Stability of the 2D Boussinesq System with Partial Dissipation
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Stability of the 2D Boussinesq System with Partial Dissipation Youhua Wei1 · Dan Li1
Received: 11 December 2019 / Revised: 30 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper establishes the global stability for the 2D Boussinesq system with partial dissipation and horizontal thermal diffusion. When there is no thermal diffusion, the stability of the temperature gradient remains an open problem. We extend the H 1 -stability in [7] to H 2 -stability which we care about and obtain the large time behavior of the linearized system. Keywords Boussinesq equation · Hydrostatic equilibrium · Partial dissipation · Large-time behavior · Stability Mathematics Subject Classification 35A05 · 35Q35 · 76D03
1 Introduction This paper concerns itself with the stability for the following Boussinesq system with partial dissipation ⎧ ∂t u 1 + u · ∇u 1 = −∂1 P + ν ∂22 u 1 , x ∈ R2 , t > 0, ⎪ ⎪ ⎨ ∂t u 2 + u · ∇u 2 = −∂2 P + ν ∂11 u 2 + , x ∈ R2 , t > 0, (1.1) ∂ + u · ∇ = η ∂11 , x ∈ R2 , t > 0, ⎪ ⎪ ⎩ t ∇ · u = 0, x ∈ R2 , t > 0, where u(x) = (u 1 (x1 , x2 ), u 2 (x1 , x2 )) denotes the velocity field, P the pressure, the temperature, ν > 0 and η > 0 are the viscosity and the thermal diffusivity, respectively. Here, we use ∂i for ∂xi (i=1,2) briefly. The 2D Boussinesq system studied here is physically relevant for certain anisotropic fluids driven by buoyancy. In certain physical regimes and under suitable scaling, the coefficient of ∂11 u 1 is small compared to that of ∂22 u 1 and the coefficient of ∂22 u 2 is small compared to that of ∂11 u 2 . The thermal diffusion is mainly in the horizontal direction.
B
Dan Li [email protected] Youhua Wei [email protected]
1
Department of Mathematics, Sichuan University, Chengdu 610064, PR China
123
Journal of Dynamics and Differential Equations
We know that (1.1) admits the following steady state solution 1 2 x , (1.2) 2 2 this special solution represents the hydrostatic equilibrium or balance. The hydrostatic balance refers to the status when the fluid is static and all forces acting on the fluid are balanced out. Our atmosphere is mostly at hydrostatic balance with the pressure gradient balanced out by the gravitation force. The hydrostatic balance plays many important roles in geophysics and astrophysics [2,5,10,12]. Understanding the stability of perturbations near the hydrostatic balance is important both mathematically and physically. In this paper, we concern about the stability and large-time behavior of perturbations near the hydrostatic equilibrium in (1.2). To this end, we consider the perturbation (u, θ ) with θ = − x2 . It’s easy to check that, if (u, ) satisfies (1.1), then (u, θ ) satisfies ⎧ 2 ⎪ ⎪ ∂t u 1 + u · ∇u 1 = −∂1 P + ν ∂22 u 1 , x ∈ R , t 2> 0, ⎨ ∂t u 2 + u · ∇u 2 = −∂2 P + ν ∂11 u 2 + θ, x ∈ R , t > 0, (1.3) ∂t θ + u · ∇θ + u 2 = η ∂11 θ, x ∈ R2 , t > 0, ⎪ ⎪ ⎩ 2 ∇ · u = 0, x ∈ R , t > 0, u (0) ≡ 0, (0) = x2 ,
P (0) =
the stability problem on (1.3) is equivalent to assessing the small data global we
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