TY - GEN

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

AU - Shibata, Yoshihiro

PY - 2010/12/1

Y1 - 2010/12/1

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

T3 - Advances in Mathematical Fluid Mechanics - Dedicated to Giovanni Paolo Galdi on the Occasion of His 60th Birthday

SP - 513

EP - 551

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

T2 - 2007 International Conference on Mathematical Fluid Mechanics

Y2 - 21 May 2007 through 25 May 2007

ER -