Generalized singular integral with rough kernel and approximation of surface quasi-geostrophic equation

Yanping Chen; Zihua Guo

doi:10.1016/j.jde.2022.11.035

Generalized singular integral with rough kernel and approximation of surface quasi-geostrophic equation

Yanping Chen, Zihua Guo

School of Mathematics

Research output: Contribution to journal › Article › Research › peer-review

12 Citations (Scopus)

Abstract

This paper is concerned with the generalized singular integral operator with rough kernel and the approximation problem for the generalized surface quasi-geostrophic equation. For the generalized singular integral operator, we obtain uniform L^p−L^q estimates with respect to a parameter β. From this one can cover the L^p-boundedness of the Calderón-Zygmund operator with rough kernel by letting β→0. We applied this estimate to study the Cauchy problem of the generalized surface quasi-geostrophic (SQG) equation. Local well-posedness in the Besov space B_p,q^s and some limit behaviour of the solutions are obtained. Our results improve the previous ones by Yu-Zheng-Jiu in 2019 and by Yu-Jiu-Li in 2021.

Original language	English
Pages (from-to)	205-228
Number of pages	24
Journal	Journal of Differential Equations
Volume	346
DOIs	https://doi.org/10.1016/j.jde.2022.11.035
Publication status	Published - 15 Feb 2023

Keywords

Approximation
Rough kernel
Singular integral
Surface quasi-geostrophic equation(SQG)

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10.1016/j.jde.2022.11.035

Cite this

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title = "Generalized singular integral with rough kernel and approximation of surface quasi-geostrophic equation",

abstract = "This paper is concerned with the generalized singular integral operator with rough kernel and the approximation problem for the generalized surface quasi-geostrophic equation. For the generalized singular integral operator, we obtain uniform Lp−Lq estimates with respect to a parameter β. From this one can cover the Lp-boundedness of the Calder{\'o}n-Zygmund operator with rough kernel by letting β→0. We applied this estimate to study the Cauchy problem of the generalized surface quasi-geostrophic (SQG) equation. Local well-posedness in the Besov space Bp,qs and some limit behaviour of the solutions are obtained. Our results improve the previous ones by Yu-Zheng-Jiu in 2019 and by Yu-Jiu-Li in 2021.",

keywords = "Approximation, Rough kernel, Singular integral, Surface quasi-geostrophic equation(SQG)",

author = "Yanping Chen and Zihua Guo",

note = "Funding Information: Y. Chen was partially supported by the National Natural Science Foundation of China (No. 11871096 ); Z. Guo was partially supported by ARC DP200101065 . Publisher Copyright: {\textcopyright} 2022 Elsevier Inc.",

year = "2023",

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day = "15",

doi = "10.1016/j.jde.2022.11.035",

language = "English",

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journal = "Journal of Differential Equations",

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TY - JOUR

T1 - Generalized singular integral with rough kernel and approximation of surface quasi-geostrophic equation

AU - Chen, Yanping

AU - Guo, Zihua

N1 - Funding Information: Y. Chen was partially supported by the National Natural Science Foundation of China (No. 11871096 ); Z. Guo was partially supported by ARC DP200101065 . Publisher Copyright: © 2022 Elsevier Inc.

PY - 2023/2/15

Y1 - 2023/2/15

N2 - This paper is concerned with the generalized singular integral operator with rough kernel and the approximation problem for the generalized surface quasi-geostrophic equation. For the generalized singular integral operator, we obtain uniform Lp−Lq estimates with respect to a parameter β. From this one can cover the Lp-boundedness of the Calderón-Zygmund operator with rough kernel by letting β→0. We applied this estimate to study the Cauchy problem of the generalized surface quasi-geostrophic (SQG) equation. Local well-posedness in the Besov space Bp,qs and some limit behaviour of the solutions are obtained. Our results improve the previous ones by Yu-Zheng-Jiu in 2019 and by Yu-Jiu-Li in 2021.

AB - This paper is concerned with the generalized singular integral operator with rough kernel and the approximation problem for the generalized surface quasi-geostrophic equation. For the generalized singular integral operator, we obtain uniform Lp−Lq estimates with respect to a parameter β. From this one can cover the Lp-boundedness of the Calderón-Zygmund operator with rough kernel by letting β→0. We applied this estimate to study the Cauchy problem of the generalized surface quasi-geostrophic (SQG) equation. Local well-posedness in the Besov space Bp,qs and some limit behaviour of the solutions are obtained. Our results improve the previous ones by Yu-Zheng-Jiu in 2019 and by Yu-Jiu-Li in 2021.

KW - Approximation

KW - Rough kernel

KW - Singular integral

KW - Surface quasi-geostrophic equation(SQG)

UR - https://www.scopus.com/pages/publications/85142676327

U2 - 10.1016/j.jde.2022.11.035

DO - 10.1016/j.jde.2022.11.035

M3 - Article

AN - SCOPUS:85142676327

SN - 0022-0396

VL - 346

SP - 205

EP - 228

JO - Journal of Differential Equations

JF - Journal of Differential Equations

ER -

Generalized singular integral with rough kernel and approximation of surface quasi-geostrophic equation

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Keywords

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Nonlinear harmonic analysis and dispersive partial differential equations

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Generalized singular integral with rough kernel and approximation of surface quasi-geostrophic equation

Abstract

Keywords

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Nonlinear harmonic analysis and dispersive partial differential equations

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