Continuous alignment of vorticity direction prevents the blow-up of the Navier–Stokes flow under the no-slip boundary condition

Giga, Yoshikazu; Gu, Zhongyang; Hsu, Pen-Yuan

doi:10.1016/j.na.2019.111579

Cited by 7 publications

(7 citation statements)

References 26 publications

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“…Since u ∈ W 2,4 𝜎 (Ω), u|u| 𝜆 and ∇u|u| 𝜆 lie in L ∞ (Ω) ⊂ L r (Ω) for every r ∈ (1, ∞). The application of the Helmholtz decomposition yields the existence of exactly one Q which satisfies (19). Further, Q is a weak solution of system…”

Section: Auxiliary Resultsmentioning

confidence: 99%

“…𝜎 (Ω) and Q ∈ W 1, 2 (Ω) be functions which satisfy (19), u satisfies conditions (4) in the sense of traces and 𝜆 > 0. Then,…”

Section: Auxiliary Resultsmentioning

confidence: 99%

“…As of now, the power 1/2 is the best (smallest) which has been achieved, and its possible improvement (decrease) seems to be a hard open problem. For the exclusion of the blow‐ups of type I, the 1/2−Hölder continuity assumption can be replaced by a more general uniform continuity assumption, see Giga and Miura 18 and Giga and Sohr 19 …”

Section: Introductionmentioning

confidence: 99%

“…For the exclusion of the blow-ups of type I, the 1/2−Hölder continuity assumption can be replaced by a more general uniform continuity assumption, see Giga and Miura 18 and Giga and Sohr. 19 Vasseur 20 proved a remarkable criterion for Ω = R 3 in terms of the velocity direction. Unlike Constantin and Fefferman, he did not use the straightforward condition for the angles (between the vectors u(x, t) and u(y, t)), but expressed his result in the language of the Bochner spaces: Theorem 1.…”

Section: Introductionmentioning

confidence: 99%

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Regularity criteria for the Navier–Stokes equations in terms of the velocity direction and the flow of energy

Guo

Kučera

Skalák

2022

Math Methods in App Sciences

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We deal with the conditional regularity of the weak solutions to the Navier–Stokes equations. We discuss a famous criterion by Vasseur in terms of divfalse(bold-italicufalse/false|bold-italicufalse|false) and extend this criterion to bounded domains with Navier and Navier‐type boundary conditions. Inspired by the equality bold-italicu·∇false|bold-italicufalse|λ=−λfalse|bold-italicufalse|λ+1divfalse(bold-italicufalse/false|bold-italicufalse|false),0.1emλ≥1, we further prove an optimal regularity criterion in terms of bold-italicu·∇false|bold-italicufalse|λ both for the whole three‐dimensional space and bounded domains with Navier's, Navier‐type, and Dirichlet boundary conditions. It specially means for λ=2 that the control of the energy flow in the critical norms LtpLxr provides the regularity of solutions. This criterion is proved by two different techniques.

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Section: Auxiliary Resultsmentioning

confidence: 99%

“…𝜎 (Ω) and Q ∈ W 1, 2 (Ω) be functions which satisfy (19), u satisfies conditions (4) in the sense of traces and 𝜆 > 0. Then,…”

Section: Auxiliary Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Regularity criteria for the Navier–Stokes equations in terms of the velocity direction and the flow of energy

Guo

Kučera

Skalák

2022

Math Methods in App Sciences

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“…We then follow the similar idea that proves [15,Lemma 3.2]. Assume that |x | ≤ |y | and connect x and y by a geodesic curve in B |x | (0 ) c .…”

Section: Criterion For a Class Of Functions To Be Inmentioning

confidence: 97%

The Helmholtz Decomposition of a BMO Type Vector Field in a Slightly Perturbed Half Space

Giga

2023

J. Math. Fluid Mech.

Self Cite

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We introduce a space of $$L^2$$ L 2 vector fields with bounded mean oscillation whose “normal” component to the boundary is well-controlled. We establish its Helmholtz decomposition in the case when the domain is a perturbed $$C^3$$ C 3 half space in $${\textbf{R}}^n$$ R n $$(n \ge 3)$$ ( n ≥ 3 ) with small perturbation.

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Geometric regularity criteria for the Navier–Stokes equations in terms of velocity direction

Skalak

2023

Math Methods in App Sciences

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In this paper, we are inspired by a famous result by Constantin and Fefferman who proved that a simple geometrical assumption on the direction of the vorticity leads to the regularity of weak solutions of the 3D Navier–Stokes equations. We show that the same result can be achieved if the vorticity direction is replaced by the velocity direction. We further strengthen this result and prove that in fact it is not necessary to consider the velocity direction in all close space points but only in the points whose distance equals to a small positive number dependant on the data. In the second part of the paper, we extend a result by Berselli and C rdoba concerning the role of the helicity for the regularity of the weak solutions of the Navier–Stokes equations.

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Continuous alignment of vorticity direction prevents the blow-up of the Navier–Stokes flow under the no-slip boundary condition

Cited by 7 publications

References 26 publications

Regularity criteria for the Navier–Stokes equations in terms of the velocity direction and the flow of energy

Regularity criteria for the Navier–Stokes equations in terms of the velocity direction and the flow of energy

The Helmholtz Decomposition of a BMO Type Vector Field in a Slightly Perturbed Half Space

Geometric regularity criteria for the Navier–Stokes equations in terms of velocity direction

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