On Maximum Modulus Estimates of the Navier--Stokes Equations with Nonzero Boundary Data

Chang, Tongkeun; Choe, Hi Jun; Kang, Kyungkeun

doi:10.1137/17m1152565

Cited by 5 publications

(7 citation statements)

References 10 publications

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“…for any r ∈ (1, ∞) and any m ∈ Remark 4.3. The Hölder continuity of v can be also proved by hand, but it is easier to refer to Chang-Choe-Kang [2].…”

Section: Navier-stokes Flowsmentioning

confidence: 99%

“…By the pointwise bounds |u(x)|x 1−n from (4.3), F (x) ∈ L p (R n + × (0, 2)) for any p ∈ [1, ∞]. By Proposition 2.3 of Chang-Choe-Kang[2] and taking p < ∞ arbitrarily large, we getv ∈ C 2c,c (R n + × [0, 2]) for any c < 1/2. Hence u = αv + v ∈ C 2b,b (R n + × [0, 2]).…”

mentioning

confidence: 94%

“…This property fails in the boundary case, as shown by the examples of [10,22]. For the theory of boundary regularity, see for example Seregin [16,17,18,19], Kang-Gustafson-Tsai [9], Mikhailov [13,14] and Dong-Gu [6] for regularity criteria, Chernobay [5] and Seregin [20] for type I blowups near the boundary, Chang-Jin [3] for Hölder continuity, and Chang-Choe-Kang [2] for Dini continuity. See also the survey paper [21] of Seregin and Shilkin.…”

mentioning

confidence: 99%

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Finite energy Navier-Stokes flows with unbounded gradients induced by localized flux in the half-space

Kang¹,

Lai²,

Lai³

et al. 2021

Preprint

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For the Stokes system in the half space, Kang [Math. Ann. 2005] showed that a solution generated by a compactly supported, Hölder continuous boundary flux may have unbounded normal derivatives near the boundary. In this paper we first prove explicit global pointwise estimates of the above solution, showing in particular that it has finite global energy and its derivatives blow up everywhere on the boundary away from the flux. We then use the above solution as a profile to construct solutions of the Navier-Stokes equations which also have finite global energy and unbounded normal derivatives due to the flux. Our main tool is the pointwise estimates of the Green tensor of the Stokes system proved by us in [11] (arXiv:2011.00134). We also examine the Stokes flows generated by dipole bumps boundary flux, and identify the regions where the normal derivatives of the solutions tend to positive or negative infinity near the boundary.

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“…for any r ∈ (1, ∞) and any m ∈ Remark 4.3. The Hölder continuity of v can be also proved by hand, but it is easier to refer to Chang-Choe-Kang [2].…”

Section: Navier-stokes Flowsmentioning

confidence: 99%

mentioning

confidence: 94%

mentioning

confidence: 99%

See 1 more Smart Citation

Finite energy Navier-Stokes flows with unbounded gradients induced by localized flux in the half-space

Kang¹,

Lai²,

Lai³

et al. 2021

Preprint

Self Cite

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show abstract

“…where 3 4 < t < 5 4 , 0 < s < t and k ≥ 0. Furthermore, there exist c 0 > 0 and t 0 ∈ (0, 1 4 ) such that if…”

Section: )mentioning

confidence: 99%

“…It is worth to mention that the estimates (1.5) is optimal, since the authors constructed in [8] an example that satisfies (1.5) but any higher integrability of the example fails (see also [12]).…”

Section: Introductionmentioning

confidence: 99%

On Caccioppoli's inequalities of Stokes equations and Navier-Stokes equations near boundary

Chang

Kang

2020

Journal of Differential Equations

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Finite energy Navier-Stokes flows with unbounded gradients induced by localized flux in the half-space

Kang

Lai

et al. 2022

Trans. Amer. Math. Soc.

View full text Add to dashboard Cite

For the Stokes system in the half space, Kang [Math. Ann. 331 (2005), pp. 87-109] showed that a solution generated by a compactly supported, Hölder continuous boundary flux may have unbounded normal derivatives near the boundary. In this paper we first prove explicit global pointwise estimates of the above solution, showing in particular that it has finite global energy and its derivatives blow up everywhere on the boundary away from the flux. We then use the above solution as a profile to construct solutions of the Navier-Stokes equations which also have finite global energy and unbounded normal derivatives due to the flux. Our main tool is the pointwise estimates of the Green tensor of the Stokes system proved by us in an earlier paper. We also examine the Stokes flows generated by dipole bumps’ boundary flux, and identify the regions where the normal derivatives of the solutions tend to positive or negative infinity near the boundary.

show abstract

On Maximum Modulus Estimates of the Navier--Stokes Equations with Nonzero Boundary Data

Cited by 5 publications

References 10 publications

Finite energy Navier-Stokes flows with unbounded gradients induced by localized flux in the half-space

Finite energy Navier-Stokes flows with unbounded gradients induced by localized flux in the half-space

On Caccioppoli's inequalities of Stokes equations and Navier-Stokes equations near boundary

Finite energy Navier-Stokes flows with unbounded gradients induced by localized flux in the half-space

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