On singularities in the quaternionic Burgers equation

Vladimir Sverak

doi:10.1007/s40316-021-00175-5

On singularities in the quaternionic Burgers equation

Vladimir Sverak

School of Mathematics

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

1 Scopus citations

Abstract

We consider the equation q_t+ qq_x= q_xx for q: R× (0 , ∞) → H (the quaternions), and show that while singularities can develop from smooth compactly supported data, such situations are non-generic. The singularities will disappear under an arbitrary small “generic” smooth perturbation of the initial data. Similar results are also established for the same equation in S¹× (0 , ∞) , where S¹ is the standard one-dimensional circle.

Original language	English (US)
Pages (from-to)	41-54
Number of pages	14
Journal	Annales Mathematiques du Quebec
Volume	46
Issue number	1
DOIs	https://doi.org/10.1007/s40316-021-00175-5
State	Published - Apr 2022

Bibliographical note

Funding Information:
The research of the author was supported in part by grant DMS 1956092 from the National Science Foundation.

Publisher Copyright:
© 2021, Fondation Carl-Herz and Springer Nature Switzerland AG.

Keywords

Generic well-posedness
Quaternionic Burgers equation
Singularities

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10.1007/s40316-021-00175-5

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abstract = "We consider the equation qt+ qqx= qxx for q: R× (0 , ∞) → H (the quaternions), and show that while singularities can develop from smooth compactly supported data, such situations are non-generic. The singularities will disappear under an arbitrary small “generic” smooth perturbation of the initial data. Similar results are also established for the same equation in S1× (0 , ∞) , where S1 is the standard one-dimensional circle.",

keywords = "Generic well-posedness, Quaternionic Burgers equation, Singularities",

author = "Vladimir Sverak",

note = "Funding Information: The research of the author was supported in part by grant DMS 1956092 from the National Science Foundation. Publisher Copyright: {\textcopyright} 2021, Fondation Carl-Herz and Springer Nature Switzerland AG.",

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AU - Sverak, Vladimir

N1 - Funding Information: The research of the author was supported in part by grant DMS 1956092 from the National Science Foundation. Publisher Copyright: © 2021, Fondation Carl-Herz and Springer Nature Switzerland AG.

PY - 2022/4

Y1 - 2022/4

N2 - We consider the equation qt+ qqx= qxx for q: R× (0 , ∞) → H (the quaternions), and show that while singularities can develop from smooth compactly supported data, such situations are non-generic. The singularities will disappear under an arbitrary small “generic” smooth perturbation of the initial data. Similar results are also established for the same equation in S1× (0 , ∞) , where S1 is the standard one-dimensional circle.

AB - We consider the equation qt+ qqx= qxx for q: R× (0 , ∞) → H (the quaternions), and show that while singularities can develop from smooth compactly supported data, such situations are non-generic. The singularities will disappear under an arbitrary small “generic” smooth perturbation of the initial data. Similar results are also established for the same equation in S1× (0 , ∞) , where S1 is the standard one-dimensional circle.

KW - Generic well-posedness

KW - Quaternionic Burgers equation

KW - Singularities

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JO - Annales Mathematiques du Quebec

JF - Annales Mathematiques du Quebec

IS - 1

ER -

On singularities in the quaternionic Burgers equation

Abstract

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