Recovering discontinuous viscosity coefficients for inverse Stokes problems by boundary measurements

TL;DR

This work tackles the inverse Stokes problem for determining a discontinuous viscosity $\mu$ from boundary Cauchy data in a bounded domain. The authors develop a novel strategy that combines an $H^1$-norm analysis of Dirichlet Green's functions with a localized Stokes-Brinkman coupling to achieve global uniqueness of $\mu$ from boundary measurements. They show that the interior inclusion $D$ and the boundary values $\mu|_{\partial D}$ are uniquely recoverable, and, under mild regularity, that tangential derivatives $\partial^{\alpha}\mu|_{\partial\Omega}$ with $|\alpha|=1$ are also determined. The approach does not require prior smoothness of $\mu$ and extends to related two-dimensional settings with minor modifications, providing a boundary-driven identifiability framework for discontinuous coefficients in Stokes-type systems.

Abstract

In this paper, we investigate the inverse Stokes problem of determining a discontinuous viscosity coefficient $μ$ in a bounded domain $Ω\subset\mathbb{R}^3$. By analyzing the singularity of the Dirichlet Green's functions in $H^1$-norm and constructing a specifically coupled Stokes-Brinkman system in a localized domain, we prove a global uniqueness theorem that the viscosity coefficient $μ$ can be uniquely determined from boundary measurements.

Theorems & Definitions (10)

• Theorem 1.1
• Remark 1.2
• Theorem 2.1
• Remark 2.2
• Lemma 3.1
• proof
• Lemma 3.2
• proof
• Lemma 3.3
• proof