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

TL;DR

The paper establishes sharp higher-order derivative estimates for the Stokes equations in a 3D domain with two closely spaced rigid inclusions, focusing on the narrow gap between them. A novel sequence of divergence-free auxiliary functions, tied to a 2D singular-coefficient PDE, isolates leading singular terms and yields pointwise bounds for derivatives up to order seven for general inclusions; symmetry further provides optimal bounds for all orders. The authors also prove lower bound blow-up rates under symmetry, confirming the optimality of the upper bounds for derivatives and the Cauchy stress. Overall, the work extends 2D techniques to 3D, offering a quantitative description of field enhancement in narrow fluid-solid gaps with potential applications to fluid-structure interactions and material failure analysis.

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.

Theorems & Definitions (44)

• Proposition 1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Proposition 2.1
• Lemma 2.2
• Lemma 2.2
• proof : Proof of Lemma \ref{['delta-center']}