Long Time Behavior of Solutions to the 2D Boussinesq Equations with Zero Diffusivity
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Long Time Behavior of Solutions to the 2D Boussinesq Equations with Zero Diffusivity Igor Kukavica1 · Weinan Wang1 Received: 9 April 2019 / Revised: 2 August 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract We address long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity in the cases of the torus, R2 , and on a bounded domain with Lions or Dirichlet boundary conditions. In all the cases, we obtain bounds on the long time behavior for the norms of the velocity and the vorticity. In particular, we obtain that the norm (u, ρ) H 2 ×H 1 is bounded by a single exponential, improving earlier bounds. Keywords Boussinesq system · Navier–Stokes equations · Asymptotic behavior · Persistence · Regularity
1 Introduction We consider the asymptotic behavior of solutions to the Boussinesq equations without diffusivity u t − u + u · ∇u + ∇π = ρe2
(1.1)
ρt + u · ∇ρ = 0
(1.2)
∇ ·u =0
(1.3)
in a bounded domain ⊆ R2 , T2 , and R2 . Here, u is the velocity satisfying the 2D Navier– Stokes equations [6,10,13,30,33–35] driven by ρ, which represents the density or temperature of the fluid, depending on the physical context. Also, e2 = (0, 1) is the unit vector in the vertical direction. Recently, there has been a lot of progress made on the existence, uniqueness, and persistence of regularity, mostly in the case of positive viscosity and vanishing diffusivity, considered here, while the same question with both vanishing viscosity and diffusivity is an important open problem. The initial results on the global existence in the regularity class have been obtained by Hou and Li [17], who proved the global existence and persistence in the class H s × H s−1 for integer s ≥ 3. Independently, Chae [4] considered the class H s × H s
B
Igor Kukavica [email protected] Weinan Wang [email protected]
1
Department of Mathematics, University of Southern California, Los Angeles, CA 90089, USA
123
Journal of Dynamics and Differential Equations
and proved the global persistence in H 3 × H 3 . The class H s × H s−1 has subsequently been studied in the case of a bounded domain, where Larios et al proved in [27] the global existence and uniqueness for s = 1 and then by Hu et al, who proved in [18] the persistence for s = 2. The remaining range 1 < s < 3 was then resolved in [19] in the case of periodic boundary conditions. For other works on the global existence and persistence in Sobolev and Besov classes, see [1–3,5,7–9,11,12,14,15,20,23,26,28]. In a recent paper [22], Ju addressed the important question of long time behavior of solutions. He proved that in the case of Dirichlet boundary conditions on a bounded domain , 2 the H 2 () × H 1 () norm grows at most as CeCt , where C > 0 is constant. In the present paper, we consider this question for this and other boundary conditions. When the domain is finite, we prove that actually the H 2 × H 1 norm is increasing as a single exponential. We conjecture that this bound is sharp. This is because it is not expected that the solu
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