### Abstract

The Cauchy problem of the Euler equations in the whole space is considered with non-decaying initial velocity in the frame work of B_{∞,1}^{1}. It is proved that if the initial velocity is real analytic then the solution is also real analytic in spatial variables. Furthermore, a new estimate for the size of the radius of convergence of Taylor's expansion is established. The key of the proof is to derive the suitable estimates for the higher order derivatives of the bilinear terms. It is also shown the propagation of the almost periodicity in spatial variables.

Original language | English |
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Pages (from-to) | 2148-2162 |

Number of pages | 15 |

Journal | Journal of Functional Analysis |

Volume | 260 |

Issue number | 7 |

DOIs | |

Publication status | Published - Apr 1 2011 |

Externally published | Yes |

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### All Science Journal Classification (ASJC) codes

- Analysis

### Cite this

**On the analyticity and the almost periodicity of the solution to the Euler equations with non-decaying initial velocity.** / Sawada, Okihiro; Takada, Ryo.

Research output: Contribution to journal › Article

*Journal of Functional Analysis*, vol. 260, no. 7, pp. 2148-2162. https://doi.org/10.1016/j.jfa.2010.12.011

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TY - JOUR

T1 - On the analyticity and the almost periodicity of the solution to the Euler equations with non-decaying initial velocity

AU - Sawada, Okihiro

AU - Takada, Ryo

PY - 2011/4/1

Y1 - 2011/4/1

N2 - The Cauchy problem of the Euler equations in the whole space is considered with non-decaying initial velocity in the frame work of B∞,11. It is proved that if the initial velocity is real analytic then the solution is also real analytic in spatial variables. Furthermore, a new estimate for the size of the radius of convergence of Taylor's expansion is established. The key of the proof is to derive the suitable estimates for the higher order derivatives of the bilinear terms. It is also shown the propagation of the almost periodicity in spatial variables.

AB - The Cauchy problem of the Euler equations in the whole space is considered with non-decaying initial velocity in the frame work of B∞,11. It is proved that if the initial velocity is real analytic then the solution is also real analytic in spatial variables. Furthermore, a new estimate for the size of the radius of convergence of Taylor's expansion is established. The key of the proof is to derive the suitable estimates for the higher order derivatives of the bilinear terms. It is also shown the propagation of the almost periodicity in spatial variables.

UR - http://www.scopus.com/inward/record.url?scp=78651455983&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=78651455983&partnerID=8YFLogxK

U2 - 10.1016/j.jfa.2010.12.011

DO - 10.1016/j.jfa.2010.12.011

M3 - Article

AN - SCOPUS:78651455983

VL - 260

SP - 2148

EP - 2162

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

SN - 0022-1236

IS - 7

ER -