction, Kukavica-Massatt-Ziane [20] and Aydin-Kukavica-Ziane [2] showed that ∇u L 2 and ν∆u − P(ρe 2 ) L 2 , where P denotes the Leray projection, converge to 0 as t → ∞ in a bounded domain.…”

“…Remark 3.2. When considering (1.1) in a bounded domain, in [2,20], it was shown that for a general initial data, ν∆u − P(ρe 2 ) L 2 converges to 0 as t → ∞, where P is the Leray projection, which is equivalent to νω − ∂ 1 ∆ −1 ρ Ḣ1 → 0. When considering the Boussinesq system in an unbounded domain, obtaining such a result for general initial data becomes nontrivial.…”

Small scale formation for the 2D Boussinesq equation

“…When Ω = T 2 , they also obtained the uniform-in-time bound u W 2,p (T 2 ) ≤ C(p) for all p ∈ [2, ∞). In a recent work by Kukavica-Massatt-Ziane [34], when Ω is a bounded domain, the upper bound of the norm of ρ has been improved to ρ H 2 (Ω) ≤ C e t for all > 0, and they also showed u H 3 ≤ C e t for all > 0.…”