Long Time Behavior of Solutions to the 2D Boussinesq Equations with Zero Diffusivity

Abstract

We address long time behavior of solutions to the 2D Boussinesq equations with zero diffusivity in the cases of the torus, R 2 , 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.Here, u is the velocity satisfying the 2D Navier-Stokes equations [CF, DG, FMT, R, T1, T2, T3] driven by ρ, which represents the density or temperature of the fluid, depending on the physical context. Also, e 2 = (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 [HL], who proved the global existence and persistence in the class H s × H s−1 for integer s ≥ 3. Independently, Chae [C] considered the class H s × H s 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 [LLT] the global existence and uniqueness for s = 1 and then by Hu et al, who proved in [HKZ1] the persistence

Asymptotic properties of the Boussinesq Equations with Dirichlet Boundary Conditions

“…(2.5) below, where A is the Stokes operator. Regarding the growth of the density, we prove that the first Sobolev norm of the density is bounded, up to a constant, by e ǫt for an arbitrarily small ǫ > 0, thus improving a result from [KW2] where the bound of the type e Ct was proven. Since the growth of the Sobolev norms of the density is controlled by the time integral of ∇u L ∞ , it is reasonable to expect that the bound was optimal; however, here we prove that the optimal bound is in fact e ǫt .…”

“…Subsequently, Ju obtained in [J] that Ce Ct 2 is an upper bound for the H 1 norm for the density, also for the Dirichlet boundary conditions. The bound was lowered to e Ct in [KW2], where also more precise results were obtained for periodic boundary conditions. In particular, [KW2,Theorem 2.1] contains a uniform in time upper bound for the quantity D 2 u L p for all p ≥ 2 in the periodic case.…”

“…The bound was lowered to e Ct in [KW2], where also more precise results were obtained for periodic boundary conditions. In particular, [KW2,Theorem 2.1] contains a uniform in time upper bound for the quantity D 2 u L p for all p ≥ 2 in the periodic case. In a recent paper by Doering et al [DWZZ], the global existence, uniqueness, and regularity for the Boussinesq for the Lions boundary condition on a Lipschitz domain Ω, was proven along with the dissipation of the L 2 norm of the velocity and its gradient.…”

“…In a recent paper by Doering et al [DWZZ], the global existence, uniqueness, and regularity for the Boussinesq for the Lions boundary condition on a Lipschitz domain Ω, was proven along with the dissipation of the L 2 norm of the velocity and its gradient. For other papers on the global existence and the regularity in Sobolev and Besov spaces, see [ACW,ACS..,BFL,BS,BrS,CD,CG,CN,CW,DP,HK1,HK2,HKR,HKZ2,HS,JMWZ,KTW,KW2,KWZ,LPZ,SW].…”

Norm inflation for the Boussinesq system

“…In [4,19], the authors addressed the global well-posedness in Sobolev spaces H s . Kukavica and the second author of this paper addressed the global Sobolev persistence of regularity in W s,q × W s,q for the fractional Boussinesq system in [15] and the long time behavior of solutions in [16], respectively. For other global well-posedness results on the Boussinesq equations, cf.…”