Boundary-layer effects for the 2-D Boussinesq equations with vanishing diffusivity limit in the half plane

Jiang, Song; Zhang, Jianwen; Zhao, Junning

doi:10.1016/j.jde.2011.01.002

Cited by 19 publications

(7 citation statements)

References 35 publications

(54 reference statements)

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“…This model plays an important role in atmospheric and oceanographic sciences (see [13,16]). Furthermore, because of its close connection to the incompressible Euler and Navier-Stokes equations, it has received significant attention in the mathematical fluid dynamics community (see [2,3,9,10,12,17]). As is stated in [14], problems related to the vanishing viscosity limit (ε → 0 and κ > 0), vanishing diffusivity limit (κ → 0 and ε > 0) or zero dissipation limit (ε, κ → 0) are important and challenging for (1.1)-(1.3).…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…For the Cauchy problem, the vanishing viscosity limit and the vanishing diffusivity limit in the two-dimensional case are established in [3]. For the initial boundary-value problem, the vanishing diffusivity limit for (1.1)-(1.3) in a half plane is investigated in [10], where the existence of a boundary layer for the temperature is proved.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

Zero dissipation limit and stability of boundary layers for the heat conductive Boussinesq equations in a bounded domain

Wang¹,

Xie²

2015

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this paper, we study the zero dissipation limit of the initial boundary-value problem of the multi-dimensional Boussinesq equations with viscosity and heat conductivity. Such equations are used as models for the motion of multi-dimensional incompressible fluids in atmospheric and oceanographic turbulence. In particular, they describe the thermal convection of an incompressible flow, and constitute the relations between the velocity field, the pressure and the local temperature. Under the Navier slip boundary condition in the velocity field and the thermal isolation boundary condition for the temperature, we prove the existence of weak amplitude characteristic boundary layers. Then, by a standard energy method, we prove the L2 convergence of the solutions when both the viscosity and the heat conductivity coefficients tend to 0.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Zero dissipation limit and stability of boundary layers for the heat conductive Boussinesq equations in a bounded domain

Wang¹,

Xie²

2015

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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“…Economic science suggests that the development of contemporary economic science has expanded from the study of things to the study of the complex large system, from the single numerical research and development to a variety of numerical composite research, from the qualitative or quantitative research development of single to complex qualitative and quantitative research, it is not only expanded the research scope from the inevitable phenomenon to accidental phenomena, from accurate phenomenon is extended to the fuzzy phenomenon, but also further research emerge in large Numbers in the objective world of incompatible problem [7][8]. And extension decision is based on the matter-element transformation can with its extensive research object and the research methods of unique gradually formed its own set of research theory and method, in order to further develop the idea of people to provide a new approach of ideal decision.…”

Section: Discussionmentioning

confidence: 99%

The Extension Decision Analysis Method based on Object-element Transformation

Xiao¹

2018

Proceedings of the 2018 2nd International Conference on Education Science and Economic Management (ICESEM 2018)

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In order to solve the problem of incompatibility in the real world, it must be based on dialectical thinking to develop a unique and creative means and methods compatible with incompatible ones. Therefore, in this paper, by means of mathematical tools and extension set matter-element transformation analysis method, the extension set are discussed deeply the practical background and its relationship with the classic set, fuzzy set, on the basis of further discussed from the philosophical foundation and mathematical system extension set is not empty, extension domain can be changed by mapping, transformation incompatible problems for incompatible logical foundation and theoretical basis, and then seek the law to solve the problem of incompatibility and extension decision analysis method, to achieve the modernization of the enterprise and the best economic benefits for enterprises to provide the best scientific decision.

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“…In this paper, we consider a boundary layer problem between two horizontal parallel plates. Such kinds of boundary layer problems were also studied in [6,[12][13][14]26]. Precisely speaking, we consider the following initial-boundary value problem of Hsieh's equation with conservative nonlinearity related to Lorenz system on the strip [0, 1] × [0, ∞)…”

Section: Introductionmentioning

confidence: 99%

Boundary layer of Hsieh's equation with conservative nonlinearity

Gao¹,

Mou²,

Ruan³

et al. 2022

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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In this paper, we consider an initial-boundary value problem of Hsieh's equation with conservative nonlinearity. The global unique solvability in the framework of Sobolev is established. In particular, one of our main motivations is to investigate the boundary layer (BL) effect and the convergence rates as the diffusion parameter $\beta$ goes zero. It is shown that the BL-thickness is of the order $O(\beta ^{\gamma })$ with $0<\gamma <\frac {1}{2}$ . We need to point out that, different from the previous work on nonconservative form of Hsieh's equations, the conservative nonlinearity $(\psi ^{\beta }\theta ^{\beta })_x$ implies that new nonlinear term $\psi _x^{\beta }\theta ^{\beta }$ needs to be handled. It is important that more regularities on the solution to the limit problem are required to obtain the convergence rates and BL-thickness. It is more difficult for initial-boundary problem due to the lack of boundary conditions (especially, higher-order derivatives) prevents us from applying the integration by part to derive the energy estimates directly. Thus it is more complicated than the case of nonconservative form. Consequently more subtle mathematical analysis needs to be introduced to overcome the difficulties.

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Boundary-layer effects for the 2-D Boussinesq equations with vanishing diffusivity limit in the half plane

Cited by 19 publications

References 35 publications

Zero dissipation limit and stability of boundary layers for the heat conductive Boussinesq equations in a bounded domain

Zero dissipation limit and stability of boundary layers for the heat conductive Boussinesq equations in a bounded domain

The Extension Decision Analysis Method based on Object-element Transformation

Boundary layer of Hsieh's equation with conservative nonlinearity

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