Extension criterion via partial components of vorticity on strong solutions to the Navier–Stokes equations in higher dimensions

Tsurumi, Hiroyuki

doi:10.1016/j.jde.2017.05.016

Cited by 3 publications

(6 citation statements)

References 9 publications

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“…This result indicates that, among the

\frac{n false(n - 1 false)}{2}

components of the vorticity,

([], \frac{n}{2})

components are negligible for the criterion whether the time local solutions can be extended beyond the critical time, which can be viewed as the generalization to the higher dimensional case of Chae and Choe . On the other hand, the author suggested difficulties in relaxing to

{\overset{B}{true}}_{\infty, \infty}^{0}

like in the work of Zhang and Chen (see remark 2.2 (ii), p. 4012 in the work of Tsurumi). The purpose of this paper is to give an affirmative answer to this problem.…”

Section: Introductionmentioning

confidence: 90%

“…Consequently, one may verify that X ( t ) is a monotonically nondecreasing function. Our next target is to show that, under the assumption , it holds true

\lim_{t \to T^{-}} X (t) \leq C (u_{0}, T, X (T_{0})) < \infty .

Thanks to ▽· u = 0 and the definition of ω , we have

u_{k} = - {(- Δ)}^{- 1} \sum_{i = 1}^{n} \partial_{i} ω_{i k},

which along with yields

{‖ \tilde{u} ‖}_{L^{2}} \leq C {‖ u ‖}_{L^{2}}, {‖ {normalΛ}^{s} \tilde{u} ‖}_{L^{2}} \leq C {‖ {normalΛ}^{s} u ‖}_{L^{2}} .

Noticing the following fact (see (3.6) in the work of Tsurumi)

\partial_{j} {\tilde{u}}_{k} = - {scriptR}_{j} \sum_{i = 1}^{n} {scriptR}_{i} {\tilde{ω}}_{i k},

where

R_{l} = \partial_{l} {false(- normalΔ false)}^{- \frac{1}{2}}

, l = 1,2,…, n denote the classical Riesz transforms, we thus have

{‖ \nabla \tilde{u} ‖}_{\dot{B}}

…”

Section: The Proof Of Theoremmentioning

confidence: 97%

“…More precisely, they showed that, if

\overset{ω}{true}

satisfies

\tilde{ω} \in L^{1} (0, T; {\dot{B}}_{\infty, \infty}^{0} ({double-struckR}^{3})),

then the solution u is regular on [0, T ]. In this direction, Tsurumi considered the higher spatial dimensions case. In order to state the result of the work of Tsurumi, we need some necessary notations.…”

Section: Introductionmentioning

confidence: 99%

“…In this direction, Tsurumi considered the higher spatial dimensions case. In order to state the result of the work of Tsurumi, we need some necessary notations. We first take a set of natural number pairs

normalΓ = {({}, ((), j false(l false), k false(l false)))}_{l = 1}^{([], \frac{n}{2})} \subset {false{1, 2, …, n false}}^{2}

satisfying

j false(l false) < k false(l false), l = 1, 2, …, ([], \frac{n}{2})

and

({}, ((), j false(l false), k false(l false))) \cap ({}, ((), j false(m false), k false(m false))) = \emptyset, l \neq m, 1 \leq l, m \leq ([], \frac{n}{2}),

where [ a ] denotes the largest integer less than or equal to a .…”

Section: Introductionmentioning

confidence: 99%

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Remark on the regularity criteria via partial components of vorticity to the Navier‐Stokes equations

2018

Math Methods in App Sciences

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In this paper, we establish several improved regularity criteria via partial components of vorticity to the incompressible Navier-Stokes equations in higher dimensions. Our result is a further improvement of the recent result given by Tsurumi [extension criterion via partial components of vorticity on strong solutions to the Navier-Stokes equations in higher dimensions, J. Differential Equations 263 (2017), 4007-4022].

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“…This result indicates that, among the

\frac{n false(n - 1 false)}{2}

components of the vorticity,

([], \frac{n}{2})

{\overset{B}{true}}_{\infty, \infty}^{0}

like in the work of Zhang and Chen (see remark 2.2 (ii), p. 4012 in the work of Tsurumi). The purpose of this paper is to give an affirmative answer to this problem.…”

Section: Introductionmentioning

confidence: 90%

“…Consequently, one may verify that X ( t ) is a monotonically nondecreasing function. Our next target is to show that, under the assumption , it holds true

\lim_{t \to T^{-}} X (t) \leq C (u_{0}, T, X (T_{0})) < \infty .

Thanks to ▽· u = 0 and the definition of ω , we have

u_{k} = - {(- Δ)}^{- 1} \sum_{i = 1}^{n} \partial_{i} ω_{i k},

which along with yields

{‖ \tilde{u} ‖}_{L^{2}} \leq C {‖ u ‖}_{L^{2}}, {‖ {normalΛ}^{s} \tilde{u} ‖}_{L^{2}} \leq C {‖ {normalΛ}^{s} u ‖}_{L^{2}} .

Noticing the following fact (see (3.6) in the work of Tsurumi)

\partial_{j} {\tilde{u}}_{k} = - {scriptR}_{j} \sum_{i = 1}^{n} {scriptR}_{i} {\tilde{ω}}_{i k},

where

R_{l} = \partial_{l} {false(- normalΔ false)}^{- \frac{1}{2}}

, l = 1,2,…, n denote the classical Riesz transforms, we thus have

{‖ \nabla \tilde{u} ‖}_{\dot{B}}

…”

Section: The Proof Of Theoremmentioning

confidence: 97%

“…More precisely, they showed that, if

\overset{ω}{true}

satisfies

\tilde{ω} \in L^{1} (0, T; {\dot{B}}_{\infty, \infty}^{0} ({double-struckR}^{3})),

Section: Introductionmentioning

confidence: 99%

normalΓ = {({}, ((), j false(l false), k false(l false)))}_{l = 1}^{([], \frac{n}{2})} \subset {false{1, 2, …, n false}}^{2}

satisfying

j false(l false) < k false(l false), l = 1, 2, …, ([], \frac{n}{2})

and

({}, ((), j false(l false), k false(l false))) \cap ({}, ((), j false(m false), k false(m false))) = \emptyset, l \neq m, 1 \leq l, m \leq ([], \frac{n}{2}),

where [ a ] denotes the largest integer less than or equal to a .…”

Section: Introductionmentioning

confidence: 99%

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Remark on the regularity criteria via partial components of vorticity to the Navier‐Stokes equations

2018

Math Methods in App Sciences

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“…in [15], and provided two components of vorticity ω = (𝜔 1 , 𝜔 2 , 0) satisfy condition (1.2) in [16], and ω ∈ L 1 (0, T; BMO) in [17]. For more advances, we refer to [18][19][20][21][22][23][24]. Very recently, for system (1.1) with p ≠ 2, there have been studied the regularity criteria in terms of velocity or direction of vorticity in [8,[25][26][27][28].…”

Section: Introductionmentioning

confidence: 99%

Regularity criteria for 3D shear‐thinning fluids in terms of two components of vorticity

Sin

Pak²,

Baranovskii

2023

Math Methods in App Sciences

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Components reduction regularity results for the Navier–Stokes equations in general dimensions

Zhang

Yao

2019

Journal of Mathematical Analysis and Applications

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Extension criterion via partial components of vorticity on strong solutions to the Navier–Stokes equations in higher dimensions

Cited by 3 publications

References 9 publications

Remark on the regularity criteria via partial components of vorticity to the Navier‐Stokes equations

Remark on the regularity criteria via partial components of vorticity to the Navier‐Stokes equations

Regularity criteria for 3D shear‐thinning fluids in terms of two components of vorticity

Components reduction regularity results for the Navier–Stokes equations in general dimensions

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