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

Chang, Tongkeun; Kang, Kyungkeun

doi:10.1016/j.jde.2020.05.013

Cited by 8 publications

(12 citation statements)

References 15 publications

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“…Following similar constructions as in [13], even further, the authors constructed an example that u is integrable in L 4 t L p x but ∇u is not square integrable (see [7]). More precisely, there exists a very weak solution of the Stokes equations (1.1) or the Navier-Stokes equations (1.3) with (1.2) such that…”

Section: Introductionmentioning

confidence: 99%

“…With the aid of this construction, the authors also proved that Caccioppoli's inequalities of Stokes equations and Navier-Stokes equations in general may fails near boundary when only local boundary problems are considered (see [7,Theorem 1.1]). It is a very important distinction in comparison to the interior case, where Caccioppoli's (type) inequalities turn out to be true (compare to [11] and [26], and refer also [12] for generalized Navier-Stokes flow).…”

Section: Introductionmentioning

confidence: 99%

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Local regularity near boundary for the Stokes and Navier-Stokes equations

Chang¹,

Kang²

2021

Preprint

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We are concerned with local regularity of the solutions for the Stokes and Navier-Stokes equations near boundary. Firstly, we construct a bounded solution but its normal derivatives are singular in any L p with 1 < p locally near boundary. On the other hand, we present criteria of solutions of the Stokes equations near boundary to imply that the gradients of solutions are bounded (in fact, even further Hölder continuous). Finally, we provide examples of solutions whose local regularity near boundary is optimal.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Local regularity near boundary for the Stokes and Navier-Stokes equations

Chang¹,

Kang²

2021

Preprint

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“…Due to the multiplication by the cut-off function, the test function is not divergence-free anymore so that the pressure appears in the inequalities and has to be treated. Recently, Chang and Kang [3] proved that it is impossible to establish a parabolic Caccioppoli inequality for the Stokes system on the half-space that has the same form as its elliptic counterpart. Up to now, there has not been a satisfactory way of how to handle this pressure term and the purpose of this work is provide an argument for the treatment of the pressure.…”

Section: Introductionmentioning

confidence: 99%

A non-local approach to the generalized Stokes operator with bounded measurable coefficients

Tolksdorf¹

2020

Preprint

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We establish functional analytic properties of the Stokes operator with bounded measurable coefficients on L p σ (R d ), d ≥ 2, for |1/p−1/2| < 1/d. These include optimal resolvent bounds and the property of maximal L q -regularity. We further give regularity estimates on the gradient of the solution to the Stokes resolvent problem with bounded measurable coefficients. As a key to these results we establish the validity of a non-local Caccioppoli inequality to solutions of the Stokes resolvent problem.

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“…A more singular (very weak) solution of the Stokes system, also generated by boundary flux, was constructed in Chang and the first author [4,Proposition 3.2] for n ≥ 2 so that ∇u L 2 t,x (B + 1 ×(0,1)) = ∞, although u L 4 t L p…”

mentioning

confidence: 99%

“…x (R n + ×(0,1)) is bounded for all p > n/(n − 1), with compactly supported boundary data belonging to L 4 t L ∞ x (Σ × (0, 1)). It is used as the profile to construct non-smooth solutions near the boundary of the Navier-Stokes equations in [4,Section 4.2]. This construction is possible because it only involves L q -type estimates.…”

mentioning

confidence: 99%

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

Kang¹,

Lai²,

Lai³

et al. 2021

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

Cited by 8 publications

References 15 publications

Local regularity near boundary for the Stokes and Navier-Stokes equations

Local regularity near boundary for the Stokes and Navier-Stokes equations

A non-local approach to the generalized Stokes operator with bounded measurable coefficients

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

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