Short time regularity of Navier-Stokes flows with locally $L^3$ initial data and applications

Kang, Kyungkeun; Miura, Hideyuki; Tsai, Tai‐Peng

doi:10.48550/arxiv.1812.10509

Cited by 4 publications

(15 citation statements)

References 19 publications

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“…The class M 2,1 C excludes DSS data and leaves DSS solutions as the borderline candidate for the failure of eventual regularity even for our non-uniform version. Note that the shape of the regular set is consistent with that obtained for DSS solutions in [15]. In contrast, all self-similar solutions in this class are regular by [12].…”

Section: Cnsupporting

confidence: 81%

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Existence of global weak solutions to the Navier-Stokes equations in weighted spaces

Bradshaw,

Kukavica,

Tsai

2019

Preprint

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We obtain a global existence result for the three-dimensional Navier-Stokes equations with a large class of data allowing growth at spatial infinity. Namely, we show the global existence of suitable weak solutions when the initial data belongs to the weighted space M 2,2. This class is strictly larger than currently available spaces of initial data for global existence and includes all locally square integrable discretely selfsimilar data. We also identify a sub-class of data for which solutions exhibit eventual regularity on a parabolic set in space-time.

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Section: Cnsupporting

confidence: 81%

“…Local in time existence of mild solutions for large data in some subcritical weak Herz spaces (which include the Herz spaces) has been established by Tsutsui [31] (p > 3 is required). These spaces also appear in [15] The paper is organized as follows. In Section 2 we reformulate the assumption…”

Section: Cnmentioning

confidence: 99%

Existence of global weak solutions to the Navier-Stokes equations in weighted spaces

Bradshaw,

Kukavica,

Tsai

2019

Preprint

Self Cite

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“…This condition implies a pressure representation like (1.2) is valid (this is mentioned in [21] and explicitly proven in [38,23]). If the data is only in L 2 uloc , the above decay condition is unavailable and, therefore, we must build the pressure formula into the definition.…”

Section: Introductionmentioning

confidence: 81%

“…A self-similar solution, on the other hand, would have at least a 1 dimensional (in space-time) singular set which violates conditions in [11]. Smoothness has recently been established in [23] when u 0 ∈ L 3,∞ is λ-DSS and λ is close to 1. Our next result establishes smoothness for discretely self-similar solutions evolving from small initial data in L 2 uloc .…”

Section: Letmentioning

confidence: 99%

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Global existence, regularity, and uniqueness of infinite energy solutions to the Navier-Stokes equations

Bradshaw¹,

Tsai²

2019

Preprint

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This paper addresses several problems associated to local energy solutions (in the sense of Lemarié-Rieusset) to the Navier-Stokes equations with initial data which is sufficiently small at large or small scales as measured using truncated Morrey-type quantities, namely: (1) global existence for a class of data including the critical L 2based Morrey space; (2) initial and eventual regularity of local energy solutions to the Navier-Stokes equations with initial data sufficiently small at small or large scales; (3) small-large uniqueness of local energy solutions for data in the critical L 2 -based Morrey space. A number of interesting corollaries are included, including eventual regularity in familiar Lebesgue, Lorentz, and Morrey spaces, a new local generalized Von Wahl uniqueness criteria, as well as regularity and uniqueness for local energy solutions with small discretely self-similar data.1. If u 0 ∈ M 2,r where 0 ≤ r < 1, then u has eventual regularity and t 1/2 u(•, t) L ∞ is bounded for sufficiently large t. If u 0 ∈ M 2,r and 1 < r ≤ 3, then u has initial regularity and t 1/2 u(•, t) L ∞ is bounded for sufficiently small t. If u, then u has initial and eventual regularity and t 1/2 u(•, t) L ∞ is bounded for sufficiently small and large t. In particular, this is true if u 0 ∈ L 3,q for 1 ≤ q < ∞.

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Localized Smoothing for the Navier–Stokes Equations and Concentration of Critical Norms Near Singularities

Barker

Prange

2020

Arch Rational Mech Anal

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This paper is concerned with two dual aspects of the regularity question of the Navier-Stokes equations. First, we prove a local in time localized smoothing effect for local energy solutions. More precisely, if the initial data restricted to the unit ball belongs to the scale-critical space L 3 , then the solution is locally smooth in space for some short time, which is quantified. This builds upon the work of Jia andŠverák, who considered the subcritical case. Second, we apply these localized smoothing estimates to prove a concentration phenomenon near a possible Type I blow-up. Namely, we show if (0, T * ) is a singular point thenThis result is inspired by and improves concentration results established by Li, Ozawa, and Wang and Maekawa, Miura, and Prange. We also extend our results to other critical spaces, namely L 3,∞ and the Besov spaceḂ −1+ 3 p p,∞ , p ∈ (3, ∞).

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Short time regularity of Navier-Stokes flows with locally $L^3$ initial data and applications

Cited by 4 publications

References 19 publications

Existence of global weak solutions to the Navier-Stokes equations in weighted spaces

Existence of global weak solutions to the Navier-Stokes equations in weighted spaces

Global existence, regularity, and uniqueness of infinite energy solutions to the Navier-Stokes equations

Localized Smoothing for the Navier–Stokes Equations and Concentration of Critical Norms Near Singularities

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