Persistence of regularity for the viscous Boussinesq equations with zero diffusivity

Hu, Weiwei; Kukavica, Igor; Ziane, Mohammed

doi:10.3233/asy-141261

Cited by 37 publications

(27 citation statements)

References 31 publications

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“…Recently, the well-posedness of the 2D Boussinesq equations has attracted attention of many mathematicians, see [1]- [4], [6]- [11], [15], [17], [16], [20], [24], [25]. In particular, when Ω = R 2 , the Cauchy problem of (1.1) has been well studied.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Global regularity of the two-dimensional Boussinesq equations without diffusivity in bounded domains

Zhou

2018

Nonlinear Analysis: Real World Applications

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Global regularity of the two-dimensional Boussinesq equations without diffusivity in bounded domains

Zhou

2018

Nonlinear Analysis: Real World Applications

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“…Proof: From the previous theorem θ remains as a C 1+γ patch. Since the characteristic function of a set with regular boundary belongs to H µ for any µ < 1 2, the result in [29] yields the existence and uniqueness of solutions…”

Section: Control Of Curvaturementioning

confidence: 95%

“…Proof: As u 0 ∈ H 1+γ+s and θ 0 ∈ H δ , for any δ ∈ (0, 1 2), from [29] one gets that u ∈ L ∞ (0, T ; H 1+µ ) ∩ L 2 (0, T ; H 2+µ ) with µ < 1 2, µ ≤ γ + s. Using the splitting (11) to bootstrap, for the initial data we find that…”

Section: Persistence Of C 2+γ Regularitymentioning

confidence: 96%

“…In particular, Abidi and Hmidi [1] established the global well-posedness for initial data in the Besov space B 0 2,1 (see Appendix for definition), then Hmidi and Keraani [26] proved global existence and regularity for initial data θ 0 ∈ L 2 , u 0 ∈ H s , s ∈ [0, 2) and finally Danchin and Paicu obtained the uniqueness [21]. The persistence of regularity in Sobolev spaces was completed by Hu, Kukavica and Ziane in [29]. Available global-in-time results in three dimensions require the initial data to be small [21], as the system includes Navier-Stokes equations as a particular case.…”

Section: Introductionmentioning

confidence: 97%

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Global Regularity for 2D Boussinesq Temperature Patches with No Diffusion

Gancedo

Garcia-Juarez

2017

Ann. PDE

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This paper considers the temperature patch problem for the incompressible Boussinesq system with no diffusion and viscosity in the whole space R 2 . We prove that for initial patches with W 2,∞ boundary the curvature remains bounded for all time. The proof explores new cancellations that allow us to bound ∇ 2 u, even for those components given by time dependent singular integrals with kernels with nonzero mean on circles. In addition, we give a different proof of the C 1+γ regularity result in [23], 0 < γ < 1, using the scale of Sobolev spaces for the velocity. Furthermore, taking advantage of the new cancellations, we go beyond to show the persistence of regularity for C 2+γ patches.

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“…We consider the asymptotic behavior of solutions to the Boussinesq equations without diffusivity u t − ∆u + u · ∇u + ∇π = ρe 2 (1.1) for s = 2. The remaining range 1 < s < 3 was then resolved in [HKZ2] in the case of periodic boundary conditions. For other works on the global existence and persistence in Sobolev and Besov classes, see [ACW,BS,BrS,CD,CG,CN,CW,DP1,DP2,DWZZ,HK1,HK2,HS,KTW,KWZ,LPZ].…”

Section: Introductionmentioning

confidence: 99%

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

Kukavica

Wang

2019

J Dyn Diff Equat

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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

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Persistence of regularity for the viscous Boussinesq equations with zero diffusivity

Cited by 37 publications

References 31 publications

Global regularity of the two-dimensional Boussinesq equations without diffusivity in bounded domains

Global regularity of the two-dimensional Boussinesq equations without diffusivity in bounded domains

Global Regularity for 2D Boussinesq Temperature Patches with No Diffusion

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

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