On the De Gregorio Modification of the Constantin–Lax–Majda Model

H. Jia; S. Stewart; V. Sverak

doi:10.1007/s00205-018-1298-1

On the De Gregorio Modification of the Constantin–Lax–Majda Model

H. Jia, S. Stewart, V. Sverak

School of Mathematics

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17 Scopus citations

Abstract

We study a modification due to De Gregorio of the Constantin–Lax–Majda (CLM) model ω_t = ωHω on the unit circle. The De Gregorio equation is ω_t+ uω_x- u_xω= 0 , u_x= Hω. In contrast with the CLM model, numerical simulations suggest that the solutions of the De Gregorio model with smooth initial data exist globally for all time, and generically converge to equilibria when t→ ± ∞, in a way resembling inviscid damping. We prove that such behavior takes place near a manifold of equilibria.

Original language	English (US)
Pages (from-to)	1269-1304
Number of pages	36
Journal	Archive For Rational Mechanics And Analysis
Volume	231
Issue number	2
DOIs	https://doi.org/10.1007/s00205-018-1298-1
State	Published - Feb 7 2019

Bibliographical note

Funding Information:
Acknowledgements. The authors are grateful for helpful discussions with ANGEL CASTRO , TAREK ELGINDI , ALEX KISELEV , and STEVE PRESTON . The research of VS was supported in part by Grants DMS 1362467 and DMS 1664297 from the National Science Foundation. HJ was supported in part by DMS-1600779 and he is also grateful to IAS where part of the work was carried out. The research of SS has been supported in part by a NDSEG fellowship. The authors thank the anonymous referee for valuable comments.

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© 2018, Springer-Verlag GmbH Germany, part of Springer Nature.

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title = "On the De Gregorio Modification of the Constantin–Lax–Majda Model",

abstract = "We study a modification due to De Gregorio of the Constantin–Lax–Majda (CLM) model ωt = ωHω on the unit circle. The De Gregorio equation is ωt+ uωx- uxω= 0 , ux= Hω. In contrast with the CLM model, numerical simulations suggest that the solutions of the De Gregorio model with smooth initial data exist globally for all time, and generically converge to equilibria when t→ ± ∞, in a way resembling inviscid damping. We prove that such behavior takes place near a manifold of equilibria.",

author = "H. Jia and S. Stewart and V. Sverak",

note = "Funding Information: Acknowledgements. The authors are grateful for helpful discussions with ANGEL CASTRO , TAREK ELGINDI , ALEX KISELEV , and STEVE PRESTON . The research of VS was supported in part by Grants DMS 1362467 and DMS 1664297 from the National Science Foundation. HJ was supported in part by DMS-1600779 and he is also grateful to IAS where part of the work was carried out. The research of SS has been supported in part by a NDSEG fellowship. The authors thank the anonymous referee for valuable comments. Publisher Copyright: {\textcopyright} 2018, Springer-Verlag GmbH Germany, part of Springer Nature.",

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AU - Stewart, S.

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N2 - We study a modification due to De Gregorio of the Constantin–Lax–Majda (CLM) model ωt = ωHω on the unit circle. The De Gregorio equation is ωt+ uωx- uxω= 0 , ux= Hω. In contrast with the CLM model, numerical simulations suggest that the solutions of the De Gregorio model with smooth initial data exist globally for all time, and generically converge to equilibria when t→ ± ∞, in a way resembling inviscid damping. We prove that such behavior takes place near a manifold of equilibria.

AB - We study a modification due to De Gregorio of the Constantin–Lax–Majda (CLM) model ωt = ωHω on the unit circle. The De Gregorio equation is ωt+ uωx- uxω= 0 , ux= Hω. In contrast with the CLM model, numerical simulations suggest that the solutions of the De Gregorio model with smooth initial data exist globally for all time, and generically converge to equilibria when t→ ± ∞, in a way resembling inviscid damping. We prove that such behavior takes place near a manifold of equilibria.

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On the De Gregorio Modification of the Constantin–Lax–Majda Model

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