The Stokes semigroup on non-decaying spaces
健 阿部 · Institutional Repositories DataBase (IRDB) · 2013
Uniqueness in a half spaceIn this chapter, we study the uniqueness of the Stokes equations in a half space in a space of bounded functions, which will be used later in Chapter 3 in order to prove the a priori L ∞ -estimate for the non-stationary Stokes equations.The uniqueness of the Stokes equations is well known for decaying velocity at infinity in spatial variables, e.g., v(•, t) ∈ L p , p ∈ (1, ∞).However, for merely bounded velocity, the uniqueness results is less known even for a half space.We prove the uniqueness of the Stokes equations for bounded velocity with assuming the decay condition for the tangential component of the pressure gradient, i.e., ∇ tan q → 0 as x n → ∞.Such the decay condition is necessary since there exist non-trivial Poiseuille flow-type solutions.The proof is by a duality argument based on the L 1 -estimate for spatial derivatives of the Stokes semigroup.