Higher derivative estimates for Stokes equations with closely spaced rigid inclusions in three dimensions

TL;DR

This paper derives higher-order derivative estimates for the Stokes equations in three dimensions with closely spaced rigid inclusions, providing detailed bounds and blow-up rates for stress derivatives in narrow regions.

Contribution

It introduces a novel inductive construction of auxiliary functions to estimate derivatives and extends the analysis to three-dimensional convex inclusions with optimal bounds.

Findings

Pointwise bounds up to seventh order derivatives

Optimal estimates under symmetry conditions

Precise blow-up rates for stress derivatives

Abstract

In this paper, we establish higher-order derivative estimates for the Stokes equations in a three-dimensional domain containing two closely spaced rigid inclusions. We construct a sequence of auxiliary functions via an inductive process to isolate the leading singular terms of higher-order derivatives within the narrow region between the inclusions. For a class of convex inclusions of general shapes, the construction of three-dimensional auxiliary functions -- unlike the two-dimensional case -- relies on the decay properties of solutions to a class of two-dimensional partial differential equations with singular coefficients. Taking advantage of this, we obtain pointwise upper bounds of derivatives up to the seventh order for general inclusions. Under additional symmetry conditions, we derive optimal estimates for derivatives of arbitrary order. Consequently, we obtain precise blow-up rates for the Cauchy stress and its higher-order derivatives in the narrow region between the inclusions.