### Abstract

We prove time local existence and uniqueness of solutions to a boundary layer problem in a rotating frame around the stationary solution called the Ekman spiral. We choose initial data in the vector-valued homogeneous Besov space Ḃ_{∞,1,σ}
^{0} (ℝ^{2}; L ^{p} (ℝ_{+})) for 2 < p < ∞. Here the L ^{p} -integrability is imposed in the normal direction, while we may have no decay in tangential components, since the Besov space Ḃ_{∞,1}
^{0} contains nondecaying functions such as almost periodic functions. A crucial ingredient is theory for vector-valued homogeneous Besov spaces. For instance we provide and apply an operator-valued bounded H ^{∞}-calculus for the Laplacian in Ḃ _{∞,1}
^{0}(ℝ^{n}; E) for a general Banach space E.

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

Pages (from-to) | 177-224 |

Number of pages | 48 |

Journal | Archive for Rational Mechanics and Analysis |

Volume | 186 |

Issue number | 2 |

DOIs | |

State | Published - Nov 2007 |

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### ASJC Scopus subject areas

- Mechanics of Materials
- Computational Mechanics
- Mathematics(all)
- Mathematics (miscellaneous)

### Cite this

_{+}

^{3}with initial data nondecreasing at infinity: The Ekman boundary layer problem.

*Archive for Rational Mechanics and Analysis*,

*186*(2), 177-224. https://doi.org/10.1007/s00205-007-0053-9

**Rotating Navier-Stokes equations in ℝ _{+}
^{3} with initial data nondecreasing at infinity : The Ekman boundary layer problem.** / Giga, Yoshikazu; Inui, Katsuya; Mahalov, Alex; Matsui, Shin'ya; Saal, Jürgen.

Research output: Contribution to journal › Article

_{+}

^{3}with initial data nondecreasing at infinity: The Ekman boundary layer problem',

*Archive for Rational Mechanics and Analysis*, vol. 186, no. 2, pp. 177-224. https://doi.org/10.1007/s00205-007-0053-9

_{+}

^{3}with initial data nondecreasing at infinity: The Ekman boundary layer problem. Archive for Rational Mechanics and Analysis. 2007 Nov;186(2):177-224. https://doi.org/10.1007/s00205-007-0053-9

}

TY - JOUR

T1 - Rotating Navier-Stokes equations in ℝ+ 3 with initial data nondecreasing at infinity

T2 - The Ekman boundary layer problem

AU - Giga, Yoshikazu

AU - Inui, Katsuya

AU - Mahalov, Alex

AU - Matsui, Shin'ya

AU - Saal, Jürgen

PY - 2007/11

Y1 - 2007/11

N2 - We prove time local existence and uniqueness of solutions to a boundary layer problem in a rotating frame around the stationary solution called the Ekman spiral. We choose initial data in the vector-valued homogeneous Besov space Ḃ∞,1,σ 0 (ℝ2; L p (ℝ+)) for 2 < p < ∞. Here the L p -integrability is imposed in the normal direction, while we may have no decay in tangential components, since the Besov space Ḃ∞,1 0 contains nondecaying functions such as almost periodic functions. A crucial ingredient is theory for vector-valued homogeneous Besov spaces. For instance we provide and apply an operator-valued bounded H ∞-calculus for the Laplacian in Ḃ ∞,1 0(ℝn; E) for a general Banach space E.

AB - We prove time local existence and uniqueness of solutions to a boundary layer problem in a rotating frame around the stationary solution called the Ekman spiral. We choose initial data in the vector-valued homogeneous Besov space Ḃ∞,1,σ 0 (ℝ2; L p (ℝ+)) for 2 < p < ∞. Here the L p -integrability is imposed in the normal direction, while we may have no decay in tangential components, since the Besov space Ḃ∞,1 0 contains nondecaying functions such as almost periodic functions. A crucial ingredient is theory for vector-valued homogeneous Besov spaces. For instance we provide and apply an operator-valued bounded H ∞-calculus for the Laplacian in Ḃ ∞,1 0(ℝn; E) for a general Banach space E.

UR - http://www.scopus.com/inward/record.url?scp=34848825320&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34848825320&partnerID=8YFLogxK

U2 - 10.1007/s00205-007-0053-9

DO - 10.1007/s00205-007-0053-9

M3 - Article

AN - SCOPUS:34848825320

VL - 186

SP - 177

EP - 224

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 2

ER -