# Asymptotic approximations of the solution to a boundary-value problem in a thin aneurysm-type domain

## TL;DR

This paper develops a rigorous asymptotic expansion for the solution of a Poisson boundary-value problem in a thin aneurysm-type domain, capturing the effects of the aneurysm as the domain's thickness tends to zero.

## Contribution

It introduces a comprehensive asymptotic analysis method for a boundary-value problem in a complex thin domain with an aneurysm, including inner, regular, and boundary-layer parts, and derives the limit problem.

## Key findings

- Asymptotic expansion constructed with regular, boundary-layer, and inner parts.
- Limit problem formulated on the corresponding graph as e9 o 0.
- Energetic and uniform estimates established to quantify aneurysm impact.

## Abstract

A nonuniform Neumann boundary-value problem is considered for the Poisson equation in a thin $3D$ aneurysm-type domain that consists of thin curvilinear cylinders that are joined through an aneurysm of diameter $\mathcal{O}(\varepsilon).$ A rigorous procedure is developed to construct the complete asymptotic expansion for the solution as the parameter $\varepsilon \to 0.$ The asymptotic expansion consists of a regular part that is located inside of each cylinder, a boundary-layer part near the base of each cylinder, and an inner part discovered in a neighborhood of the aneurysm. Terms of the inner part of the asymptotics are special solutions of boundary-value problems in an unbounded domain with different outlets at infinity. It turns out that they have polynomial growth at infinity. By matching these parts, we derive the limit problem $(\varepsilon =0)$ in the corresponding graph and a recurrence procedure to determine all terms of the asymptotic expansion. Energetic and uniform pointwise estimates are proved. These estimates allow us to observe the impact of the aneurysm.