Spatial decay bounds for the channel flow of the Boussinesq equations

Liu, Yan; Li, Yuanfei; Lin, Yiwu; Yao, Zheng‐an

doi:10.1016/j.jmaa.2011.02.066

Cited by 5 publications

(6 citation statements)

References 21 publications

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“…Eðξ; tÞdξ ≤ 4τM 1 Eðz; tÞ; (5:26) which has been well studied by Liu et al 25 and Horgan and Wheeler. 34 By applying the argument of these papers, from (5.26), it follows that…”

Section: Energy Decay Estimatesmentioning

confidence: 95%

“…Combining , , , and , we obtain an integro‐differential inequality of the form

\frac{E false(z, t false)}{\partial z} + 4 τ^{2} M_{2} \int_{z}^{\infty} E false(ξ, t false) d ξ \leq 4 τ M_{1} E false(z, t false),

which has been well studied by Liu et al and Horgan and Wheeler . By applying the argument of these papers, from , it follows that

E false(z, t false) \leq m_{1} E false(0, t false) e^{- m_{2} z},

where

m_{1} = \frac{2 \sqrt{M_{1}^{2} - M_{2}}}{\sqrt{M_{1}^{2} + M_{2}} - M_{1}}, m_{2} = 2 τ (\sqrt{M_{1}^{2} + M_{2}} - M_{1}) .

Now, we finish the proof of Theorem .…”

Section: Energy Decay Estimatesmentioning

confidence: 99%

“…The results of the present paper are regarded as a version of Saint-Venant type results. Numerous authors have dealt with Saint-Venant type decay estimate for solutions of equations in a semi-infinite channel or cylinder; we mention in particular the papers of Ames et al, 19 Lin and Payne,20,23,24 Liu et al, 21,25 and Song. 22 When κ depends on the temperature, the problem becomes much more complicated.…”

Section: Introductionmentioning

confidence: 99%

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Spatial decay bounds in a channel flow of 2‐D Boussinesq fluid with variable thermal diffusivity

Cheng

2018

Math Methods in App Sciences

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In this paper, the authors investigate the flow of an incompressible Boussinesq fluid in a semi‐infinite plane channel. An explicit spatial decay estimate in terms of the distance from finite end of the channel is obtained from an integro‐differential inequality for an energy integral. To make the result explicit, an upper bound for total energy is also obtained. The results of this paper may be thought of as a version of Saint‐Venant principle.

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“…Eðξ; tÞdξ ≤ 4τM 1 Eðz; tÞ; (5:26) which has been well studied by Liu et al 25 and Horgan and Wheeler. 34 By applying the argument of these papers, from (5.26), it follows that…”

Section: Energy Decay Estimatesmentioning

confidence: 95%

“…Combining , , , and , we obtain an integro‐differential inequality of the form

\frac{E false(z, t false)}{\partial z} + 4 τ^{2} M_{2} \int_{z}^{\infty} E false(ξ, t false) d ξ \leq 4 τ M_{1} E false(z, t false),

which has been well studied by Liu et al and Horgan and Wheeler . By applying the argument of these papers, from , it follows that

E false(z, t false) \leq m_{1} E false(0, t false) e^{- m_{2} z},

where

m_{1} = \frac{2 \sqrt{M_{1}^{2} - M_{2}}}{\sqrt{M_{1}^{2} + M_{2}} - M_{1}}, m_{2} = 2 τ (\sqrt{M_{1}^{2} + M_{2}} - M_{1}) .

Now, we finish the proof of Theorem .…”

Section: Energy Decay Estimatesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Spatial decay bounds in a channel flow of 2‐D Boussinesq fluid with variable thermal diffusivity

Cheng

2018

Math Methods in App Sciences

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“…In another paper, Song 5 considered the time‐dependent double‐diffusive convective Darcy flow in a semi‐infinite channel and the Saint–Venant type decay of the solutions on a is obtained. For more works, one can see previous studies 6–15 . However, these papers need to assume that the solutions satisfy certain a priori assumptions at the infinity of the cylinder or the channel.…”

Section: Introductionmentioning

confidence: 99%

Phragmén–Lindelöf alternative results in time‐dependent double‐diffusive Darcy plane flow

Chen

2022

Math Methods in App Sciences

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This paper investigates the spatial behavior of the solutions of the double-diffusive Darcy plane flow in a semi-infinite channel. Using the energy estimate method and the differential inequality technology, a differential inequality about the solutions is derived. By solving this differential inequality, it is proved that the solutions grow polynomially or decay exponentially with spatial variable. In the case of decay, we obtain the upper bound for the total energy. We also give some remarks to generalize the results of this paper.

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“…This paper considers the 2-D Boussinesq system in a semi-infinite channel. By making use of the earlier work of [19] and some Soblev inequalities, the continuous dependence on the coefficient of the system is obtained. The authors also show how to bound the total energy.…”

mentioning

confidence: 99%