### Abstract

In this paper, we show the unique existence of solutions to the nonstationary problem for the modified Oseen equation with rotating effect in Ω: D_{t}u-Δu + kD_{3}u + M_{a}u +∇p = 0, div u = 0 inΩ × (0,∞), u|∂Ω = 0, u| _{t=0} = f, (OS) where Ω is an exterior domain in R^{3}, M_{a}u = -a(e_{3} × x). ∇u + ae_{3} × u, x = (x1, x2, x3) ∈ R^{q} and e_{3} = (0, 0, 1). This problem arises from a linearization of the Navier Stokes equations describing an incompressible viscous fluid flow past a rotating obstacle. If 1 < q < ∞ and initial data f satisfies the conditions: f ∈ W^{2} _{q} (Ω), div f = 0 in Ω, f |∂Ω = 0 and M _{a} f ∈ L_{q} (Ω), then problem (OS) admits a unique solution (u(t), p(t)) which satisfies the following conditions: u(t) ∈ C^{1}([0,∞), L_{q} (Ω)) ∩ C^{0}([0, ∞),W^{2} _{q} (Ω)), p(t) ∈ C ^{0}([0,∞), Ŵ ^{1} _{q} (Ω)), ∥(u(t), t^{1/2}∇u(t), t∇^{2}u(t),∇ p(t)) ∥ L_{q} (Ω) ≤ C_{a0,k0,γ} E ^{γt}∥ f ∥ _{Lq} (Ω) , t ^{(1/2)(1+(1/q))} ∥p^{(t)} ∥ _{Lq(Ωb)} ≤ _{Ca0,k0,b,γ} E^{γ t}∥ f ∥ _{Lq(Ω)} , ∥D_{t}u(t) ∥ _{Lq(Ω)} + ∥u(t) ∥ W^{2} _{q(Ω)} + ∥∇p(t) ∥ _{Lq(Ω)} ≤ _{Ca0,k0,γ} E^{γt} (∥ f ∥ _{W2q(Ω)} + ∥M_{a}f ∥ _{Lq(Ω)} ) for any t > 0 and large positive γ , where b is any number such that B_{b} ⊃ R^{3}\Ω and Ω_{b} = B_{b} ∩ Ω with B_{b} = {x ∈ R^{3}

Original language | English |
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Title of host publication | Advances in Mathematical Fluid Mechanics - Dedicated to Giovanni Paolo Galdi on the Occasion of His 60th Birthday |

Pages | 513-551 |

Number of pages | 39 |

DOIs | |

Publication status | Published - 2010 |

Event | 2007 International Conference on Mathematical Fluid Mechanics - Estoril Duration: 2007 May 21 → 2007 May 25 |

### Other

Other | 2007 International Conference on Mathematical Fluid Mechanics |
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City | Estoril |

Period | 07/5/21 → 07/5/25 |

### Fingerprint

### Keywords

- Continuous semigroup
- L Framework
- Oseen operator
- Rotating effect

### ASJC Scopus subject areas

- Fluid Flow and Transfer Processes

### Cite this

*Advances in Mathematical Fluid Mechanics - Dedicated to Giovanni Paolo Galdi on the Occasion of His 60th Birthday*(pp. 513-551) https://doi.org/10.1007/978-3-642-04068-9-29

**On a C0 semigroup associated with a modified oseen equation with rotating effect.** / Shibata, Yoshihiro.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Advances in Mathematical Fluid Mechanics - Dedicated to Giovanni Paolo Galdi on the Occasion of His 60th Birthday.*pp. 513-551, 2007 International Conference on Mathematical Fluid Mechanics, Estoril, 07/5/21. https://doi.org/10.1007/978-3-642-04068-9-29

}

TY - GEN

T1 - On a C0 semigroup associated with a modified oseen equation with rotating effect

AU - Shibata, Yoshihiro

PY - 2010

Y1 - 2010

N2 - In this paper, we show the unique existence of solutions to the nonstationary problem for the modified Oseen equation with rotating effect in Ω: Dtu-Δu + kD3u + Mau +∇p = 0, div u = 0 inΩ × (0,∞), u|∂Ω = 0, u| t=0 = f, (OS) where Ω is an exterior domain in R3, Mau = -a(e3 × x). ∇u + ae3 × u, x = (x1, x2, x3) ∈ Rq and e3 = (0, 0, 1). This problem arises from a linearization of the Navier Stokes equations describing an incompressible viscous fluid flow past a rotating obstacle. If 1 < q < ∞ and initial data f satisfies the conditions: f ∈ W2 q (Ω), div f = 0 in Ω, f |∂Ω = 0 and M a f ∈ Lq (Ω), then problem (OS) admits a unique solution (u(t), p(t)) which satisfies the following conditions: u(t) ∈ C1([0,∞), Lq (Ω)) ∩ C0([0, ∞),W2 q (Ω)), p(t) ∈ C 0([0,∞), Ŵ 1 q (Ω)), ∥(u(t), t1/2∇u(t), t∇2u(t),∇ p(t)) ∥ Lq (Ω) ≤ Ca0,k0,γ E γt∥ f ∥ Lq (Ω) , t (1/2)(1+(1/q)) ∥p(t) ∥ Lq(Ωb) ≤ Ca0,k0,b,γ Eγ t∥ f ∥ Lq(Ω) , ∥Dtu(t) ∥ Lq(Ω) + ∥u(t) ∥ W2 q(Ω) + ∥∇p(t) ∥ Lq(Ω) ≤ Ca0,k0,γ Eγt (∥ f ∥ W2q(Ω) + ∥Maf ∥ Lq(Ω) ) for any t > 0 and large positive γ , where b is any number such that Bb ⊃ R3\Ω and Ωb = Bb ∩ Ω with Bb = {x ∈ R3

AB - In this paper, we show the unique existence of solutions to the nonstationary problem for the modified Oseen equation with rotating effect in Ω: Dtu-Δu + kD3u + Mau +∇p = 0, div u = 0 inΩ × (0,∞), u|∂Ω = 0, u| t=0 = f, (OS) where Ω is an exterior domain in R3, Mau = -a(e3 × x). ∇u + ae3 × u, x = (x1, x2, x3) ∈ Rq and e3 = (0, 0, 1). This problem arises from a linearization of the Navier Stokes equations describing an incompressible viscous fluid flow past a rotating obstacle. If 1 < q < ∞ and initial data f satisfies the conditions: f ∈ W2 q (Ω), div f = 0 in Ω, f |∂Ω = 0 and M a f ∈ Lq (Ω), then problem (OS) admits a unique solution (u(t), p(t)) which satisfies the following conditions: u(t) ∈ C1([0,∞), Lq (Ω)) ∩ C0([0, ∞),W2 q (Ω)), p(t) ∈ C 0([0,∞), Ŵ 1 q (Ω)), ∥(u(t), t1/2∇u(t), t∇2u(t),∇ p(t)) ∥ Lq (Ω) ≤ Ca0,k0,γ E γt∥ f ∥ Lq (Ω) , t (1/2)(1+(1/q)) ∥p(t) ∥ Lq(Ωb) ≤ Ca0,k0,b,γ Eγ t∥ f ∥ Lq(Ω) , ∥Dtu(t) ∥ Lq(Ω) + ∥u(t) ∥ W2 q(Ω) + ∥∇p(t) ∥ Lq(Ω) ≤ Ca0,k0,γ Eγt (∥ f ∥ W2q(Ω) + ∥Maf ∥ Lq(Ω) ) for any t > 0 and large positive γ , where b is any number such that Bb ⊃ R3\Ω and Ωb = Bb ∩ Ω with Bb = {x ∈ R3

KW - Continuous semigroup

KW - L Framework

KW - Oseen operator

KW - Rotating effect

UR - http://www.scopus.com/inward/record.url?scp=84896799179&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84896799179&partnerID=8YFLogxK

U2 - 10.1007/978-3-642-04068-9-29

DO - 10.1007/978-3-642-04068-9-29

M3 - Conference contribution

AN - SCOPUS:84896799179

SN - 9783642040672

SP - 513

EP - 551

BT - Advances in Mathematical Fluid Mechanics - Dedicated to Giovanni Paolo Galdi on the Occasion of His 60th Birthday

ER -