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

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_tu-Δu + kD₃u + M_au +∇p = 0, div u = 0 inΩ × (0,∞), u|∂Ω = 0, u| _t=0 = f, (OS) where Ω is an exterior domain in R³, M_au = -a(e₃ × x). ∇u + ae₃ × u, x = (x1, x2, x3) ∈ R^q 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 ∈ L_q (Ω), then problem (OS) admits a unique solution (u(t), p(t)) which satisfies the following conditions: u(t) ∈ C¹([0,∞), L_q (Ω)) ∩ C⁰([0, ∞),W² _q (Ω)), p(t) ∈ C ⁰([0,∞), Ŵ ¹ _q (Ω)), ∥(u(t), t^1/2∇u(t), t∇²u(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_tu(t) ∥ _Lq(Ω) + ∥u(t) ∥ W² _q(Ω) + ∥∇p(t) ∥ _Lq(Ω) ≤ _Ca0,k0,γ E^γt (∥ f ∥ _W2q(Ω) + ∥M_af ∥ _Lq(Ω) ) for any t > 0 and large positive γ , where b is any number such that B_b ⊃ R³\Ω and Ω_b = B_b ∩ Ω with B_b = {x ∈ R³