Asymptotics of the Solution of a Boundary Value Problem in a Thin Cylinder With Nonsmooth Lateral Surface

Asymptotics of solutions of the Neumann problem in a domain with closely posed components of the boundary

“…For n 3 the singularity of the boundary in the vicinity of the point O cannot be converted into a conical one and the general theory does not furnish an answer on the behavior of the solution u 0 for x → O. The asymptotic decomposition of solutions in the vicinity of the point O are constructed and justified only for some classes of equations in mathematical physics (see [20,22,33] and also [18,19,24,25,29,37] for related problems in layered domains) and the analysis is performed with the so-called procedure of dimension reduction (see e.g., [21,27]). The same procedure is used in Section 3 to describe the phenomenon of boundary layer, stretching the coordinates in two different scales ε −1 and −1 = ε −1/2m for transversal and longitudal directions (see (1.10)) results in the second limit problem which becomes (n − 1)-dimensional…”

“…Asymptotic behavior at infinity of solutions of similar problems is investigated in [18,19,24,25,29,37]; the derivation of supplementary limit problem is given in [31].…”

The Asymptotic Properties of the Solution to the Stokes Problem in Domains That Are Layer-Like at Infinity

“…Therefore, the methods used here are closed to those of the papers [12] and [10], [13], where, in particular, the Stokes problem was considered in the case of the boundary components being tangent. We should also mention the papers [8], [9], where the similar asymptotic constructions were found for the Stokes and Lame systems in a sector of the layer and the papers [11], [14] and [16] where the Neumann problem for second order elliptic equation was investigated in Π.…”

Artificial Boundary Conditions for the Stokes and Navier–Stokes Equations in Domains that are Layer-Like at Infinity

“…In view of the asymptotic procedure developed in [15,18,20] this solution of the Navier-Stokes problem (1.3) can be decomposed further into a formal series in powers of r. Let us show that, in contrast to (2.4), in this series there appear functions in y which are not harmonic. By a proper choice of the angular variable ϕ, the main asymptotic term in (5.3) can be always reduced to the expression …”

“…These asymptotic expansions (see formulae (2.2)-(2.5)) were obtained in [20] with the help of a method developed in [15][16][17][18], they contain the plane harmonics P N . The approximation problem in the bounded domain Ω R is composed from the Stokes (or Navier-Stokes) equations, the Dirichlet conditions restricted to Σ R = {x ∈ ∂Ω : r < R}, and the ABC on the truncation boundary Γ R = {x : r = R, |z| < 1/2}, in the linear case this means 4) where the operator M R has to be chosen properly.…”