Analyticity and large time behavior for the Burgers equation and the quasi-geostrophic equation, the both with the critical dissipation

Tsukasa Iwabuchi

doi:10.1016/j.anihpc.2020.02.003

Analyticity and large time behavior for the Burgers equation and the quasi-geostrophic equation, the both with the critical dissipation

Tsukasa Iwabuchi

SCI - Mathematics

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

10 Citations (Scopus)

Abstract

This paper is concerned with the Cauchy problem of the Burgers equation with the critical dissipation. The well-posedness and analyticity in both of the space and the time variables are studied based on the frequency decomposition method. The large time behavior is revealed for any large initial data. As a result, it is shown that any smooth and integrable solution is analytic in space and time as long as time is positive and behaves like the Poisson kernel as time tends to infinity. The corresponding results are also obtained for the quasi-geostrophic equation.

Original language	English
Pages (from-to)	855-876
Number of pages	22
Journal	Annales de l'Institut Henri Poincare (C) Analyse Non Lineaire
Volume	37
Issue number	4
DOIs	https://doi.org/10.1016/j.anihpc.2020.02.003
Publication status	Published - 2020 Jul 1

Keywords

Analyticity in space and time
Burgers equation
Critical dissipation
Large data
Large time behavior
Quasi-geostrophic equation

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10.1016/j.anihpc.2020.02.003

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title = "Analyticity and large time behavior for the Burgers equation and the quasi-geostrophic equation, the both with the critical dissipation",

abstract = "This paper is concerned with the Cauchy problem of the Burgers equation with the critical dissipation. The well-posedness and analyticity in both of the space and the time variables are studied based on the frequency decomposition method. The large time behavior is revealed for any large initial data. As a result, it is shown that any smooth and integrable solution is analytic in space and time as long as time is positive and behaves like the Poisson kernel as time tends to infinity. The corresponding results are also obtained for the quasi-geostrophic equation.",

keywords = "Analyticity in space and time, Burgers equation, Critical dissipation, Large data, Large time behavior, Quasi-geostrophic equation",

author = "Tsukasa Iwabuchi",

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doi = "10.1016/j.anihpc.2020.02.003",

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N1 - Funding Information: The author would like to thank the referees for their thoughtful comments. The author was supported by the Grant-in-Aid for Young Scientists (A) (No. 17H04824 ) from JSPS and by Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers from JSPS . Publisher Copyright: © 2020 The Author

PY - 2020/7/1

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N2 - This paper is concerned with the Cauchy problem of the Burgers equation with the critical dissipation. The well-posedness and analyticity in both of the space and the time variables are studied based on the frequency decomposition method. The large time behavior is revealed for any large initial data. As a result, it is shown that any smooth and integrable solution is analytic in space and time as long as time is positive and behaves like the Poisson kernel as time tends to infinity. The corresponding results are also obtained for the quasi-geostrophic equation.

AB - This paper is concerned with the Cauchy problem of the Burgers equation with the critical dissipation. The well-posedness and analyticity in both of the space and the time variables are studied based on the frequency decomposition method. The large time behavior is revealed for any large initial data. As a result, it is shown that any smooth and integrable solution is analytic in space and time as long as time is positive and behaves like the Poisson kernel as time tends to infinity. The corresponding results are also obtained for the quasi-geostrophic equation.

KW - Analyticity in space and time

KW - Burgers equation

KW - Critical dissipation

KW - Large data

KW - Large time behavior

KW - Quasi-geostrophic equation

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Analyticity and large time behavior for the Burgers equation and the quasi-geostrophic equation, the both with the critical dissipation

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Keywords

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