Estimates for the Navier–Stokes equations in the half-space for nonlocalized data

Maekawa, Yasunori; Miura, Hideyuki; Prange, Christophe

doi:10.2140/apde.2020.13.945

Cited by 16 publications

(39 citation statements)

References 35 publications

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“…Indeed, the theorem is quite general, as it requires only local information near a proposed singularity. Additionally, Theorem 1.1 improves on certain norm concentration results in [37,34,35]. For instance, Neustupa obtained in [37] that at a singular point (x * , T * ),…”

Section: Introductionmentioning

confidence: 62%

“…However, a careful inspection of the proof shows that the behavior near x 3 = 0 is not important. In fact, a requirement of the form ∇ ′ P → 0 as x 3 → ∞ is enough to rule out parasitic solutions, see [35,Theorem 5].…”

Section: 99)mentioning

confidence: 99%

“…for all R > 0 and an absolute constant ε > 0. In [34,35], it is demonstrated that one may take R ∼ √ T * − t in (1.7) in exchange for writing sup x * ∈R 3 . It would be interesting to obtain R ∼ √ 1 − t in (1.6), but it is not clear to the authors whether the result remains valid.…”

Section: Introductionmentioning

confidence: 99%

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Localised necessary conditions for singularity formation in the Navier-Stokes equations with curved boundary

Albritton¹,

Barker²

2018

Preprint

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We prove two results concerning local regularity theory of the Navier-Stokes equations near a curved portion Γ ⊂ ∂ Ω of the boundary. Suppose that u is a boundary suitable weak solution with singularity (x * , T * ), where x * ∈ Ω ∪ Γ. Then (under a weak background assumption) the L 3 norm of u tends to infinity in every ball centered at x * :Furthermore, u gives rise to a non-trivial mild bounded ancient solution in R 3 or R 3 + through a rescaling procedure that zooms in on the singularity. This generalizes results of Seregin et al. into the local setting with curved boundary. Our proof relies on a truncation procedure for boundary suitable weak solutions. The former result is based on energy estimates for L 3 initial data and a Liouville theorem. For the latter result, we apply perturbation theory for L ∞ initial data due to K. Abe and Y. Giga. The curved boundary is treated in two different ways.

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Section: Introductionmentioning

confidence: 62%

Section: 99)mentioning

confidence: 99%

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Localised necessary conditions for singularity formation in the Navier-Stokes equations with curved boundary

Albritton¹,

Barker²

2018

Preprint

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“…This result was improved by Maekawa, Miura and Prange. They [25] proved that for every t ∈ (0, T ) there esists x(t) ∈ R 3 such that…”

Section: Introductionmentioning

confidence: 99%

The localized characterization for the singularity formation in the Navier-Stokes equations

Tan

2021

Preprint

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This paper is concerned with the localized behaviors of the solution u to the Navier-Stokes equations near the potential singular points. We establish the concentration rate for the L p,∞ norm of u with 3 ≤ p ≤ ∞. Namely, we show that if z 0 = (t 0 , x 0 ) is a singular point, then for any r > 0, it holds lim sup t→t − 0 where δ * is a positive constant independent of p and ν. Our main tools are some εregularity criteria in L p,∞ spaces and an embedding theorem from L p,∞ space into a Morrey type space. These are of independent interests.

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“…Later, Maekawa-Miura-Prange in [28] improved (1.7) and showed that for every t ∈ (0, T * ) (not just a sequence t n → T * ) there exists x(t) ∈ R 3 such that…”

mentioning

confidence: 99%

The local characterizations of the singularity formation for the MHD equations

Tan¹,

Wang²

2021

Preprint

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This paper characterizes the possible blow-up of solutions for the 3D magneto-hydrodynamics (MHD for short) equations. We first establish some ǫ-regularity criteria in L q,∞ spaces for suitable weak solutions, and then together with an embedding theorem from L p,∞ space into a Morrey type space to characterize the local behaviors of solutions near a potential singular point. More precisely, we show that if z 0 = (t 0 , x 0 ) is a singular point, then for any r > 0 it holds that where δ * is a positive constant independent on ν and p.

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Estimates for the Navier–Stokes equations in the half-space for nonlocalized data

Cited by 16 publications

References 35 publications

Localised necessary conditions for singularity formation in the Navier-Stokes equations with curved boundary

Localised necessary conditions for singularity formation in the Navier-Stokes equations with curved boundary

The localized characterization for the singularity formation in the Navier-Stokes equations

The local characterizations of the singularity formation for the MHD equations

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