Minimal periods of semilinear evolution equations with Lipschitz nonlinearity

James C. Robinson; Alejandro Vidal-López

doi:10.1016/j.jde.2005.04.009

Minimal periods of semilinear evolution equations with Lipschitz nonlinearity

James C. Robinson^*
, Alejandro Vidal-López

^*Corresponding author for this work

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

7 Citations (Scopus)

Abstract

It is known that any periodic orbit of a Lipschitz ordinary differential equation ẋ = f(x) must have period at least 2π/L, where L is the Lipschitz constant of f. In this paper, we prove a similar result for the semilinear evolution equation du/dt = -Au + f(u): for each α with 0 ≤ α ≤ 1/2 there exists a constant K_α such that if L is the Lipschitz constant of f as a map from D(A^α) into H then any periodic orbit has period at least K_αL^-1/(1-α). As a concrete application we recover a result of Kukavica giving a lower bound on the period for the 2d Navier-Stokes equations with periodic boundary conditions.

Original language	English
Pages (from-to)	396-406
Number of pages	11
Journal	Journal of Differential Equations
Volume	220
Issue number	2
DOIs	https://doi.org/10.1016/j.jde.2005.04.009
Publication status	Published - 15 Jan 2006
Externally published	Yes

Keywords

Minimal period
Navier-Stokes equations
Period orbits
Semilinear evolution equations

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10.1016/j.jde.2005.04.009

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title = "Minimal periods of semilinear evolution equations with Lipschitz nonlinearity",

abstract = "It is known that any periodic orbit of a Lipschitz ordinary differential equation ẋ = f(x) must have period at least 2π/L, where L is the Lipschitz constant of f. In this paper, we prove a similar result for the semilinear evolution equation du/dt = -Au + f(u): for each α with 0 ≤ α ≤ 1/2 there exists a constant Kα such that if L is the Lipschitz constant of f as a map from D(Aα) into H then any periodic orbit has period at least KαL-1/(1-α). As a concrete application we recover a result of Kukavica giving a lower bound on the period for the 2d Navier-Stokes equations with periodic boundary conditions.",

keywords = "Minimal period, Navier-Stokes equations, Period orbits, Semilinear evolution equations",

author = "Robinson, \{James C.\} and Alejandro Vidal-L{\'o}pez",

note = "Funding Information: JCR is a Royal Society University Research Fellow and would like to thank the Society for all their support, and Igor Kukavica for his helpful emails concerning estimates on |Au| for use in Section 4. AVL has been partially supported by the Grant FPI 2000-6835 associated with Proyecto BFM2000-0798 and Proyecto BFM2003-03810 from the DGES, and a Marie Curie visiting studentship.",

year = "2006",

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doi = "10.1016/j.jde.2005.04.009",

language = "English",

volume = "220",

pages = "396--406",

journal = "Journal of Differential Equations",

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T1 - Minimal periods of semilinear evolution equations with Lipschitz nonlinearity

AU - Robinson, James C.

AU - Vidal-López, Alejandro

N1 - Funding Information: JCR is a Royal Society University Research Fellow and would like to thank the Society for all their support, and Igor Kukavica for his helpful emails concerning estimates on |Au| for use in Section 4. AVL has been partially supported by the Grant FPI 2000-6835 associated with Proyecto BFM2000-0798 and Proyecto BFM2003-03810 from the DGES, and a Marie Curie visiting studentship.

PY - 2006/1/15

Y1 - 2006/1/15

N2 - It is known that any periodic orbit of a Lipschitz ordinary differential equation ẋ = f(x) must have period at least 2π/L, where L is the Lipschitz constant of f. In this paper, we prove a similar result for the semilinear evolution equation du/dt = -Au + f(u): for each α with 0 ≤ α ≤ 1/2 there exists a constant Kα such that if L is the Lipschitz constant of f as a map from D(Aα) into H then any periodic orbit has period at least KαL-1/(1-α). As a concrete application we recover a result of Kukavica giving a lower bound on the period for the 2d Navier-Stokes equations with periodic boundary conditions.

AB - It is known that any periodic orbit of a Lipschitz ordinary differential equation ẋ = f(x) must have period at least 2π/L, where L is the Lipschitz constant of f. In this paper, we prove a similar result for the semilinear evolution equation du/dt = -Au + f(u): for each α with 0 ≤ α ≤ 1/2 there exists a constant Kα such that if L is the Lipschitz constant of f as a map from D(Aα) into H then any periodic orbit has period at least KαL-1/(1-α). As a concrete application we recover a result of Kukavica giving a lower bound on the period for the 2d Navier-Stokes equations with periodic boundary conditions.

KW - Minimal period

KW - Navier-Stokes equations

KW - Period orbits

KW - Semilinear evolution equations

UR - https://www.scopus.com/pages/publications/29144484760

U2 - 10.1016/j.jde.2005.04.009

DO - 10.1016/j.jde.2005.04.009

M3 - Article

AN - SCOPUS:29144484760

SN - 0022-0396

VL - 220

SP - 396

EP - 406

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 2

ER -

Minimal periods of semilinear evolution equations with Lipschitz nonlinearity

Abstract

Keywords

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