The Resolvent Problem for the Stokes Equations on Halfspace in Lₚ

The resolvent problem for the Stokes equations on halfspace in R³ is considered. Letting H = \ (x₁ ,x₂ ,x₃ ) ∈ R³ | x₃ < 0 \ and given f ∈ Lₚ (H), we find u, ∇ P such that \[{gathered} . {gathered} λ { u}(x) - ν Δ { u}(x) + ∇ _p (x) = { f}(x), \\ ∇ · u(c) = 0, \\ {gathered} \} x ∈ H, \\ . { u} |∂ H = 0 \\ {gathered} \] We show that if λ 0 and if ν > 0 and 1 < p < ∞, the solution is unique and u ∈ W2,p satisfies \[ | λ | \| { u} \|L_p(H) + ν \| {Δ { u}} \|L_p(H) \| { f} \|L_p(H) \] where c depends on p and arg λ only. This enables us to prove that the nonstationary Stokes equations generate a bounded analytic semigroup on Lₚ (H), 1 < p < ∞. That is, given u₀ ∈ Lₚ (H), the problem \[ {gathered} . {gathered} {{∂ { u}}} {{∂ t}}(x,t) - ν Δ _x { u}(x,t) + ∇ _x p(x,t) = 0, \\ ∇ · { u}(x,t) = 0, \\ {gathered} \} x ∈ H, \\ . { u} |∂ H = 0,\,\,\, \\ { u}(x,0) = { u}_0 (x) \\ {gathered} \] has a unique solution u satisfying the conditions that \| u \|Lₚ (H) M\| u₀ \|Lₚ (H), that u is an analytic function of t, and other properties of analytic semigroups.