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

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    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: 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

    Original languageEnglish
    Title of host publicationAdvances in Mathematical Fluid Mechanics - Dedicated to Giovanni Paolo Galdi on the Occasion of His 60th Birthday
    Pages513-551
    Number of pages39
    DOIs
    Publication statusPublished - 2010
    Event2007 International Conference on Mathematical Fluid Mechanics - Estoril
    Duration: 2007 May 212007 May 25

    Other

    Other2007 International Conference on Mathematical Fluid Mechanics
    CityEstoril
    Period07/5/2107/5/25

    Fingerprint

    Linearization
    Navier Stokes equations
    Flow of fluids

    Keywords

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

    ASJC Scopus subject areas

    • Fluid Flow and Transfer Processes

    Cite this

    Shibata, Y. (2010). On a C0 semigroup associated with a modified oseen equation with rotating effect. In 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.

    Advances in Mathematical Fluid Mechanics - Dedicated to Giovanni Paolo Galdi on the Occasion of His 60th Birthday. 2010. p. 513-551.

    Research output: Chapter in Book/Report/Conference proceedingConference contribution

    Shibata, Y 2010, On a C0 semigroup associated with a modified oseen equation with rotating effect. in 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
    Shibata Y. On a C0 semigroup associated with a modified oseen equation with rotating effect. In Advances in Mathematical Fluid Mechanics - Dedicated to Giovanni Paolo Galdi on the Occasion of His 60th Birthday. 2010. p. 513-551 https://doi.org/10.1007/978-3-642-04068-9-29
    Shibata, Yoshihiro. / On a C0 semigroup associated with a modified oseen equation with rotating effect. Advances in Mathematical Fluid Mechanics - Dedicated to Giovanni Paolo Galdi on the Occasion of His 60th Birthday. 2010. pp. 513-551
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    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

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