An L^q(L²)-theory of the generalized Stokes resolvent system in infinite cylinders

Abstract

Abstract. Estimates of the generalized Stokes resolvent system, i.e. with prescribed divergence, in an infinite cylinderAs a preparation, spectral decompositions of vector-valued homogeneous Sobolev spaces are studied. The main theorem is proved using the techniques of Schauder decompositions, operator-valued multiplier functions and R-boundedness of operator families.

The Navier–Stokes equations with the Neumann boundary condition in an infinite cylinder

“…Moreover, the operator admits a bounded H ∞ -calculus. See also [15], [17], [18]. The higher regularity estimate (6.3) is also known for the Stokes operator subject to the Dirichlet boundary condition [8], [19,IV].…”

Stokes resolvent systems in an infinite cylinder

“…Let us show that 14) for some (u, p) ∈ D(S r,λ,ξ ) as j → ∞. Here we used several times the compact embedding W 1,r (Σ) ⊂⊂ L r (Σ) on a bounded domain Σ.…”

“…Note that even for bounded domains (Theorem 3.4) we consider the non-solenoidal case and show Fréchet differentiability of operator-valued multiplier functions concerned with (R λ,ξ ) (Corollary 3.6). This more general approach allows to analyze the so-called generalized Stokes resolvent system, i.e., with prescribed divergence g, in an infinite straight cylinder with application to unbounded cylindrical domains with several exits to infinity, see a forthcoming article [14].…”

“…However, in the forthcoming paper [14], using a 2 , b 2 , we will analyze the generalized Stokes resolvent system with prescribed divergence g in an infinite cylinder of R n , which can be applied to study the Stokes resolvent system on unbounded cylindrical domains with several outlets to infinity.…”

The L p -Helmholtz projection in finite cylinders

“…Results on resolvent estimates, maximal regularity, and boundedness of the H ∞ -calculus for the Stokes operator in L p σ (Ω) are serialized in [11], [12], and [13]. Again, Ω is assumed to be an infinite layer, an infinite cylinder or the union of finitely many of these with a bounded domain.…”