### Abstract

As a generalization of a vector field on a manifold, the notion of an arc field on a locally complete metric space was introduced in Bleecker and Calcaterra (J Math Anal Appl, 248: 645-677, 2000). In that paper, the authors proved an analogue of the Cauchy-Lipschitz Theorem, i. e they showed the existence and uniqueness of solution curves for a time independent arc field. In this paper, we extend the result to the time dependent case, namely we show the existence and uniqueness of solution curves for a time dependent arc field. We also introduce the notion of the sum of two time dependent arc fields and show existence and uniqueness of solution curves for this sum.

Original language | English (US) |
---|---|

Pages (from-to) | 231-256 |

Number of pages | 26 |

Journal | Journal of Dynamics and Differential Equations |

Volume | 25 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2013 |

### Fingerprint

### Keywords

- Arc fields
- Cauchy-Lipschitz Theorem
- Metric spaces
- Sum of arc fields

### ASJC Scopus subject areas

- Analysis

### Cite this

**Generating and Adding Flows on Locally Complete Metric Spaces.** / Kim, Hwa Kil; Masmoudi, Nader.

Research output: Contribution to journal › Article

*Journal of Dynamics and Differential Equations*, vol. 25, no. 1, pp. 231-256. https://doi.org/10.1007/s10884-012-9280-3

}

TY - JOUR

T1 - Generating and Adding Flows on Locally Complete Metric Spaces

AU - Kim, Hwa Kil

AU - Masmoudi, Nader

PY - 2013/3

Y1 - 2013/3

N2 - As a generalization of a vector field on a manifold, the notion of an arc field on a locally complete metric space was introduced in Bleecker and Calcaterra (J Math Anal Appl, 248: 645-677, 2000). In that paper, the authors proved an analogue of the Cauchy-Lipschitz Theorem, i. e they showed the existence and uniqueness of solution curves for a time independent arc field. In this paper, we extend the result to the time dependent case, namely we show the existence and uniqueness of solution curves for a time dependent arc field. We also introduce the notion of the sum of two time dependent arc fields and show existence and uniqueness of solution curves for this sum.

AB - As a generalization of a vector field on a manifold, the notion of an arc field on a locally complete metric space was introduced in Bleecker and Calcaterra (J Math Anal Appl, 248: 645-677, 2000). In that paper, the authors proved an analogue of the Cauchy-Lipschitz Theorem, i. e they showed the existence and uniqueness of solution curves for a time independent arc field. In this paper, we extend the result to the time dependent case, namely we show the existence and uniqueness of solution curves for a time dependent arc field. We also introduce the notion of the sum of two time dependent arc fields and show existence and uniqueness of solution curves for this sum.

KW - Arc fields

KW - Cauchy-Lipschitz Theorem

KW - Metric spaces

KW - Sum of arc fields

UR - http://www.scopus.com/inward/record.url?scp=84874576594&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84874576594&partnerID=8YFLogxK

U2 - 10.1007/s10884-012-9280-3

DO - 10.1007/s10884-012-9280-3

M3 - Article

AN - SCOPUS:84874576594

VL - 25

SP - 231

EP - 256

JO - Journal of Dynamics and Differential Equations

JF - Journal of Dynamics and Differential Equations

SN - 1040-7294

IS - 1

ER -