On Nonstationary Stokes Problem in Exterior Domains

The paper is concerned with Lp -estimates for solutions of n -dimensional exterior Stokes problem. The main result of the paper are new Lp - Lq estimates (1) |v(t)|q ≤ C1 |v0|p t-μ, (2) |vt(t)|q ≤ C2 |v0|p t-μ' (3) |∇v(t)|q ≤ C3 |v0|p t-μ, for the solution of a homogeneous Stokes problem with the initial condition v(x, 0) = v0(x); | · |p is the Lp -norm in an exterior domain Ω ⊆ ℝn We prove that estimate (1) holds with μ =n/2 (1/p - 1/q), for arbitrary p, q satisfying the conditions 1 ≤ p ≤, q ≤ ∞ p + q > 2 for 2 n > 2, 1 < p ≤ q < ∞ for n = 2. Estimate (2) holds with μ′ = 1 + μ and n ≥ 3. Finally, inequality (3) holds with μ = 1/2 + μ for q ∈ [p, n] and μ = n/2p for q ∈ (n, ∞). The constants Ci are independent of t > 0. We show also that in formulas (1) and (3) μ, μ are exact, in particular, that μ < 1/2 for q > n > 2. The method of the proof of (1)-(3) is quite elementary and relies on energy estimates, imbedding theorems, Lp - Lq estimates for the Cauchy problem and duality arguments. In addition, we give a new proof of Wp,r2,1 (QT) - estimates of derivatives of the solution of the Stokes problem (here QT = Ω × (0, T), p, r > 1), obtained by Y. Giga and H. Sohr [13], [14]. Inequality (1) allows us to show that the constant in this estimate can be taken independent of T, if n > 2, p < n/2, and we prove that the condition p < n/2 can not be relaxed.