Localized smoothing and concentration for the Navier-Stokes equations in the half space

Dallas Albritton; Tobias Barker; Christophe Prange

doi:10.1016/j.jfa.2022.109729

Localized smoothing and concentration for the Navier-Stokes equations in the half space

Dallas Albritton, Tobias Barker, Christophe Prange

Department of Mathematical Sciences

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Abstract

We establish a local-in-space short-time smoothing effect for the Navier-Stokes equations in the half space. The whole space analogue, due to Jia and Šverák [21], is a central tool in two of the authors' recent work on quantitative L_x³ blow-up criteria [7]. The main difficulty is that the non-local effects of the pressure in the half space are much stronger than in the whole space. As an application, we demonstrate that the critical L_x³ norm must concentrate at scales ∼T^⁎−t in the presence of a Type I blow-up.

Original language	English
Article number	109729
Journal	Journal of Functional Analysis
Volume	284
Issue number	1
Early online date	4 Oct 2022
DOIs	https://doi.org/10.1016/j.jfa.2022.109729
Publication status	Published - 1 Jan 2023

Bibliographical note

Funding Information:
DA was supported by NSF Postdoctoral Fellowship Grant No. 2002023 and Simons Foundation Grant No. 816048 . DA is also grateful to ENS Paris for partially supporting his academic visit to Paris during which this research was initiated. CP is partially supported by the Agence Nationale de la Recherche , project BORDS, grant ANR-16-CE40-0027-01 , project SINGFLOWS, grant ANR-18-CE40-0027-01 , project CRISIS, grant ANR-20-CE40-0020-01 and by the CY Initiative of Excellence , project CYNA. Finally, we thank the referee for his or her valuable work.

Data Availability
No data was used for the research described in this article

Keywords

Epsilon regularity
Navier-Stokes equations

ASJC Scopus subject areas

Analysis

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10.1016/j.jfa.2022.109729

journalrevisionnocolor.PDFAccepted author manuscript, 567 KBLicence: CC BY-NC-ND

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title = "Localized smoothing and concentration for the Navier-Stokes equations in the half space",

abstract = "We establish a local-in-space short-time smoothing effect for the Navier-Stokes equations in the half space. The whole space analogue, due to Jia and {\v S}ver{\'a}k [21], is a central tool in two of the authors' recent work on quantitative Lx3 blow-up criteria [7]. The main difficulty is that the non-local effects of the pressure in the half space are much stronger than in the whole space. As an application, we demonstrate that the critical Lx3 norm must concentrate at scales ∼T⁎−t in the presence of a Type I blow-up.",

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note = "Funding Information: DA was supported by NSF Postdoctoral Fellowship Grant No. 2002023 and Simons Foundation Grant No. 816048 . DA is also grateful to ENS Paris for partially supporting his academic visit to Paris during which this research was initiated. CP is partially supported by the Agence Nationale de la Recherche , project BORDS, grant ANR-16-CE40-0027-01 , project SINGFLOWS, grant ANR-18-CE40-0027-01 , project CRISIS, grant ANR-20-CE40-0020-01 and by the CY Initiative of Excellence , project CYNA. Finally, we thank the referee for his or her valuable work. Data Availability No data was used for the research described in this article",

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AU - Barker, Tobias

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N1 - Funding Information: DA was supported by NSF Postdoctoral Fellowship Grant No. 2002023 and Simons Foundation Grant No. 816048 . DA is also grateful to ENS Paris for partially supporting his academic visit to Paris during which this research was initiated. CP is partially supported by the Agence Nationale de la Recherche , project BORDS, grant ANR-16-CE40-0027-01 , project SINGFLOWS, grant ANR-18-CE40-0027-01 , project CRISIS, grant ANR-20-CE40-0020-01 and by the CY Initiative of Excellence , project CYNA. Finally, we thank the referee for his or her valuable work. Data Availability No data was used for the research described in this article

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N2 - We establish a local-in-space short-time smoothing effect for the Navier-Stokes equations in the half space. The whole space analogue, due to Jia and Šverák [21], is a central tool in two of the authors' recent work on quantitative Lx3 blow-up criteria [7]. The main difficulty is that the non-local effects of the pressure in the half space are much stronger than in the whole space. As an application, we demonstrate that the critical Lx3 norm must concentrate at scales ∼T⁎−t in the presence of a Type I blow-up.

AB - We establish a local-in-space short-time smoothing effect for the Navier-Stokes equations in the half space. The whole space analogue, due to Jia and Šverák [21], is a central tool in two of the authors' recent work on quantitative Lx3 blow-up criteria [7]. The main difficulty is that the non-local effects of the pressure in the half space are much stronger than in the whole space. As an application, we demonstrate that the critical Lx3 norm must concentrate at scales ∼T⁎−t in the presence of a Type I blow-up.

KW - Epsilon regularity

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Localized smoothing and concentration for the Navier-Stokes equations in the half space

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

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